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Cm Math Chaos

1 · Prerequisites

2 · Summary

The chaotic layer of the non-relativistic classical mechanics collection, all of it mathematics: the coded invariant set of the area-preserving quadratic map, its conjugacy with the two-sign shift, its transverse homoclinic points, and the persistence of all of this for maps $C^{1}$ -close to the quadratic family, realised by the stroboscopic map of a smooth periodically pulsed oscillator. Every item carries domain: mathematics, depends only on mathematical suppliers, and contains no postulate, measurement or empirical statement; the mechanical reading of the pulsed oscillator belongs to the consuming physics pages, which supply their own units and scope.

Symbolic coding. The two-sign bilateral shift space, its coding map, and the ball of bounded bilateral real sequences introduces the two-sign bilateral shift space $X_{\pm} = {- 1, + 1}^{Z}$ , its shift $S_{\pm}$ , the isometry $φ$ onto the published two-symbol shift space, and the closed ball $B_{R}$ of bounded bilateral sequences. The bounded functions on a set form a complete metric space in the supremum metric, and the coding map of the two-sign shift space is an isometry proves that the bounded functions on an arbitrary nonempty set form a complete metric space in the supremum metric, that every closed ball $B_{R}$ is complete, and that $φ$ is a bijective isometry; this is the only point of contact with the published shift space, through which all its metric and chaos properties transfer. The square-root recurrence $T_{s} (x)_{n} = s_{n} \sqrt{a + x_{n - 1} + x_{n + 1}}$ is a contraction of $B_{R}$ with a unique coded solution then shows that the square-root recurrence $x_{n} = s_{n} \sqrt{a + x_{n - 1} + x_{n + 1}}$ defines, for $a \geq 100$ , a contraction of $B_{R}$ with constant $1 / L$ , $L = \sqrt{a - 2 R}$ : each code $s$ has a unique solution $x (s)$ with $sign x_{n} (s) = s_{n}$ , $∣ x_{n} (s) ∣ \geq L$ and $x_{n + 1} + x_{n - 1} = x_{n}^{2} - a$ . Finite symbol windows determine the code: locality of the iterates, coding continuity and injectivity of $h (σ) = (x_{- 1} (σ), x_{0} (σ))$ shows that the $m$ -th iterate from the zero sequence sees only a window of $2 m - 1$ symbols, with uniform error $2 R L^{- m}$ , and deduces that the coding map $h (s) = (x_{- 1} (s), x_{0} (s))$ is continuous and injective.

The coded invariant set. The quadratic map $F (x, y) = (y, y^{2} - a - x)$ has a compact coded invariant set conjugate to the two-sign shift, with chaos under ${A C}_{ω}$ proves that $F (x, y) = (y, y^{2} - a - x)$ satisfies $F \circ h = h \circ S_{\pm}$ and that $Λ = h (X_{\pm})$ is invariant. Granting the Axiom of Countable Choice once, through the equivalence between closed and sequentially closed subsets of $R^{2}$ , $h$ is a homeomorphism onto $Λ$ , so mixing and density of periodic points transfer from the shift; the theorem states these properties and the resulting chaos condition under ${A C}_{ω}$ . The set $Λ$ is sequentially compact without choice and compact under ${A C}_{ω} + D C$ . Sensitivity with constant $2 L$ is proved by an explicit symbol flip. Constant codes give the fixed points, and finite nonconstant word codes give homoclinic orbits converging to a fixed point at both ends identifies the constant codes with the hyperbolic fixed points $(r_{u}, r_{u})$ , $r_{u} = 1 + u \sqrt{a + 1}$ , and shows that every code that is constant outside a finite set but not constant gives a homoclinic point of the corresponding fixed point, with uniform two-sided convergence and infinitely many distinct such points.

Cones, slope fields and coded curves. Stable and unstable cones, invariant line slopes, stable and unstable sets, and transverse homoclinic points of the quadratic family defines hyperbolic fixed points, the cones $C_{k}^{u} = {∣ u ∣ \leq k ∣ v ∣}$ and $C_{k}^{s} = {∣ v ∣ \leq k ∣ u ∣}$ with $C_{k}^{u} \cap C_{k}^{s} = {0}$ , and slope maps $Φ_{y} (h) = 1 / (2 y - h)$ and $Ψ_{x} (r) = 1 / (2 x - r)$ . It distinguishes $D (F_{a}^{- 1})$ from the pointwise inverse matrix $[D F_{a} (x, y)]^{- 1}$ , defines invariant line fields along a sequence with the inverse derivative evaluated at the correct orbit point, and defines stable and unstable sets for global bijections and local planar diffeomorphisms. Transversality is defined using $C^{1}$ curves in the plane. Cone invariance and expansion, and convergence of the slope graph transforms to unique invariant stable and unstable line fields proves cone invariance and expansion under the aperture condition $k (2 L - k) \geq 1$ , the slope contractions with factor $λ^{- 2}$ , and uniform convergence of the slope iterations to the unique invariant line fields with slopes in $[- k, k]$ . Weighted half-tail contractions for $q_{n} = M_{s_{n}} (q_{n - 1}, q_{n + 1})$ give a unique $C^{1}$ curve with tangent slope bounded by $c / (1 - q)$ gives the abstract half-tail construction: for compact update intervals it applies the Banach fixed-point theorem on a closed weighted ball and a local parameter interval to produce a $C^{1}$ family of solutions with derivative bounded by $c / (1 - q)$ . The coded curves are genuine $C^{1}$ stable and unstable curves of $F$ , with transverse intersection at every finite-word homoclinic point applies it to the quadratic family to obtain $C^{1}$ stable and unstable curves through every finite-word homoclinic point, with tangents in cones of aperture $1 / 2$ , and identifies the local stable set of each fixed point $(r_{u}, r_{u})$ with the constructed graph over $∣ b - r_{u} ∣ \leq 1 / 8$ in the coordinate max norm. The local unstable set is its reflection. Transversality of the tangent lines then makes every finite-word point a transverse homoclinic point.

Persistence and the smooth realisation. A $C^{2}$ map $C^{1}$ -close to $F$ has a unique inverse branch for its first component, with $O (ε)$ derivative bounds solves the first component of a map close to $F$ by a contraction, giving a unique inverse branch $V$ with $O (ε)$ estimates. Root bounds and $C^{1}$ dependence for the perturbed scalar recurrence $v^{2} - a - u - w + e (u, v, w) = 0$ shows that the perturbed scalar recurrence $v^{2} - a - u - w + e (u, v, w) = 0$ has, for each sign, exactly one root in the corresponding window, with explicit margins $\geq 3 / 4$ , derivative bounds $d \leq 2 / 13$ , and exact orbit form $P_{u, w} (v) = 0 ⟺ B (u, V (u, v)) = V (v, w)$ . Explicit persistence under the stated forward and conjugated-inverse bounds: coded chaos, hyperbolic fixed points and transverse homoclinic points constructs, for every $C^{2}$ diffeomorphism $G$ satisfying six explicit closeness inequalities with $0 < ε \leq 1 / 6$ and $C (a, R) ε \leq 1$ , where $C$ is the defect-bound constant of Root bounds and $C^{1}$ dependence for the perturbed scalar recurrence $v^{2} - a - u - w + e (u, v, w) = 0$ (and the same bounds for the distinct map $Σ \circ G^{- 1} \circ Σ$ ), a coded invariant set $Λ_{G} = h_{G} (X_{\pm})$ on which $G$ is conjugate to the shift. Under ${A C}_{ω}$ it is mixing with dense periodic points and chaotic with sensitivity constant $2 l$ ; its constant codes give hyperbolic fixed points, and its finite-word codes give transverse homoclinic points. No structural-stability theorem is invoked. The smooth $P$ -periodic pulsed oscillator: the normalized pulse, its Hamiltonian and its stroboscopic (period) map defines the smooth $P$ -periodic pulsed oscillator $q^{''} = - q + β_{δ} (t) (q^{2} - a)$ with phase form $ω = d q \land d p$ and Hamiltonian convention $ι_{X_{H}} ω = d H$ , using a normalized bump pulse of strength $\int β_{δ} = 1$ . The rescaled pulse endpoint map converges to the kick $K (q, p) = (q, p + q^{2} - a)$ in $C^{1}$ , and the period map is smooth, symplectic and $C^{1}$ -close to $F$ proves that the rescaled pulse endpoint map converges to the kick $K (q, p) = (q, p + q^{2} - a)$ in $C^{1}$ as $δ \to 0$ , that the period map $G_{δ} = K_{δ} \circ R_{0}$ is a symplectic $C^{\infty}$ diffeomorphism close to $F$ with its conjugate map also close to $F$ , and A smooth periodic one-degree-of-freedom oscillator with polynomial pulse forcing has stroboscopic chaos and a transverse homoclinic point concludes that, for all sufficiently small $δ$ , this stroboscopic map has a coded chaotic invariant set and transverse homoclinic points, with the inherited Axioms of Countable and Dependent Choice for the coding homeomorphism and compactness.

The companion page cm-math-chaos-examples records explicit finite checks: the constants and constant-code solutions at $a = 100$ , the invariant slope and cone margins, the single-flip homoclinic code, the $δ = 0$ endpoint computation $K \circ R_{0} = F$ , and the perturbed root and margin numbers.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

DefinitionDefinition: AI-adaptedProof: Not applicableprecheck passOpen item page →

The two-sign bilateral shift space, its coding map, and the ball of bounded bilateral real sequences

Definition

Write $Σ_{\pm} : = {- 1, + 1}$ and

$X_{\pm} : = Σ_{\pm}^{Z} = {s : s is a function Z \to Σ_{\pm}},$

with the function-set convention of The vector space $F^{X}$ of all functions $X \to F$ with pointwise operations, and $F^{n}$ as the case $X = n = {0, 1, \dots, n - 1}$ ; for $s \in X_{\pm}$ write $s_{k} : = s (k)$ . The elements of $X_{\pm}$ are called two-sign bilateral sequences.

Coding correspondence with the two-symbol space. Let

$φ : X_{\pm} ⟶ {0, 1}^{Z}, φ (s)_{k} : = \frac{s_{k} + 1}{2} \in {0, 1},$

and let $X_{2} : = {0, 1}^{Z}$ , $d_{2}$ and $σ_{2}$ be the set, the metric expression and the shift map of The bilateral two-symbol shift space, its metric, its shift map and its cylinder sets. The formulas $φ (s)_{k} = (s_{k} + 1) / 2$ and $φ^{- 1} (t)_{k} = 2 t_{k} - 1$ define maps $X_{\pm} \to X_{2}$ and $X_{2} \to X_{\pm}$ that are inverse to each other, coordinate by coordinate.

Metric expression on $X_{\pm}$ . Define

$D_{\pm} (s, t) : = d_{2} (φ (s), φ (t)) (s, t \in X_{\pm}) .$

Unwinding the series of The bilateral two-symbol shift space, its metric, its shift map and its cylinder sets and using $∣ φ (s)_{k} - φ (t)_{k} ∣ = \frac{1}{2} ∣ s_{k} - t_{k} ∣$ for every $k \in Z$ gives the explicit expression

$D_{\pm} (s, t) = \sum_{k \geq 0} 2^{- (k + 1)} ∣ s_{k} - t_{k} ∣ + \sum_{k \geq 1} 2^{- (k + 1)} ∣ s_{- k} - t_{- k} ∣ .$

Both series converge by comparison with the geometric series $\sum_{k \geq 0} 2^{- k} = 2$ (Series, partial sums, convergence and the sum, divergence, and the tail series, For $∣ r ∣ < 1$ , $\sum_{k \geq 0} r^{k} = 1 / (1 - r)$ , and for $∣ r ∣ \geq 1$ the series diverges, Integer powers $a^{m}$ , Absolute value in an ordered field), because $∣ s_{k} - t_{k} ∣ \in {0, 2}$ for every $k$ ; hence $D_{\pm} (s, t)$ is a well-defined nonnegative real number. Whether $D_{\pm}$ is a metric is not asserted here; it is proved in the companion lemma on this page, together with the statement that $φ$ is an isometry of $(X_{\pm}, D_{\pm})$ onto $(X_{2}, d_{2})$ .

The shift. The (bilateral) shift of $X_{\pm}$ is

$S_{\pm} : X_{\pm} ⟶ X_{\pm}, S_{\pm} (s)_{k} : = s_{k + 1} (k \in Z, s \in X_{\pm}) .$

For $m \in Z$ its iterate satisfies $S_{\pm}^{m} (s)_{k} = s_{k + m}$ ; in particular $S_{\pm}$ is a bijection with inverse $S_{\pm}^{- 1} (s)_{k} = s_{k - 1}$ , and $S_{\pm} = φ^{- 1} \circ σ_{2} \circ φ$ .

Bounded bilateral sequences. A function $x : Z \to R$ is bounded when its range ${x_{k} : k \in Z}$ is a bounded subset of $R$ (Lower bound, bounded below, bounded set). Write

$ℓ^{\infty} (Z) : = {x : Z \to R : x is bounded}, ∥ x ∥_{\infty} : = \sup {∣ x_{k} ∣ : k \in Z} .$

The set ${∣ x_{k} ∣ : k \in Z}$ whose supremum is taken is nonempty and bounded above (Lower bound, bounded below, bounded set), so $∥ x ∥_{\infty}$ is a well-defined nonnegative real number (Suprema and infima are unique, Complete ordered field (least-upper-bound property)). For $R \geq 0$ put

$B_{R} : = {x \in ℓ^{\infty} (Z) : ∥ x ∥_{\infty} \leq R} .$

Thus $B_{R}$ is the closed ball of radius $R$ in the supremum norm, and $∥ x ∥_{\infty} \leq R$ is by definition the statement that $∣ x_{k} ∣ \leq R$ for every $k \in Z$ (Lower bound, bounded below, bounded set, Absolute value in an ordered field).

Remarks

The only difference between $(X_{\pm}, D_{\pm}, S_{\pm})$ and the $(X_{2}, d_{2}, σ_{2})$ is the relabelling $0 \leftrightarrow - 1$ , $1 \leftrightarrow + 1$ of the two symbols: $φ$ is a bijection satisfying $φ \circ S_{\pm} = σ_{2} \circ φ$ . Every metric, compactness or chaos statement about the two-symbol shift therefore transfers through $φ$ once it is known that $φ$ is an isometry, which is the content of The bounded functions on a set form a complete metric space in the supremum metric, and the coding map of the two-sign shift space is an isometry clause 3.
No measure, probability or invariant measure on $X_{\pm}$ is introduced; all statements on this page are topological or metric.
The space $ℓ^{\infty} (Z)$ and its closed ball $B_{R}$ are used to state the square-root recurrence of The square-root recurrence $T_{s} (x)_{n} = s_{n} \sqrt{a + x_{n - 1} + x_{n + 1}}$ is a contraction of $B_{R}$ with a unique coded solution; nothing in this definition asserts that $B_{R}$ is complete, which is a separate lemma.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

The bounded functions on a set form a complete metric space in the supremum metric, and the coding map of the two-sign shift space is an isometry

Statement

Let $S$ be a nonempty set, let $B (S)$ be the set of bounded functions $S \to R$ and let $d_{\infty}$ be the supremum metric on $B (S)$ (The supremum metric $d_{\infty} (f, g) = \sup_{x} ∣ f (x) - g (x) ∣$ is a metric on the bounded real-valued functions on a nonempty set); write $0$ for the zero function. For $R \geq 0$ put

$B_{R} (S) : = {f \in B (S) : ∣ f (s) ∣ \leq R for every s \in S} .$

Let $X_{\pm}$ , $φ$ , $X_{2}$ , $d_{2}$ , $D_{\pm}$ be as in The two-sign bilateral shift space, its coding map, and the ball of bounded bilateral real sequences. Then:

$(B (S), d_{\infty})$ is a complete metric space (Complete metric space: every Cauchy sequence converges in the space);
for every $R \geq 0$ , $(B_{R} (S), d_{\infty})$ is a complete metric space;
$D_{\pm} (s, t) = d_{2} (φ (s), φ (t))$ is a metric on $X_{\pm}$ (Metric space: $d (x, y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric), and $φ : (X_{\pm}, D_{\pm}) \to (X_{2}, d_{2})$ is a bijective isometry (Convergence of a sequence in a metric space: $x_{k} \to x$ iff $d (x_{k}, x) \to 0$ in $R$ for the underlying notion of distance);
consequently $ℓ^{\infty} (Z)$ with the distance $∥ x - y ∥_{\infty}$ is a complete metric space, $B_{R}$ is complete for every $R \geq 0$ , and the same holds for the one-sided bounded sequences, that is for $S = N$ .

Facts & Assumptions

Given: A nonempty set $S$ , the space $B (S)$ of bounded real functions on $S$ with supremum metric $d_{\infty}$ , the numbers $R \geq 0$ , and the objects $X_{\pm}, φ, X_{2}, d_{2}, D_{\pm}$ of The two-sign bilateral shift space, its coding map, and the ball of bounded bilateral real sequences.

[F1]

$d_{\infty}$ is a metric on $B (S)$ and $d_{\infty} (f, g) = \sup {∣ f (s) - g (s) ∣ : s \in S}$ (The supremum metric $d_{\infty} (f, g) = \sup_{x} ∣ f (x) - g (x) ∣$ is a metric on the bounded real-valued functions on a nonempty set, Metric space: $d (x, y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric); the defining set is nonempty and bounded above, its supremum exists and is unique (Suprema and infima are unique, Complete ordered field (least-upper-bound property), Lower bound, bounded below, bounded set).

[F2]

$d_{2}$ is a metric on $X_{2}$ (The bilateral shift metric is a metric, cylinders are its basic balls, and the shift space is sequentially compact, clause 1) and $D_{\pm} (s, t) = d_{2} (φ (s), φ (t))$ by definition of $D_{\pm}$ ; $φ$ is a bijection with inverse $t \mapsto (2 t_{k} - 1)_{k \in Z}$ (The two-sign bilateral shift space, its coding map, and the ball of bounded bilateral real sequences).

[F3]

Cauchy sequences, convergence and completeness are those of Cauchy sequence in a metric space, Convergence of a sequence in a metric space: $x_{k} \to x$ iff $d (x_{k}, x) \to 0$ in $R$ and Complete metric space: every Cauchy sequence converges in the space; a sequence in a metric space converges to $z$ exactly when the distances to $z$ tend to $0$ .

[F4]

Every Cauchy sequence of reals converges to a real number (The Cauchy criterion from the least-upper-bound property: in a complete ordered field every Cauchy sequence converges, Sequences of reals: bounded, eventually, frequently, tails, subsequences).

[F5]

Limits preserve eventual inequalities: if the real sequences $(x_{j})$ and $(y_{j})$ converge and $x_{j} \leq y_{j}$ eventually, then $\lim_{j} x_{j} \leq \lim_{j} y_{j}$ (Limits preserve non-strict inequalities, Sequences of reals: bounded, eventually, frequently, tails, subsequences).

[F6]

Absolute value: $∣ a ∣ \geq 0$ , $∣ a ∣ = 0$ exactly for $a = 0$ , $∣ a - b ∣ = ∣ b - a ∣$ , $∣ a + b ∣ \leq ∣ a ∣ + ∣ b ∣$ and $∣ a ∣ \leq M$ exactly when $- M \leq a \leq M$ for $M \geq 0$ (Absolute value in an ordered field, Basic properties of the absolute value, The triangle inequality).

Proof

technique · direct; take coordinatewise limits of a Cauchy sequence and estimate the supremum distance by an eventual coordinate bound

1.1givenF1F3F4F6

Let $(f_{j})_{j \in N}$ be a Cauchy sequence in $(B (S), d_{\infty})$ and fix $s \in S$ . For all $j, k$ one has $∣ f_{j} (s) - f_{k} (s) ∣ \in {∣ f_{j} (s^{'}) - f_{k} (s^{'}) ∣ : s^{'} \in S}$ and therefore $∣ f_{j} (s) - f_{k} (s) ∣ \leq d_{\infty} (f_{j}, f_{k})$ by [F1] and [F6]. Hence $(f_{j} (s))_{j}$ is a Cauchy sequence of reals, so it converges; define $f (s) : = \lim_{j} f_{j} (s) \in R$ . This defines a function $f : S \to R$ .

1.2givenF2F6

Isometry property: for $s, t \in X_{\pm}$ put $Φ (s, t) : = d_{2} (φ (s), φ (t))$ . Then $Φ (s, t) = 0$ if and only if $φ (s) = φ (t)$ (by [F2], [F6] and definiteness of $d_{2}$ ), if and only if $s = t$ since $φ$ is injective ([F2]). Symmetry holds because $d_{2}$ is symmetric, and the triangle inequality follows from that of $d_{2}$ : $D_{\pm} (s, u) = d_{2} (φ s, φ u) \leq d_{2} (φ s, φ t) + d_{2} (φ t, φ u) = D_{\pm} (s, t) + D_{\pm} (t, u)$ ([F2], [F6]). Finally $D_{\pm} (s, t) = d_{2} (φ s, φ t)$ says exactly that the bijection $φ$ preserves distances, so it is an isometry onto $(X_{2}, d_{2})$ .

2.1step 1.1F1F5F6

Boundedness of $f$ : since $(f_{j})$ is Cauchy, there is $N$ with $d_{\infty} (f_{j}, f_{N}) \leq 1$ for all $j \geq N$ ; then $∣ f_{j} (s) ∣ \leq ∣ f_{N} (s) ∣ + 1 \leq \sup_{s^{'}} ∣ f_{N} (s^{'}) ∣ + 1 = : M$ for all $j \geq N$ and all $s$ , where $M < \infty$ because $f_{N}$ is bounded ([F1], [F6]). For fixed $s$ , the bounds $- M \leq f_{j} (s) \leq M$ pass to the limit by [F5], giving $- M \leq f (s) \leq M$ , hence $∣ f (s) ∣ \leq M$ by [F6]. Thus $f \in B (S)$ .

3.1step 1.1step 2.1F1F3F5F6

Uniform error estimate: given $ε > 0$ , choose $N$ with $d_{\infty} (f_{j}, f_{k}) \leq ε$ for all $j, k \geq N$ . Fix $j \geq N$ and $s \in S$ ; then $∣ f_{k} (s) - f_{j} (s) ∣ \leq d_{\infty} (f_{k}, f_{j}) \leq ε$ for every $k \geq N$ ([F1], [F6]), thus $f_{j} (s) - ε \leq f_{k} (s) \leq f_{j} (s) + ε$ . Passing these inequalities to the coordinate limit with [F5] gives $f_{j} (s) - ε \leq f (s) \leq f_{j} (s) + ε$ , or $∣ f (s) - f_{j} (s) ∣ \leq ε$ by [F6]. Taking the supremum over $s$ in [F1] gives $d_{\infty} (f, f_{j}) \leq ε$ for every $j \geq N$ , so $f_{j} \to f$ in $(B (S), d_{\infty})$ .

4.1step 2.1step 3.1F1F5F6

The space $(B_{R} (S), d_{\infty})$ is complete: if $(f_{j})$ is Cauchy in $(B_{R} (S), d_{\infty})$ , then by steps 2.1 and 3.1 its pointwise limit $f$ lies in $B (S)$ and $f_{j} \to f$ ; for each $s$ one has $- R \leq f_{j} (s) \leq R$ for all $j$ , so [F5] gives $- R \leq f (s) \leq R$ , hence $∣ f (s) ∣ \leq R$ by [F6]; hence $f \in B_{R} (S)$ and the Cauchy sequence converges inside $B_{R} (S)$ .

5.1step 3.1step 4.1step 1.2givenF1∎

Conclusion. Clause 1 is proved by steps 1.1, 2.1 and 3.1: the pointwise limit of a Cauchy sequence of bounded functions is bounded (step 2.1) and is the uniform limit (step 3.1). Clause 2 is step 4.1, and clause 3 is step 1.2. For clause 4, take $S = Z$ : by the definitions of The two-sign bilateral shift space, its coding map, and the ball of bounded bilateral real sequences the space $B (Z)$ is exactly $ℓ^{\infty} (Z)$ and $d_{\infty} (f, g) = \sup_{k} ∣ f_{k} - g_{k} ∣ = ∥ f - g ∥_{\infty}$ , so clauses 1 and 2 give completeness of $ℓ^{\infty} (Z)$ and of each $B_{R}$ , $R \geq 0$ ; with $S = N$ the same identifications give the one-sided statement, the argument being identical since it never uses any structure of the index set beyond its nonemptiness.

Remarks

The proof is a coordinatewise limit argument plus one uniform two-index estimate; no choice principle is used, since the limit function is defined by the uniquely determined real limits of the coordinate sequences.
Clause 3 is the only point of contact between the two-sign space and the published two-symbol space: because $φ$ is a bijective isometry, every metric statement about $(X_{2}, d_{2})$ transfers verbatim, and it will be used in this form for sequential compactness and for the chaos properties of the shift.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

The square-root recurrence $T_{s} (x)_{n} = s_{n} \sqrt{a + x_{n - 1} + x_{n + 1}}$ is a contraction of $B_{R}$ with a unique coded solution

Statement

Fix $a \geq 100$ and put

$R : = \sqrt{a} + 3, L : = \sqrt{a - 2 R} .$

Let $X_{\pm}$ and $B_{R}$ be the two-sign shift space and the closed ball of bounded bilateral sequences of The two-sign bilateral shift space, its coding map, and the ball of bounded bilateral real sequences, and for $s \in X_{\pm}$ define

$T_{s} : B_{R} ⟶ R^{Z}, (T_{s} x)_{n} : = s_{n} \sqrt{a + x_{n - 1} + x_{n + 1}} (n \in Z) .$

For a nonzero real $r$ , define $sign r : = + 1$ when $r > 0$ and $sign r : = - 1$ when $r < 0$ . Once clause 2 establishes $T_{s} (B_{R}) \subseteq B_{R}$ , its iterates are defined by $T_{s}^{0} : = {id}_{B_{R}}$ and $T_{s}^{m + 1} : = T_{s} \circ T_{s}^{m}$ for $m \in N$ .

Then:

$a - 2 R = (\sqrt{a} - 1)^{2} - 7 \geq 74$ , so $L \geq \sqrt{74} > 1$ ; moreover $a + 2 R < R^{2}$ , hence $\sqrt{a + 2 R} < R$ ;
for every $s \in X_{\pm}$ and $x \in B_{R}$ the radicands $a + x_{n - 1} + x_{n + 1}$ lie in $[a - 2 R, a + 2 R] \subseteq (0, \infty)$ , so $T_{s} x$ is a well-defined bounded sequence with $∥ T_{s} x ∥_{\infty} \leq \sqrt{a + 2 R} < R$ ; in particular $T_{s} (B_{R}) \subseteq B_{R}$ ;
for all $x, y \in B_{R}$ and every $s \in X_{\pm}$ ,

$∥ T_{s} x - T_{s} y ∥_{\infty} \leq \frac{1}{L} ∥ x - y ∥_{\infty}, \frac{1}{L} < 1;$

hence $T_{s}$ has, for each $s \in X_{\pm}$ , exactly one fixed point $x (s) \in B_{R}$ , and for every $n \in Z$

$sign x_{n} (s) = s_{n}, ∣ x_{n} (s) ∣ \geq L, x_{n + 1} (s) + x_{n - 1} (s) = x_{n} (s)^{2} - a .$

Facts & Assumptions

Given: The real number $a \geq 100$ , the constants $R = \sqrt{a} + 3$ and $L = \sqrt{a - 2 R}$ , the shift space $X_{\pm}$ and the complete ball $B_{R}$ of The two-sign bilateral shift space, its coding map, and the ball of bounded bilateral real sequences and The bounded functions on a set form a complete metric space in the supremum metric, and the coding map of the two-sign shift space is an isometry, and the operators $T_{s}$ .

[F1]

Square roots (Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ , Complete ordered field (least-upper-bound property)): every $u \geq 0$ has a unique $u^{1 / 2} : = \sqrt{u} \geq 0$ with $(\sqrt{u})^{2} = u$ ; if $0 \leq u \leq v$ then $\sqrt{u} \leq \sqrt{v}$ , since $\sqrt{u}, \sqrt{v} \geq 0$ and $(\sqrt{u})^{2} = u \leq v = (\sqrt{v})^{2}$ ; in particular $\sqrt{u^{2}} = u$ for $u \geq 0$ and $\sqrt{x} > 0$ for $x > 0$ . Also $\sqrt{a} \geq 10$ because $a \geq 100 = 10^{2}$ and $t \mapsto t^{2}$ is increasing on $[0, \infty)$ .

[F2]

The ball $B_{R}$ is nonempty and, with the supremum distance $∥ x - y ∥_{\infty}$ , a complete metric space (The bounded functions on a set form a complete metric space in the supremum metric, and the coding map of the two-sign shift space is an isometry, clauses 1, 2 and 4; Metric space: $d (x, y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric); the value $∥ x ∥_{\infty}$ satisfies $∥ x ∥_{\infty} \leq R$ exactly when $∣ x_{n} ∣ \leq R$ for all $n$ (The two-sign bilateral shift space, its coding map, and the ball of bounded bilateral real sequences, Absolute value in an ordered field).

[F3]

Banach's fixed point theorem (A contraction of a nonempty complete metric space into itself has exactly one fixed point, the limit of the iterates from any starting point, Lipschitz map, $α$ -Hölder map for rational $0 < α \leq 1$ , and contraction): a contraction of a nonempty complete metric space into itself, with constant $q < 1$ , has exactly one fixed point.

[F4]

Absolute value and order: $∣ u ∣ \leq M$ exactly when $- M \leq u \leq M$ for $M \geq 0$ ; $∣ u v ∣ = ∣ u ∣ ∣ v ∣$ ; and $s_{n} \in {- 1, + 1}$ for every $n$ (Basic properties of the absolute value, Absolute value in an ordered field, The two-sign bilateral shift space, its coding map, and the ball of bounded bilateral real sequences).

[F5]

Order arithmetic on intervals (Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length, Complete ordered field (least-upper-bound property)): if $∣ x_{j} ∣ \leq R$ for $j \in Z$ then $a - R - R \leq a + x_{n - 1} + x_{n + 1} \leq a + R + R$ ; and for reals $u \geq v > 0$ one has $\sqrt{u} \geq \sqrt{v} > 0$ by [F1].

Proof

technique · direct; bound the radicands on the ball, estimate the square-root Lipschitz constant from below by $2L$, and apply Banach's fixed point theorem

1.1givenF1algebra

Constants: by [F1], $\sqrt{a} \geq 10$ , so $R = \sqrt{a} + 3 \geq 13$ and $a - 2 R = a - 2 \sqrt{a} - 6 = (\sqrt{a} - 1)^{2} - 7 \geq 81 - 7 = 74$ ; hence $L = \sqrt{a - 2 R} \geq \sqrt{74} > 1$ by [F1]. Further $R^{2} = a + 6 \sqrt{a} + 9$ and $R^{2} - (a + 2 R) = a + 6 \sqrt{a} + 9 - a - 2 \sqrt{a} - 6 = 4 \sqrt{a} + 3 > 0$ , so $a + 2 R < R^{2}$ and [F1] gives $\sqrt{a + 2 R} < R$ .

2.1givenstep 1.1F1F2F4F5

Well-definedness and self-mapping: let $s \in X_{\pm}$ , $x \in B_{R}$ and $n \in Z$ . Since $∣ x_{n \pm 1} ∣ \leq R$ by [F2], [F5] gives $a - 2 R \leq a + x_{n - 1} + x_{n + 1} \leq a + 2 R$ ; by step 1.1 this is a positive interval, so [F1] makes $\sqrt{a + x_{n - 1} + x_{n + 1}}$ a well-defined positive real, and $T_{s} x$ is a function $Z \to R$ . Moreover $∣ (T_{s} x)_{n} ∣ = ∣ s_{n} ∣ \sqrt{a + x_{n - 1} + x_{n + 1}} = \sqrt{a + x_{n - 1} + x_{n + 1}} \leq \sqrt{a + 2 R} < R$ by [F4], [F1] and step 1.1. Taking the supremum over $n$ gives $∥ T_{s} x ∥_{\infty} \leq \sqrt{a + 2 R} < R$ , so $T_{s} (B_{R}) \subseteq B_{R}$ .

2.2givenstep 1.1F1F2F4F5algebra

Contraction: let $x, y \in B_{R}$ and $n \in Z$ , and put $u : = a + x_{n - 1} + x_{n + 1}$ , $v : = a + y_{n - 1} + y_{n + 1}$ ; by step 1.1 both lie in $[a - 2 R, a + 2 R] \subseteq [L^{2}, \infty)$ , so $\sqrt{u}, \sqrt{v} \geq L > 0$ by [F1]. Then $∣ \sqrt{u} - \sqrt{v} ∣ = \frac{∣ u - v ∣}{\sqrt{u} + \sqrt{v}} \leq \frac{∣ u - v ∣}{2 L} \leq \frac{∣ x_{n - 1} - y_{n - 1} ∣ + ∣ x_{n + 1} - y_{n + 1} ∣}{2 L} \leq \frac{∥ x - y ∥_{\infty}}{L},$ where the first identity is the algebraic identity $u - v = (\sqrt{u} - \sqrt{v}) (\sqrt{u} + \sqrt{v})$ ([F1], [F4]), the second inequality uses [F4] and [F5], and the last uses [F2]. Since $∣ (T_{s} x)_{n} - (T_{s} y)_{n} ∣ = ∣ \sqrt{u} - \sqrt{v} ∣$ by [F4], taking the supremum over $n$ gives $∥ T_{s} x - T_{s} y ∥_{\infty} \leq ∥ x - y ∥_{\infty} / L$ , and $1 / L < 1$ by step 1.1.

3.1step 2.1step 2.2F2F3

Fixed point: $B_{R}$ is nonempty ( $0 \in B_{R}$ ) and complete in the sup distance by [F2]; $T_{s}$ maps $B_{R}$ into itself by step 2.1 and is a contraction with constant $1 / L < 1$ by step 2.2; [F3] therefore gives exactly one $x (s) \in B_{R}$ with $T_{s} x (s) = x (s)$ .

4.1step 3.1step 1.1F1F4F5∎

Properties of the fixed point: fix $n$ and write $r : = \sqrt{a + x_{n - 1} (s) + x_{n + 1} (s)} > 0$ , so that $x_{n} (s) = s_{n} r$ by the fixed point equation. If $s_{n} = + 1$ then $x_{n} (s) = r > 0$ , and if $s_{n} = - 1$ then $x_{n} (s) = - r < 0$ ; hence in both cases $sign x_{n} (s) = s_{n}$ and $∣ x_{n} (s) ∣ = r$ . Squaring the fixed point equation gives $x_{n} (s)^{2} = r^{2} = a + x_{n - 1} (s) + x_{n + 1} (s)$ , which is the recurrence, and $∣ x_{n} (s) ∣ = r \geq L$ by step 1.1, [F1] and [F5].

Remarks

The choice $a \geq 100$ is a convenient explicit margin: it gives $\sqrt{a} \geq 10$ , hence $L^{2} \geq 74$ and the self-mapping inequality $\sqrt{a + 2 R} < R$ with room to spare. The argument works for every $a$ with $a - 2 R > 1$ and $a + 2 R < R^{2}$ .
No choice principle is used: the contraction and the completeness of $B_{R}$ are choice-free, and the fixed point is unique.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

Finite symbol windows determine the code: locality of the iterates, coding continuity and injectivity of $h (σ) = (x_{- 1} (σ), x_{0} (σ))$

Statement

Fix $a \geq 100$ and let $R = \sqrt{a} + 3$ , $L = \sqrt{a - 2 R}$ , $X_{\pm}$ , $D_{\pm}$ and $B_{R}$ be as in The two-sign bilateral shift space, its coding map, and the ball of bounded bilateral real sequences and The square-root recurrence $T_{s} (x)_{n} = s_{n} \sqrt{a + x_{n - 1} + x_{n + 1}}$ is a contraction of $B_{R}$ with a unique coded solution. Define the quadratic phase map

$F : R^{2} ⟶ R^{2}, F (x, y) : = (y, y^{2} - a - x),$

and for each $s \in X_{\pm}$ let $x (s)$ be the unique fixed point of $T_{s}$ in $B_{R}$ . Define the coding map

$h : X_{\pm} ⟶ R^{2}, h (s) : = (x_{- 1} (s), x_{0} (s)) .$

Then:

$F$ is a bijection with $F^{- 1} (x, y) = (x^{2} - a - y, x)$ , and for every $s \in X_{\pm}$ and $n \in Z$

$F (x_{n - 1} (s), x_{n} (s)) = (x_{n} (s), x_{n + 1} (s));$

(locality) for all $m \geq 0$ , $n \in Z$ and $s, t \in X_{\pm}$ : if $s_{j} = t_{j}$ for all $j$ with $∣ j - n ∣ \leq m - 1$ , then $(T_{s}^{m} 0)_{n} = (T_{t}^{m} 0)_{n}$ , where $T_{s}^{m}$ is the $m$ -fold iterate and $0$ the zero sequence;
(uniform error) for all $m \geq 0$ and $s \in X_{\pm}$ ,

$∥ T_{s}^{m} 0 - x (s) ∥_{\infty} \leq 2 R L^{- m};$

(coding continuity) if $s, t \in X_{\pm}$ agree at every coordinate $j$ with $∣ j ∣ \leq m$ , then $∣ x_{n} (s) - x_{n} (t) ∣ \leq 4 R L^{- m}$ for $n \in {- 1, 0}$ , and consequently

$∥ h (s) - h (t) ∥_{2} \leq 4 \sqrt{2} R L^{- m};$

in particular $h$ is continuous at every $s \in X_{\pm}$ (Continuity of a map between metric spaces, at a point and globally, in the $ε$ - $δ$ form) when $X_{\pm}$ carries $D_{\pm}$ and $R^{2}$ the Euclidean metric;

(injectivity) $h$ is injective.

Facts & Assumptions

Given: The parameter $a \geq 100$ , the constants $R, L$ , the shift spaces $X_{\pm}$ , the balls $B_{R}$ , the operators $T_{s}$ , their fixed points $x (s)$ of The square-root recurrence $T_{s} (x)_{n} = s_{n} \sqrt{a + x_{n - 1} + x_{n + 1}}$ is a contraction of $B_{R}$ with a unique coded solution, the map $F$ and the coding map $h$ displayed above.

[F1]

Fixed point data (The square-root recurrence $T_{s} (x)_{n} = s_{n} \sqrt{a + x_{n - 1} + x_{n + 1}}$ is a contraction of $B_{R}$ with a unique coded solution): for each $s \in X_{\pm}$ , $x (s) \in B_{R}$ , $T_{s} x (s) = x (s)$ , $sign x_{n} (s) = s_{n}$ , $∣ x_{n} (s) ∣ \geq L$ and $x_{n + 1} (s) + x_{n - 1} (s) = x_{n} (s)^{2} - a$ for all $n$ . The iterate convention for $T_{s}^{m}$ is the one fixed in that lemma.

[F2]

Banach error estimate (The a priori bound $d (x^{*}, x_{n}) \leq q^{n} d (x_{1}, x_{0}) / (1 - q)$ and the a posteriori bound $d (x^{*}, x_{n + 1}) \leq q d (x_{n + 1}, x_{n}) / (1 - q)$ , A contraction of a nonempty complete metric space into itself has exactly one fixed point, the limit of the iterates from any starting point, Lipschitz map, $α$ -Hölder map for rational $0 < α \leq 1$ , and contraction): for a contraction with constant $q < 1$ on a complete metric space, the iterates from any starting point $x_{0}$ satisfy $d (x^{*}, x_{m}) \leq q^{m} d (x_{1}, x_{0}) / (1 - q)$ ; here $q = 1 / L$ , $L \geq \sqrt{74} > 2$ , so $1 / (1 - 1 / L) = L / (L - 1) \leq 2$ .

[F3]

Ball and metric data (The two-sign bilateral shift space, its coding map, and the ball of bounded bilateral real sequences, The bounded functions on a set form a complete metric space in the supremum metric, and the coding map of the two-sign shift space is an isometry, The square-root recurrence $T_{s} (x)_{n} = s_{n} \sqrt{a + x_{n - 1} + x_{n + 1}}$ is a contraction of $B_{R}$ with a unique coded solution clauses 1-3, Metric space: $d (x, y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric): $0 \in B_{R}$ ; $∥ y ∥_{\infty} \leq R$ for $y \in B_{R}$ ; $T_{s}$ maps $B_{R}$ into itself and contracts with constant $1 / L < 1$ ; therefore induction gives $∥ T_{s}^{m} 0 ∥_{\infty} \leq R$ for every $m \geq 0$ ; the distance on $X_{\pm}$ is $D_{\pm} (s, t) = d_{2} (φ (s), φ (t))$ with the metric $d_{2}$ of $X_{2} = {0, 1}^{Z}$ .

[F4]

Cylinders and balls in the binary space (The bilateral shift metric is a metric, cylinders are its basic balls, and the shift space is sequentially compact, clause 2): with $u, v \in X_{2}$ and $F_{m} : = {- m, \dots, m}$ , if $d_{2} (u, v) < 2^{- m}$ then $u_{j} = v_{j}$ for $∣ j ∣ \leq m$ ; under the isometry $φ$ of The bounded functions on a set form a complete metric space in the supremum metric, and the coding map of the two-sign shift space is an isometry this says for $s, t \in X_{\pm}$ : if $D_{\pm} (s, t) < 2^{- m}$ then $s_{j} = t_{j}$ for $∣ j ∣ \leq m$ .

[F5]

Absolute value, order and the sequence $0$ (Absolute value in an ordered field, Basic properties of the absolute value, The triangle inequality, The two-sign bilateral shift space, its coding map, and the ball of bounded bilateral real sequences): $∣ u v ∣ = ∣ u ∣ ∣ v ∣$ , $∣ u + w ∣ \leq ∣ u ∣ + ∣ w ∣$ , $∣ u ∣ \leq M$ if and only if $- M \leq u \leq M$ for $M \geq 0$ , and the zero sequence satisfies $(T_{s}^{0} 0)_{n} = 0$ .

[F6]

Continuity in a metric space (Continuity of a map between metric spaces, at a point and globally, in the $ε$ - $δ$ form, Open ball, closed ball and sphere in a metric space, Convergence of a sequence in a metric space: $x_{k} \to x$ iff $d (x_{k}, x) \to 0$ in $R$ ): $h$ is continuous at $s$ when for every $ε > 0$ there is $δ > 0$ such that $D_{\pm} (s, t) < δ$ implies $∥ h (s) - h (t) ∥_{2} < ε$ .

Proof

technique · direct; an induction on $m$ for the finite window, the a priori Banach estimate for the uniform error, and the explicit inverse of $F$ for injectivity

1.1givenF1algebra

Inverse of $F$ : for every $(x, y) \in R^{2}$ , the two compositions are $F^{- 1} (F (x, y)) = F^{- 1} (y, y^{2} - a - x) = (y^{2} - a - (y^{2} - a - x), y) = (x, y)$ and $F (F^{- 1} (x, y)) = F (x^{2} - a - y, x) = (x, x^{2} - a - (x^{2} - a - y)) = (x, y) .$ Thus $F$ is a bijection with $F^{- 1} (x, y) = (x^{2} - a - y, x)$ . For $s \in X_{\pm}$ and $n \in Z$ , [F1] gives $x_{n + 1} (s) = x_{n} (s)^{2} - a - x_{n - 1} (s)$ , hence $F (x_{n - 1} (s), x_{n} (s)) = (x_{n} (s), x_{n} (s)^{2} - a - x_{n - 1} (s)) = (x_{n} (s), x_{n + 1} (s))$ .

1.2givenF1F3algebra

Locality, by induction on $m \geq 0$ : for $m = 0$ the sequence $T_{s}^{0} 0 = 0$ is independent of $s$ , so the claim is vacuous. Assume it holds for $m$ and let $s, t$ agree for $∣ j - n ∣ \leq m$ . Then $(T_{s}^{m + 1} 0)_{n} = s_{n} \sqrt{a + (T_{s}^{m} 0)_{n - 1} + (T_{s}^{m} 0)_{n + 1}}$ . The symbol $s_{n} = t_{n}$ lies in $∣ n - n ∣ = 0 \leq m$ ; by the induction hypothesis applied at $n - 1$ and $n + 1$ (for which the required windows are $∣ j - (n - 1) ∣ \leq m - 1$ and $∣ j - (n + 1) ∣ \leq m - 1$ , both contained in $∣ j - n ∣ \leq m$ ) the two radicands coincide; hence the two values coincide. This proves clause 2.

2.1givenstep 1.2F1F2F3

Uniform error: $T_{s}^{m} 0$ is the $m$ -th Banach iterate of the starting point $0 \in B_{R}$ for the contraction $T_{s}$ with constant $1 / L$ on the complete space $B_{R}$ ([F3], [F1]). By [F2], $∥ T_{s}^{m} 0 - x (s) ∥_{\infty} \leq q^{m} ∥ T_{s} 0 - 0 ∥_{\infty} / (1 - q) \leq q^{m} R / (1 - q) \leq 2 R L^{- m}$ , where $∥ T_{s} 0 ∥_{\infty} \leq R$ by [F3].

3.1givenstep 1.2step 2.1F5algebra

Coding estimates: let $s, t$ agree for $∣ j ∣ \leq m$ and let $n \in {- 1, 0}$ . For $n = 0$ , locality at level $m$ requires agreement for $∣ j ∣ \leq m - 1$ , which holds; for $n = - 1$ it requires agreement for $∣ j + 1 ∣ \leq m - 1$ , which implies $∣ j ∣ \leq m$ , so the required agreement also holds. Hence $(T_{s}^{m} 0)_{n} = (T_{t}^{m} 0)_{n}$ , and two applications of step 2.1 give $∣ x_{n} (s) - x_{n} (t) ∣ \leq ∣ (T_{s}^{m} 0)_{n} - (T_{t}^{m} 0)_{n} ∣ + 4 R L^{- m} = 4 R L^{- m}$ for $n = - 1, 0$ (Absolute value in an ordered field, [F5]). Consequently $∥ h (s) - h (t) ∥_{2} = \sqrt{∣ x_{- 1} (s) - x_{- 1} (t) ∣^{2} + ∣ x_{0} (s) - x_{0} (t) ∣^{2}} \leq \sqrt{2} \cdot 4 R L^{- m}$ .

4.1givenstep 3.1F4F6algebra

Continuity of $h$ : let $ε > 0$ and choose $m \geq 0$ with $4 \sqrt{2} R L^{- m} < ε$ , possible because $L > 1$ , hence $L^{- m} = (1 / L)^{m}$ tends to $0$ (For $∣ r ∣ < 1$ the sequence $r^{k}$ is null, and for $∣ r ∣ > 1$ the sequence $∣ r ∣^{k}$ diverges to $+ \infty$ , Series, partial sums, convergence and the sum, divergence, and the tail series). Put $δ : = 2^{- m}$ . If $D_{\pm} (s, t) < δ$ , then by [F4] $s_{j} = t_{j}$ for $∣ j ∣ \leq m$ , so step 3.1 gives $∥ h (s) - h (t) ∥_{2} < ε$ . By [F6] this is continuity of $h$ at $s$ , and $s$ was arbitrary.

5.1givenstep 1.1F1∎

Injectivity: suppose $h (s) = h (t)$ . For $u \in X_{\pm}$ and $n \in Z$ put $z_{n} (u) : = (x_{n - 1} (u), x_{n} (u))$ . By step 1.1, $z_{n + 1} (u) = F (z_{n} (u))$ and $z_{n - 1} (u) = F^{- 1} (z_{n} (u))$ for all $n$ ; induction on $n \geq 0$ gives $z_{n} (u) = F^{n} (z_{0} (u))$ and induction on $n \leq 0$ gives $z_{n} (u) = F^{n} (z_{0} (u))$ (using the bijectivity of $F$ from step 1.1). Since $z_{0} (s) = h (s) = h (t) = z_{0} (t)$ , it follows that $z_{n} (s) = z_{n} (t)$ for all $n$ , hence $x_{n} (s) = x_{n} (t)$ for all $n$ . By [F1], $s_{n} = sign x_{n} (s) = sign x_{n} (t) = t_{n}$ , so $s = t$ .

Remarks

Clause 2 is the quantitative content of "finite memory": the value of the $m$ -th approximation at coordinate $n$ sees only the $2 m - 1$ symbols centred at $n$ . Clause 4 turns this into a modulus of continuity of the coding map through the uniform error estimate.
Injectivity uses only that $F$ is invertible and that the sign of $x_{n}$ recovers $s_{n}$ ; no differentiability is used in this item.

TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

The quadratic map $F (x, y) = (y, y^{2} - a - x)$ has a compact coded invariant set conjugate to the two-sign shift, with chaos under ${A C}_{ω}$

Statement

Fix $a \geq 100$ , let $R = \sqrt{a} + 3$ , $L = \sqrt{a - 2 R}$ , let $X_{\pm}$ and $S_{\pm}$ be the two-sign bilateral shift space and its shift (The two-sign bilateral shift space, its coding map, and the ball of bounded bilateral real sequences), let $x (s) \in B_{R}$ be the coded sequence of The square-root recurrence $T_{s} (x)_{n} = s_{n} \sqrt{a + x_{n - 1} + x_{n + 1}}$ is a contraction of $B_{R}$ with a unique coded solution, let

$F (x, y) : = (y, y^{2} - a - x), h (s) : = (x_{- 1} (s), x_{0} (s)) (s \in X_{\pm}),$

and put $Λ : = h (X_{\pm}) \subseteq R^{2}$ . Then:

$F \circ h = h \circ S_{\pm}$ , i.e. $F (h (s)) = h (S_{\pm} s)$ for every $s \in X_{\pm}$ , and $Λ$ is invariant: $F (Λ) = Λ$ ;
$h$ is a continuous bijection of $(X_{\pm}, D_{\pm})$ onto $Λ$ ; it is a homeomorphism onto $Λ$ with the Euclidean metric (Continuity of a map between metric spaces, at a point and globally, in the $ε$ - $δ$ form) once the Axiom of Countable Choice is granted, since the proof uses ${A C}_{ω}$ exactly once, through the equivalence between closed and sequentially closed subsets of $R^{2}$ (A point lies in the closure of $A$ iff some sequence in $A$ converges to it, and a set is closed iff it is sequentially closed);
$Λ$ is sequentially compact (Countably compact, sequentially compact and limit point compact metric spaces) and, granting the Axioms of Countable and Dependent Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain), compact (Open cover, subcover, compact metric space, and compact subset of a metric space) and a continuous image of the compact space $X_{\pm}$ (The image of a compact metric space under a continuous map is compact, and so is the image of any compact subset);
(chaos) Granting the Axiom of Countable Choice as in clause 2, $F ∣_{Λ} : Λ \to Λ$ is continuous, topologically mixing, has dense periodic points, and has sensitive dependence on initial conditions with sensitivity constant $2 L$ ; hence $F ∣_{Λ}$ is chaotic in the stated sense (Topological mixing, dense periodic points, sensitive dependence, and the stated chaos list).

Facts & Assumptions

Given: The parameter $a \geq 100$ , the constants $R, L$ , the shift space $X_{\pm}$ with its shift $S_{\pm}$ and metric $D_{\pm}$ , the maps $T_{s}$ , $x (s)$ , $F$ and $h$ of The two-sign bilateral shift space, its coding map, and the ball of bounded bilateral real sequences, The square-root recurrence $T_{s} (x)_{n} = s_{n} \sqrt{a + x_{n - 1} + x_{n + 1}}$ is a contraction of $B_{R}$ with a unique coded solution and Finite symbol windows determine the code: locality of the iterates, coding continuity and injectivity of $h (σ) = (x_{- 1} (σ), x_{0} (σ))$ , and $Λ = h (X_{\pm})$ .

[F1]

Fixed point and coding data (The square-root recurrence $T_{s} (x)_{n} = s_{n} \sqrt{a + x_{n - 1} + x_{n + 1}}$ is a contraction of $B_{R}$ with a unique coded solution, Finite symbol windows determine the code: locality of the iterates, coding continuity and injectivity of $h (σ) = (x_{- 1} (σ), x_{0} (σ))$ ): $T_{s} (B_{R}) \subseteq B_{R}$ , $T_{s} x (s) = x (s)$ , $sign x_{n} (s) = s_{n}$ , $∣ x_{n} (s) ∣ \geq L$ , $x_{n + 1} (s) = x_{n} (s)^{2} - a - x_{n - 1} (s)$ ; $F$ is a bijection with $F^{- 1} (x, y) = (x^{2} - a - y, x)$ and $F (x_{n - 1} (s), x_{n} (s)) = (x_{n} (s), x_{n + 1} (s))$ ; $h$ is continuous and injective; and $D_{\pm} (s, t) = d_{2} (φ (s), φ (t))$ with $φ$ a bijective isometry onto the published two-symbol space $X_{2}$ (The bounded functions on a set form a complete metric space in the supremum metric, and the coding map of the two-sign shift space is an isometry, clause 3).

[F2]

The published shift data (The bilateral two-symbol shift space, its metric, its shift map and its cylinder sets, The bilateral shift metric is a metric, cylinders are its basic balls, and the shift space is sequentially compact, The bilateral two-symbol shift is continuous, topologically mixing, has dense periodic points and is sensitive; it is chaotic in the stated sense): the two-symbol shift $σ_{2}$ on $X_{2}$ is continuous, topologically mixing, has dense periodic points and is sensitive with constant $1$ ; $X_{2}$ is sequentially compact, with the diagonal construction using no choice principle, and compact under the Axioms of Countable and Dependent Choice.

[F3]

Topological vocabulary (Topological mixing, dense periodic points, sensitive dependence, and the stated chaos list, Dense, nowhere dense and codense subsets of a topological space, and the criterion by basic open sets, Continuity of a map between metric spaces, at a point and globally, in the $ε$ - $δ$ form, The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement, Convergence of a sequence in a metric space: $x_{k} \to x$ iff $d (x_{k}, x) \to 0$ in $R$ , Countably compact, sequentially compact and limit point compact metric spaces): the definitions of topological mixing, periodic points and their density, sensitivity, sequential compactness, continuity and open sets, and the fact that the closed subsets of a metric space are the complements of the open sets.

[F4]

Sequential closure (A point lies in the closure of $A$ iff some sequence in $A$ converges to it, and a set is closed iff it is sequentially closed, A sequence in a metric space has at most one limit, Convergence of a sequence in a metric space: $x_{k} \to x$ iff $d (x_{k}, x) \to 0$ in $R$ ): a subset $A$ of a metric space is closed if and only if it is sequentially closed, i.e. contains the limit of every convergent sequence with values in $A$ ; the implication "sequentially closed $\Rightarrow$ closed" is the direction proved from the Axiom of Countable Choice, while the converse direction is choice-free, and limits of convergent sequences in a metric space are unique.

[F5]

Choice scope (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain, Open cover, subcover, compact metric space, and compact subset of a metric space): ${A C}_{ω}$ is used once, in step 3.1, through the sequential-closure direction of [F4]; the compactness assertions of clause 3 use ${A C}_{ω} + D C$ , and only through the published compactness of $X_{2}$ and the general fact that continuous images of compact spaces are compact.

Proof

technique · direct; prove the conjugacy identity by uniqueness of the fixed point, then transfer compactness and the chaos properties through the coding homeomorphism and exhibit sensitivity by flipping a distant symbol

1.1givenF1algebra

Shift invariance of the code: fix $s \in X_{\pm}$ and define $y_{n} : = x_{n + 1} (s)$ for $n \in Z$ . Then, using $T_{S_{\pm} s}$ and [F1], $(T_{S_{\pm} s} y)_{n} = (S_{\pm} s)_{n} \sqrt{a + y_{n - 1} + y_{n + 1}} = s_{n + 1} \sqrt{a + x_{n} (s) + x_{n + 2} (s)} = (T_{s} x (s))_{n + 1} = x_{n + 1} (s) = y_{n};$ thus $y$ is a fixed point of $T_{S_{\pm} s}$ in $B_{R}$ , and by the uniqueness clause of [F1] (via The square-root recurrence $T_{s} (x)_{n} = s_{n} \sqrt{a + x_{n - 1} + x_{n + 1}}$ is a contraction of $B_{R}$ with a unique coded solution) $y = x (S_{\pm} s)$ , i.e. $x_{n} (S_{\pm} s) = x_{n + 1} (s)$ for all $n$ .

1.2givenF3

$F ∣_{Λ}$ is continuous: $F$ has polynomial components, hence is continuous on $R^{2}$ , and a restriction of a continuous map is continuous (Continuity of a map between metric spaces, at a point and globally, in the $ε$ - $δ$ form).

2.1step 1.1F1

Conjugacy identity: for $s \in X_{\pm}$ , [F1] and step 1.1 give $F (h (s)) = F (x_{- 1} (s), x_{0} (s)) = (x_{0} (s), x_{1} (s)) = (x_{- 1} (S_{\pm} s), x_{0} (S_{\pm} s)) = h (S_{\pm} s)$ . Hence $F (h (X_{\pm})) \subseteq h (X_{\pm})$ , and since $S_{\pm}$ is onto, for every $h (s) \in h (X_{\pm})$ the identity applied to $S_{\pm}^{- 1} s$ gives $h (s) = F (h (S_{\pm}^{- 1} s)) \in F (h (X_{\pm}))$ ; therefore $F (Λ) = Λ$ .

3.1givenstep 2.1F1F2F3F4

$h$ is a homeomorphism: $h$ is a continuous bijection of $X_{\pm}$ onto $Λ$ by [F1]. To prove that $h^{- 1}$ is continuous it suffices to prove that $h$ maps closed subsets of $X_{\pm}$ to closed subsets of $R^{2}$ . Let $C \subseteq X_{\pm}$ be closed. Then $C$ is sequentially compact: a sequence in $C$ has a subsequence converging in the sequentially compact space $X_{\pm}$ ([F2] and the isometry $φ$ of [F1]), and the limit lies in $C$ because $C$ is closed and therefore sequentially closed by the choice-free direction of [F4]. The continuous image $h (C)$ is sequentially compact: given a sequence $(h (x_{k}))$ with $x_{k} \in C$ , pick a subsequence $x_{k_{j}}$ converging to some $x \in C$ , then $h (x_{k_{j}}) \to h (x) \in h (C)$ by continuity (Countably compact, sequentially compact and limit point compact metric spaces, Continuity of a map between metric spaces, at a point and globally, in the $ε$ - $δ$ form); no choice is used beyond the single subsequence selection that sequential compactness of $C$ provides. Moreover $h (C)$ is sequentially closed: if a sequence in $h (C)$ converges, then some subsequence converges to a point of $h (C)$ by the previous paragraph (with that sequence in place of the given one), and the whole sequence has the same limit by uniqueness of limits ([F4]); hence the limit lies in $h (C)$ . By the ${A C}_{ω}$ direction of [F4] the sequentially closed set $h (C)$ is closed in $R^{2}$ . Thus $h$ is a closed map, so $h^{- 1}$ is continuous and $h$ is a homeomorphism.

3.2givenstep 2.1F1F3algebra

Sensitivity: let $z = h (s) \in Λ$ and $ε > 0$ ; choose $m \geq 0$ with $4 \sqrt{2} R L^{- m} < ε$ and let $k \geq m + 1$ . Define $t \in X_{\pm}$ by $t_{k} : = - s_{k}$ and $t_{j} : = s_{j}$ for $j \neq k$ . Then $s, t$ agree for all $∣ j ∣ \leq m$ , so by the coding-continuity clause of [F1] (Finite symbol windows determine the code: locality of the iterates, coding continuity and injectivity of $h (σ) = (x_{- 1} (σ), x_{0} (σ))$ ) one has $∥ h (s) - h (t) ∥_{2} \leq 4 \sqrt{2} R L^{- m} < ε$ . On the other hand $S_{\pm}^{k} s$ and $S_{\pm}^{k} t$ differ exactly at coordinate $0$ , so their signs at $0$ are opposite; by [F1], $∣ x_{0} (S_{\pm}^{k} s) - x_{0} (S_{\pm}^{k} t) ∣ \geq ∣ x_{0} (S_{\pm}^{k} s) ∣ + ∣ x_{0} (S_{\pm}^{k} t) ∣ \geq 2 L$ . Using step 2.1, $F^{k} (z) = F^{k} (h (s)) = h (S_{\pm}^{k} s)$ and $F^{k} (h (t)) = h (S_{\pm}^{k} t)$ , so $∥ F^{k} (z) - F^{k} (h (t)) ∥_{2} \geq ∣ x_{0} (S_{\pm}^{k} s) - x_{0} (S_{\pm}^{k} t) ∣ \geq 2 L$ . Thus $2 L$ is a sensitivity constant.

4.1givenstep 3.1F1F2F3

$Λ$ is sequentially compact: $X_{2}$ is sequentially compact without choice ([F2]); since $φ : X_{\pm} \to X_{2}$ is a bijective isometry ([F1]), a sequence in $X_{\pm}$ is carried to a sequence in $X_{2}$ with a convergent subsequence whose preimage under $φ$ converges in $X_{\pm}$ to the corresponding preimage limit; hence $X_{\pm}$ is sequentially compact. Applying the continuous-image argument of step 3.1 with $C = X_{\pm}$ shows that $Λ = h (X_{\pm})$ is sequentially compact.

4.2step 2.1step 3.1F2F3F5

$Λ$ is compact under ${A C}_{ω} + D C$ [F5]: $X_{2}$ is compact by [F2]; since $φ$ is a bijective isometry, a set $U \subseteq X_{\pm}$ is open exactly when $φ (U) \subseteq X_{2}$ is open, so the images under $φ$ of an open cover of $X_{\pm}$ form an open cover of $X_{2}$ , whose finite subcover pulls back to a finite subcover of the original cover; hence $X_{\pm}$ is compact. A continuous image of a compact space is compact by The image of a compact metric space under a continuous map is compact, and so is the image of any compact subset, so $Λ = h (X_{\pm})$ is compact.

4.3givenstep 2.1step 3.1F1F2F3

Topological mixing of $F ∣_{Λ}$ : let $U, V \subseteq Λ$ be nonempty and open in $Λ$ ; then $h^{- 1} (U)$ and $h^{- 1} (V)$ are nonempty open subsets of $X_{\pm}$ (step 3.1). Since $S_{\pm}$ is topologically mixing ([F2], transferred through the isometry $φ$ of [F1] and the conjugacy $S_{\pm} = φ^{- 1} \circ σ_{2} \circ φ$ ), there is $N$ with $S_{\pm}^{n} (h^{- 1} (U)) \cap h^{- 1} (V) \neq \emptyset$ for every $n \geq N$ ; applying $h$ and step 2.1, $F^{n} (U) \cap V = h (S_{\pm}^{n} (h^{- 1} (U)) \cap h^{- 1} (V)) \neq \emptyset$ for every $n \geq N$ .

4.4step 2.1step 3.1F1F2F3

Dense periodic points: a point $s$ is periodic for $S_{\pm}$ if and only if $h (s)$ is periodic for $F ∣_{Λ}$ , by the conjugacy of step 2.1 and the injectivity of $h$ ([F1]). Since $h$ is a bijection carrying the topology of $X_{\pm}$ onto that of $Λ$ (steps 2.1, 3.1 and Dense, nowhere dense and codense subsets of a topological space, and the criterion by basic open sets), it maps the dense set of periodic points of $S_{\pm}$ ([F2]) to a dense subset of $Λ$ consisting of periodic points of $F ∣_{Λ}$ .

5.1step 2.1step 1.2step 3.1step 4.1step 4.2step 4.3step 4.4step 3.2F3∎

Conclusion: by steps 1.2, 4.3, 4.4 and 3.2, the restriction $F ∣_{Λ}$ is continuous, topologically mixing, has dense periodic points and is sensitive with constant $2 L$ ; by the definition of Topological mixing, dense periodic points, sensitive dependence, and the stated chaos list it is chaotic in the stated sense, and clauses 1-4 are proved.

Remarks

The proof uses ${A C}_{ω}$ once in the homeomorphism claim of clause 2; the mixing and dense-periodic-point deductions in clause 4 use that homeomorphism. Sensitivity is proved directly and does not use choice. Compactness in clause 3 additionally uses ${A C}_{ω} + D C$ through the published compactness of $X_{2}$ .
Sensitivity is proved by an explicit far-symbol flip, so no compactness or uniform continuity of $h^{- 1}$ is needed. The constant $2 L$ is not claimed to be optimal.
Clause 1 identifies $Λ$ as an invariant set of the polynomial map $F$ ; no claim is made that $Λ$ is an attractor, has full measure, or exhausts the non-wandering set of $F$ .

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

Constant codes give the fixed points, and finite nonconstant word codes give homoclinic orbits converging to a fixed point at both ends

Statement

Fix $a \geq 100$ , let $R = \sqrt{a} + 3$ , $L = \sqrt{a - 2 R}$ , let $x (s)$ and $h$ be the coded sequence and coding map of The two-sign bilateral shift space, its coding map, and the ball of bounded bilateral real sequences, The square-root recurrence $T_{s} (x)_{n} = s_{n} \sqrt{a + x_{n - 1} + x_{n + 1}}$ is a contraction of $B_{R}$ with a unique coded solution and Finite symbol windows determine the code: locality of the iterates, coding continuity and injectivity of $h (σ) = (x_{- 1} (σ), x_{0} (σ))$ , and let $F (x, y) = (y, y^{2} - a - x)$ . For $u \in {- 1, + 1}$ write $\overset{ˉ}{u} \in X_{\pm}$ for the constant code ${\overset{ˉ}{u}}_{n} : = u$ , and put

$r_{u} : = 1 + u \sqrt{a + 1},$

so that $r_{+} > 0 > r_{-}$ , and write $W^{s}, W^{u}$ for the stable and unstable sets of Stable and unstable cones, invariant line slopes, stable and unstable sets, and transverse homoclinic points of the quadratic family. Then:

$x_{n} (\overset{ˉ}{u}) = r_{u}$ for every $n$ , and $r_{u}^{2} = a + 2 r_{u}$ ; consequently $h (\overset{ˉ}{u}) = (r_{u}, r_{u})$ is a fixed point of $F$ ;
if $s \in X_{\pm}$ satisfies $s_{n} = u$ for all $∣ n ∣ \geq N$ for some $N \geq 0$ , then

$F^{n} (h (s)) ⟶ (r_{u}, r_{u}) as n \to + \infty and as n \to - \infty;$

if moreover $s$ is not the constant code $\overset{ˉ}{u}$ , then $h (s) \neq (r_{u}, r_{u})$ , $F^{n} (h (s)) \neq (r_{u}, r_{u})$ for every $n \in Z$ , and $h (s) \in W^{s} (r_{u}, r_{u}) \cap W^{u} (r_{u}, r_{u})$ ; hence $h (s)$ is a homoclinic point of the fixed point $(r_{u}, r_{u})$ ;
distinct codes $s \neq t$ of the kind in clause 2 give distinct points $h (s) \neq h (t)$ ; in particular the finite-word deviations from the two constant codes produce infinitely many distinct homoclinic points.

Facts & Assumptions

Given: The parameter $a \geq 100$ , the constants $R, L$ , the coded sequence $x (s)$ and map $h$ , the phase map $F$ , the constant codes $\overset{ˉ}{u}$ , the numbers $r_{u}$ , and a code $s$ with $s_{n} = u$ for $∣ n ∣ \geq N$ .

[F1]

Coding data (The square-root recurrence $T_{s} (x)_{n} = s_{n} \sqrt{a + x_{n - 1} + x_{n + 1}}$ is a contraction of $B_{R}$ with a unique coded solution, Finite symbol windows determine the code: locality of the iterates, coding continuity and injectivity of $h (σ) = (x_{- 1} (σ), x_{0} (σ))$ ): $x (s) \in B_{R}$ is the unique fixed point of $T_{s}$ , $sign x_{n} (s) = s_{n}$ , $∣ x_{n} (s) ∣ \geq L$ , $x_{n + 1} (s) + x_{n - 1} (s) = x_{n} (s)^{2} - a$ ; locality: $(T_{s}^{m} 0)_{n}$ depends only on $s_{j}$ with $∣ j - n ∣ \leq m - 1$ ; uniform error: $∥ T_{s}^{m} 0 - x (s) ∥_{\infty} \leq 2 R L^{- m}$ ; $h$ is injective; $F (x_{n - 1} (s), x_{n} (s)) = (x_{n} (s), x_{n + 1} (s))$ .

[F2]

Conjugacy and invariant set (The quadratic map $F (x, y) = (y, y^{2} - a - x)$ has a compact coded invariant set conjugate to the two-sign shift, with chaos under ${A C}_{ω}$ ): $F (h (s)) = h (S_{\pm} s)$ for all $s$ , and the coded set is invariant.

[F3]

Square roots and the equation $r^{2} = a + 2 r$ (Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ , Absolute value in an ordered field): $a + 2 r_{u} > 0$ , $\sqrt{a + 2 r_{u}} = ∣ r_{u} ∣$ and $r_{u} = u \sqrt{a + 2 r_{u}}$ , because $r_{u}^{2} = a + 2 r_{u}$ and $sign r_{u} = u$ .

[F4]

Stable and unstable sets (Stable and unstable cones, invariant line slopes, stable and unstable sets, and transverse homoclinic points of the quadratic family): $W^{s} (\overset{ˉ}{z}) = {z : F^{n} (z) defined for all n \geq 0, F^{n} (z) \to \overset{ˉ}{z}}$ , $W^{u} (\overset{ˉ}{z}) = {z : F^{- n} (z) \to \overset{ˉ}{z}}$ , and a homoclinic point of a fixed point $\overset{ˉ}{z}$ is a point of $W^{s} (\overset{ˉ}{z}) \cap W^{u} (\overset{ˉ}{z})$ different from $\overset{ˉ}{z}$ ; convergence is metric convergence (Convergence of a sequence in a metric space: $x_{k} \to x$ iff $d (x_{k}, x) \to 0$ in $R$ , Metric space: $d (x, y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric).

[F5]

The sequence $L^{- m}$ tends to $0$ because $L > 1$ (For $∣ r ∣ < 1$ the sequence $r^{k}$ is null, and for $∣ r ∣ > 1$ the sequence $∣ r ∣^{k}$ diverges to $+ \infty$ , For $∣ r ∣ < 1$ , $\sum_{k \geq 0} r^{k} = 1 / (1 - r)$ , and for $∣ r ∣ \geq 1$ the series diverges).

[F6]

Absolute value and the base arithmetic (Basic properties of the absolute value, Absolute value in an ordered field): $∣ u v ∣ = ∣ u ∣ ∣ v ∣$ , $∣ u + w ∣ \leq ∣ u ∣ + ∣ w ∣$ , and $10^{2} = 100$ .

[F7]

Squaring is strictly increasing on the nonnegative reals (Squaring is monotone on the nonnegatives): if $0 \leq x < y$ then $x^{2} < y^{2}$ , and if $0 \leq x \leq y$ then $x^{2} \leq y^{2}$ .

Proof

technique · direct; check the constant codes against the recurrence, use the locality of the finite windows for the two-sided convergence, and invoke injectivity of the coding for the distinctness of finite-word homoclinic points

1.1givenF3F6F7algebra

The constants $r_{u}$ : since $a \geq 100$ , [F7] applied to $10^{2} \leq a = (\sqrt{a})^{2}$ gives $\sqrt{a} \geq 10$ . Also $(\sqrt{a} + 1)^{2} = a + 2 \sqrt{a} + 1 > a + 1$ , so [F7] gives $\sqrt{a + 1} < \sqrt{a} + 1$ ; and $1^{2} < a + 1$ gives $\sqrt{a + 1} > 1$ . Thus $a + 2 - 2 \sqrt{a + 1} > a - 2 \sqrt{a} = \sqrt{a} (\sqrt{a} - 2) > 0$ , so $a + 2 r_{u} = a + 2 + 2 u \sqrt{a + 1} > 0$ for both signs. The number $r_{u} = 1 + u \sqrt{a + 1}$ satisfies $r_{u}^{2} = 1 + 2 u \sqrt{a + 1} + (a + 1) = a + 2 + 2 u \sqrt{a + 1} = a + 2 r_{u}$ , and $sign r_{u} = u$ by $\sqrt{a + 1} > 1$ . Hence by [F3] $r_{u} = u \sqrt{a + 2 r_{u}}$ .

2.1step 1.1F1F2F6F7algebra

Constant codes: let $\tilde{x}$ be the constant sequence ${\tilde{x}}_{n} : = r_{u}$ . By step 1.1, $(T_{\overset{ˉ}{u}} \tilde{x})_{n} = u \sqrt{a + r_{u} + r_{u}} = u \sqrt{a + 2 r_{u}} = r_{u} = {\tilde{x}}_{n}$ for all $n$ ; so $\tilde{x}$ is a fixed point of $T_{\overset{ˉ}{u}}$ in $B_{R}$ (indeed $∣ r_{u} ∣ \leq 1 + \sqrt{a + 1} < \sqrt{a} + 2 < R$ by [F7] and $a \geq 100$ ). By uniqueness ([F1]) $x (\overset{ˉ}{u}) = \tilde{x}$ , hence $h (\overset{ˉ}{u}) = (r_{u}, r_{u})$ ; and $F (r_{u}, r_{u}) = (r_{u}, r_{u}^{2} - a - r_{u}) = (r_{u}, r_{u})$ because $r_{u}^{2} - a = 2 r_{u}$ .

3.1step 2.1F1F2F5algebra

Convergence in the forward direction: let $m \geq 0$ and let $n \geq N + m - 1$ ; then every $j$ with $∣ j - n ∣ \leq m - 1$ satisfies $j \geq n - (m - 1) \geq N$ , hence $s_{j} = u = {\overset{ˉ}{u}}_{j}$ . By [F1], $(T_{s}^{m} 0)_{n} = (T_{\overset{ˉ}{u}}^{m} 0)_{n}$ . Therefore $∣ x_{n} (s) - x_{n} (\overset{ˉ}{u}) ∣ \leq ∣ x_{n} (s) - (T_{s}^{m} 0)_{n} ∣ + ∣ (T_{\overset{ˉ}{u}}^{m} 0)_{n} - x_{n} (\overset{ˉ}{u}) ∣ \leq 4 R L^{- m}$ for $n \geq N + m - 1$ , using the uniform error bound of [F1] twice and $x (\overset{ˉ}{u}) = \tilde{x}$ of step 2.1. Given $ε > 0$ , choose $m$ with $4 R L^{- m} < ε$ , possible by [F5]; then $∣ x_{n} (s) - r_{u} ∣ < ε$ for all $n \geq N + m - 1$ . Hence $x_{n} (s) \to r_{u}$ as $n \to + \infty$ , and therefore $F^{n} (h (s)) = (x_{n - 1} (s), x_{n} (s)) \to (r_{u}, r_{u})$ by [F1].

3.2step 2.1F1F2algebra

Nonconstancy: suppose $s \neq \overset{ˉ}{u}$ ; then there is $k$ with $s_{k} \neq u$ , and by [F1] $sign x_{k} (s) = s_{k} \neq u = sign r_{u}$ , so $x_{k} (s) \neq r_{u}$ and $h (s) = (x_{- 1} (s), x_{0} (s))$ ; if $h (s) = (r_{u}, r_{u})$ then $x_{- 1} (s) = x_{0} (s) = r_{u}$ , and then by the recurrence $x_{n} (s) = r_{u}$ for all $n$ , so $s_{n} = sign x_{n} (s) = u$ for all $n$ , a contradiction. Hence $h (s) \neq (r_{u}, r_{u})$ . Since $F^{n} (h (s)) = h (S_{\pm}^{n} s)$ by [F2] and each $S_{\pm}^{n} s$ is again not constant, the same argument gives $F^{n} (h (s)) \neq (r_{u}, r_{u})$ for all $n \in Z$ .

4.1step 3.1F1algebra

Convergence in the backward direction: if $n \leq - N - (m - 1)$ and $∣ j - n ∣ \leq m - 1$ , then $j \leq n + (m - 1) \leq - N$ , so $∣ j ∣ \geq N$ and $s_{j} = u = {\overset{ˉ}{u}}_{j}$ . Locality [F1] therefore gives $(T_{s}^{m} 0)_{n} = (T_{\overset{ˉ}{u}}^{m} 0)_{n}$ , and the same two uniform error bounds as in step 3.1 yield $∣ x_{n} (s) - r_{u} ∣ \leq 4 R L^{- m}$ . Hence $x_{n} (s) \to r_{u}$ as $n \to - \infty$ . Since $F$ is a bijection ([F1]) and $F^{n} (h (s)) = (x_{n - 1} (s), x_{n} (s))$ , this says $F^{n} (h (s)) \to (r_{u}, r_{u})$ as $n \to - \infty$ .

5.1step 3.1step 4.1step 3.2F1F4

Homoclinic points: by steps 3.1 and 4.1, the point $h (s)$ lies in $W^{s} (r_{u}, r_{u}) \cap W^{u} (r_{u}, r_{u})$ (the forward and backward orbits are defined for all times because $F$ is a bijection and $F^{n} (h (s)) = (x_{n - 1} (s), x_{n} (s))$ for all $n$ ), and by step 3.2 it differs from the fixed point. By the definition [F4] it is a homoclinic point of $(r_{u}, r_{u})$ .

6.1step 5.1F1∎

Distinctness: if $s \neq t$ then $h (s) \neq h (t)$ because $h$ is injective by [F1]; the nonconstant finite-word codes are infinite in number (for example the codes differing from $\overset{ˉ}{u}$ at exactly the coordinates $0, 1, \dots, j$ for $j \geq 0$ are pairwise distinct), so there are infinitely many distinct homoclinic points.

Remarks

The convergence rates are uniform in the code: for a code that agrees with $\overset{ˉ}{u}$ for $∣ n ∣ \geq N$ , the deviation satisfies $∣ x_{n} (s) - r_{u} ∣ \leq 4 R L^{- m}$ whenever $∣ n ∣ \geq N + m - 1$ .
The two constant codes give fixed points, not homoclinic points; homoclinic points arise precisely from nonconstant codes with constant tails, which are the finite-word deviations promised by the design.

DefinitionDefinition: AI-adaptedProof: Not applicableprecheck passOpen item page →

Stable and unstable cones, invariant line slopes, stable and unstable sets, and transverse homoclinic points of the quadratic family

Definition

The quadratic family. For $a \in R$ let

$F_{a} : R^{2} ⟶ R^{2}, F_{a} (x, y) : = (y, y^{2} - a - x),$

with differential at $(x, y) \in R^{2}$

$D F_{a} (x, y) = (\begin{array}{cc} 0 & 1 \\ - 1 & 2 y \end{array}) .$

The inverse map is

$F_{a}^{- 1} (x, y) = (x^{2} - a - y, x),$

and its differential at $(x, y)$ is

$D (F_{a}^{- 1}) (x, y) = (\begin{array}{cc} 2 x & - 1 \\ 1 & 0 \end{array}) .$

Here $D F_{a} (x, y)^{- 1}$ denotes the inverse matrix of $D F_{a} (x, y)$ , namely $(\begin{array}{cc} 2 y & - 1 \\ 1 & 0 \end{array})$ ; it is not the same notation as $D (F_{a}^{- 1}) (x, y)$ , which evaluates the derivative of the inverse map at the point $(x, y)$ . All matrices act on column vectors of $R^{2}$ (The total (Fréchet) derivative $D f (a)$ as the linear first-order approximation with $o (∥ h ∥_{2})$ remainder, The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case).

Complex coordinates and modulus. Use $C : = R [x] / (x^{2} + 1)$ , with real numbers embedded as constant classes and $i : = x + (x^{2} + 1)$ , so $i^{2} = - 1$ (The complex numbers as $R [x] / (x^{2} + 1)$ , with the real embedding and imaginary unit $i$ ). This is a field and each element has a unique form $α + i β$ with $α, β \in R$ ( $C = R [x] / (x^{2} + 1)$ is a field, every element is uniquely $a + b i$ , and every nonzero element has inverse $(a - b i) / (a^{2} + b^{2})$ ). Define its modulus by

$∣ α + i β ∣ : = \sqrt{α^{2} + β^{2}},$

where the square root is the unique nonnegative real square root (Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ ). Identify $C^{2}$ with pairs of complex coordinates; a real matrix acts on it by the same matrix entries, extended $C$ -linearly.

Hyperbolic fixed points. Let $U, V \subseteq R^{2}$ be open, let $G : U \to V$ be a $C^{1}$ diffeomorphism, and let $\overset{ˉ}{z} \in U \cap V$ satisfy $G (\overset{ˉ}{z}) = \overset{ˉ}{z}$ . Its eigenvalues are the numbers $λ \in C$ for which $D G (\overset{ˉ}{z}) v = λ v$ for some $v \in C^{2} ∖ {0}$ . The fixed point $\overset{ˉ}{z}$ is hyperbolic when every such eigenvalue satisfies $∣ λ ∣ \neq 1$ . A global $C^{1}$ diffeomorphism of $R^{2}$ is the special case $U = V = R^{2}$ .

Cones. Fix $k \in (0, 1)$ . The unstable cone and stable cone of aperture $k$ are

$C_{k}^{u} : = {(u, v) \in R^{2} : ∣ u ∣ \leq k ∣ v ∣}, C_{k}^{s} : = {(u, v) \in R^{2} : ∣ v ∣ \leq k ∣ u ∣}$

(Absolute value in an ordered field); both are closed subsets of $R^{2}$ , they are closed under multiplication by real scalars, and $C_{k}^{u} \cap C_{k}^{s} = {0}$ , because a nonzero vector in the intersection would satisfy $∣ u ∣ \leq k ∣ v ∣$ and $∣ v ∣ \leq k ∣ u ∣$ , hence $∣ u ∣ \leq k^{2} ∣ u ∣ < ∣ u ∣$ .

Slopes. A line through the origin not contained in the horizontal axis is written uniquely as ${(h v, v) : v \in R}$ with the slope parameter $h = u / v$ ; here $∣ h ∣ \leq k$ characterises the lines contained in $C_{k}^{u}$ . Similarly a line not contained in the vertical axis is written uniquely as ${(u, r u) : u \in R}$ with the slope parameter $r = v / u$ , and $∣ r ∣ \leq k$ characterises the lines contained in $C_{k}^{s}$ .

For real numbers $x, y$ two maps on slope parameters are defined:

$Φ_{y} (h) : = \frac{1}{2 y - h} (h \neq 2 y) (forward slope map), Ψ_{x} (r) : = \frac{1}{2 x - r} (r \neq 2 x) (backward slope map) .$

The intended reading is the graph-transform reading: $D F_{a} (x, y)$ carries the line ${(h v, v) : v \in R}$ at $(x, y)$ to the line ${(h^{'} v, v) : v \in R}$ at $F_{a} (x, y)$ with $h^{'} = Φ_{y} (h)$ , and $D (F_{a}^{- 1}) (x, y)$ carries the line ${(u, r u) : u \in R}$ at $(x, y)$ to the line ${(u, r^{'} u) : u \in R}$ at $F_{a}^{- 1} (x, y)$ with $r^{'} = Ψ_{x} (r)$ . These two statements follow from the displayed matrices and are proved in the companion lemma on this page, not asserted as part of this definition.

Invariant line fields on a sequence. A real bilateral sequence is a function $s : Z \to R$ ; write $s_{n} : = s (n)$ . Put $z_{n} : = (s_{n - 1}, s_{n})$ . An unstable line field along $s$ is a family $(L_{n}^{u})_{n \in Z}$ of lines through the origin of $R^{2}$ such that $D F_{a} (z_{n})$ maps $L_{n}^{u}$ onto $L_{n + 1}^{u}$ for every $n$ . A stable line field along $s$ is a family $(L_{n}^{s})_{n \in Z}$ such that $D (F_{a}^{- 1}) (z_{n + 1})$ maps $L_{n + 1}^{s}$ onto $L_{n}^{s}$ for every $n$ . The latter evaluates the derivative of the inverse map at $z_{n + 1}$ . When $s$ satisfies $s_{n + 1} = s_{n}^{2} - a - s_{n - 1}$ , one has $z_{n + 1} = F_{a} (z_{n})$ , so these are the source and target orbit points. The displayed matrix conditions also define line families for arbitrary sequences, without asserting that such a sequence is an orbit. For a real bound $L \geq 2$ , sequences with $∣ s_{n} ∣ \geq L$ , and apertures satisfying $k (2 L - k) \geq 1$ , existence and uniqueness of line fields with slopes in $[- k, k]$ are proved in the companion lemma on this page.

Stable and unstable sets. Let $G : Z \to Z$ be a bijection of a metric space $(Z, d)$ (Metric space: $d (x, y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric) with fixed point $\overset{ˉ}{z}$ (so $G (\overset{ˉ}{z}) = \overset{ˉ}{z}$ ). Put

$W^{s} (\overset{ˉ}{z}) : = {z \in Z : G^{n} (z) is defined for every n \geq 0 and G^{n} (z) \to \overset{ˉ}{z} as n \to \infty},$

$W^{u} (\overset{ˉ}{z}) : = {z \in Z : G^{- n} (z) is defined for every n \geq 0 and G^{- n} (z) \to \overset{ˉ}{z} as n \to \infty},$

where convergence is that of the metric $d$ (Convergence of a sequence in a metric space: $x_{k} \to x$ iff $d (x_{k}, x) \to 0$ in $R$ ). The sets $W^{s} (\overset{ˉ}{z})$ and $W^{u} (\overset{ˉ}{z})$ are the stable set and the unstable set of $\overset{ˉ}{z}$ .

These orbit-based definitions also apply to a local map. Let $U, V \subseteq R^{2}$ be open, let $G : U \to V$ be a $C^{1}$ diffeomorphism, and let $\overset{ˉ}{z} \in U \cap V$ satisfy $G (\overset{ˉ}{z}) = \overset{ˉ}{z}$ . Define $G^{0} (z) : = z$ and recursively $G^{n + 1} (z) : = G (G^{n} (z))$ whenever $G^{n} (z) \in U$ ; define $G^{- n - 1} (z) : = G^{- 1} (G^{- n} (z))$ whenever $G^{- n} (z) \in V$ . Then

$W_{G}^{s} (\overset{ˉ}{z}) : = {z \in U : G^{n} (z) is defined for every n \geq 0 and G^{n} (z) \to \overset{ˉ}{z}},$

$W_{G}^{u} (\overset{ˉ}{z}) : = {z \in V : G^{- n} (z) is defined for every n \geq 0 and G^{- n} (z) \to \overset{ˉ}{z}},$

where convergence is Euclidean metric convergence. The explicit domain clauses require every forward iterate to lie in $U$ before applying $G$ , and every backward iterate to lie in $V$ before applying $G^{- 1}$ . In the global case above every iterate is defined automatically.

A homoclinic point is a point in both the stable and unstable sets, different from the fixed point $\overset{ˉ}{z}$ . For a global bijection on $R^{2}$ or a local planar diffeomorphism as above, it is transverse when there are $ε > 0$ and $C^{1}$ maps ( $C^{k}$ Euclidean maps and diffeomorphisms, The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set)

$γ_{s}, γ_{u} : (- ε, ε) ⟶ R^{2}$

such that $γ_{s} (0) = γ_{u} (0) = z$ , the stable curve takes values in the stable set and the unstable curve takes values in the unstable set, both are injective on $(- ε, ε)$ with $γ_{s}^{'} (0) \neq 0 \neq γ_{u}^{'} (0)$ , and the derivative vectors $γ_{s}^{'} (0), γ_{u}^{'} (0) \in R^{2}$ are linearly independent. Thus derivatives and tangent vectors in this definition are taken in the two-dimensional differentiable setting used by the results on this page; no derivative is assigned to a curve in an arbitrary metric space. The interval $(- ε, ε)$ is that of Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length.

Remarks

The definition records the two vocabularies used by the coding theorem and the transversality theorem: the linear cone and slope objects of the quadratic family, and the invariant-set objects of global bijections and local planar diffeomorphisms. It asserts no invariance of the cones, no convergence of the slope maps and no transversality; those are the contents of Cone invariance and expansion, and convergence of the slope graph transforms to unique invariant stable and unstable line fields and The coded curves are genuine $C^{1}$ stable and unstable curves of $F$ , with transverse intersection at every finite-word homoclinic point.
The transversality condition above is stated with two $C^{1}$ curves through the point, one lying in each invariant set, whose tangents span $R^{2}$ . When the local stable and unstable sets are themselves $C^{1}$ curves near $z$ (which holds at the fixed points and their finite images in the situation of The coded curves are genuine $C^{1}$ stable and unstable curves of $F$ , with transverse intersection at every finite-word homoclinic point), this agrees with the usual formulation " $W^{s}$ and $W^{u}$ meet transversely at $z$ ".
The forward slope map is defined on $[- k, k]$ when $∣ y ∣ > k$ , since $∣ 2 y - h ∣ \geq 2 ∣ y ∣ - k > 0$ ; this condition alone does not ensure contraction. In Cone invariance and expansion, and convergence of the slope graph transforms to unique invariant stable and unstable line fields, the stronger bound $∣ y ∣ \geq L \geq 2$ gives $∣ Φ_{y} (h) - Φ_{y} (h^{'}) ∣ \leq (2 L - k)^{- 2} ∣ h - h^{'} ∣$ for $h, h^{'} \in [- k, k]$ , with $(2 L - k)^{- 2} < 1$ . The additional aperture condition $k (2 L - k) \geq 1$ ensures that the image lies in $[- k, k]$ . The same conclusions hold for the backward slope map with $∣ x ∣ \geq L$ .

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

Cone invariance and expansion, and convergence of the slope graph transforms to unique invariant stable and unstable line fields

Statement

Let $a \in R$ , $k \in (0, 1)$ and $L \geq 2$ . Let $s = (s_{n})_{n \in Z}$ be a real bilateral sequence with $∣ s_{n} ∣ \geq L$ for every $n$ , put $z_{n} : = (s_{n - 1}, s_{n})$ , and let $C_{k}^{u}$ , $C_{k}^{s}$ , $Φ_{y}$ , $Ψ_{x}$ , $D F_{a}$ and $F_{a}$ be as in Stable and unstable cones, invariant line slopes, stable and unstable sets, and transverse homoclinic points of the quadratic family. Write

$λ : = 2 L - k > 1, ρ : = λ^{- 2} < 1,$

Assume in addition the aperture condition $k (2 L - k) \geq 1$ , equivalently $1 / λ \leq k$ . This condition is needed for invariance of the stated cones and slope interval; $L \geq 2$ alone does not imply it for arbitrarily small $k$ . Then:

(cone invariance and expansion) for every $n$ : $D F_{a} (z_{n})$ maps $C_{k}^{u}$ into itself, and every $(u, v) \in C_{k}^{u}$ satisfies

$\max (∣ v ∣, ∣ - u + 2 s_{n} v ∣) \geq λ \max (∣ u ∣, ∣ v ∣);$

and $D (F_{a}^{- 1}) (z_{n})$ maps $C_{k}^{s}$ into itself, with $\max (∣ 2 s_{n - 1} u - v ∣, ∣ u ∣) \geq λ \max (∣ u ∣, ∣ v ∣)$ for every $(u, v) \in C_{k}^{s}$ ; 2. (slope contractions) for every real $y$ with $∣ y ∣ \geq L$ , the map $Φ_{y}$ carries $[- k, k]$ into $[- 1 / λ, 1 / λ] \subseteq [- k, k]$ and satisfies $∣ Φ_{y} (h) - Φ_{y} (h^{'}) ∣ \leq ρ ∣ h - h^{'} ∣$ there; the same holds for $Ψ_{x}$ for every $∣ x ∣ \geq L$ ; 3. (unstable line field) define $h_{n}^{(0)} : = 0$ and $h_{n}^{(m + 1)} : = Φ_{s_{n - 1}} (h_{n - 1}^{(m)})$ for $m \geq 0$ . Then $(h_{n}^{(m)})_{m}$ converges for every $n$ to a limit $h_{n} \in [- k, k]$ , uniformly in $n$ , and

$h_{n} = \frac{1}{2 s_{n - 1} - h_{n - 1}} (n \in Z);$

furthermore $D F_{a} (z_{n})$ maps the line ${(h_{n} v, v) : v \in R}$ onto the line ${(h_{n + 1} v, v)}$ for every $n$ , so the lines form the unique unstable line field along $s$ whose slopes all lie in $[- k, k]$ ; 4. (stable line field) similarly, $r_{n}^{(0)} : = 0$ and $r_{n}^{(m + 1)} : = Ψ_{s_{n}} (r_{n + 1}^{(m)})$ converge to limits $r_{n} \in [- k, k]$ with $r_{n} = 1 / (2 s_{n} - r_{n + 1})$ , and $D (F_{a}^{- 1}) (z_{n + 1})$ maps ${(u, r_{n + 1} u) : u \in R}$ onto ${(u, r_{n} u) : u \in R}$ ; this is the unique stable line field along $s$ with slopes in $[- k, k]$ .

Facts & Assumptions

Given: The parameter $a$ , the aperture $k \in (0, 1)$ , the bound $L \geq 2$ , the sequence $s$ with $∣ s_{n} ∣ \geq L$ , the cones, slope maps and matrix data of Stable and unstable cones, invariant line slopes, stable and unstable sets, and transverse homoclinic points of the quadratic family, and the recursively defined slope families $h^{(m)}$ , $r^{(m)}$ .

[F1]

Cone, slope and matrix data (Stable and unstable cones, invariant line slopes, stable and unstable sets, and transverse homoclinic points of the quadratic family): $D F_{a} (x, y) (u, v) = (v, - u + 2 y v)$ and $D (F_{a}^{- 1}) (x, y) (u, v) = (2 x u - v, u)$ ; $C_{k}^{u} = {∣ u ∣ \leq k ∣ v ∣}$ , $C_{k}^{s} = {∣ v ∣ \leq k ∣ u ∣}$ , $C_{k}^{u} \cap C_{k}^{s} = {0}$ . For $h \neq 2 y$ , the matrix gives $D F_{a} (x, y) (h v, v) = (v, (2 y - h) v)$ , so its image line has slope $Φ_{y} (h) = 1 / (2 y - h)$ . Likewise, for $r \neq 2 x$ , $D (F_{a}^{- 1}) (x, y) (u, r u) = ((2 x - r) u, u)$ , so its image line has slope $Ψ_{x} (r) = 1 / (2 x - r)$ .

[F2]

Absolute value and order (Absolute value in an ordered field, Basic properties of the absolute value, The triangle inequality, Complete ordered field (least-upper-bound property)): $∣ u v ∣ = ∣ u ∣ ∣ v ∣$ ; $∣ u - w ∣ \geq ∣ u ∣ - ∣ w ∣$ ; if $∣ u ∣ \leq k ∣ v ∣$ and $0 < k < 1$ then $\max (∣ u ∣, ∣ v ∣) = ∣ v ∣$ ; if $∣ v ∣ \leq k ∣ u ∣$ then $\max (∣ u ∣, ∣ v ∣) = ∣ u ∣$ ; and the stated aperture condition gives $1 / λ \leq k$ .

[F3]

Series and geometric sequences (For $∣ r ∣ < 1$ , $\sum_{k \geq 0} r^{k} = 1 / (1 - r)$ , and for $∣ r ∣ \geq 1$ the series diverges, Series, partial sums, convergence and the sum, divergence, and the tail series, A series of nonnegative terms converges iff its partial sums are bounded, and then the sum is their supremum, For $∣ r ∣ < 1$ the sequence $r^{k}$ is null, and for $∣ r ∣ > 1$ the sequence $∣ r ∣^{k}$ diverges to $+ \infty$ ): for $0 \leq ρ < 1$ the series $\sum_{j \geq 0} ρ^{j}$ converges, $ρ^{m} \to 0$ , and $\sum_{j \geq m} ρ^{j} = ρ^{m} / (1 - ρ)$ .

[F4]

Completeness and limits of real sequences (The Cauchy criterion from the least-upper-bound property: in a complete ordered field every Cauchy sequence converges, Sequences of reals: bounded, eventually, frequently, tails, subsequences, Algebra of limits: sums, scalar multiples, products and quotients, Limits preserve non-strict inequalities, Convergence of a sequence in a metric space: $x_{k} \to x$ iff $d (x_{k}, x) \to 0$ in $R$ ): every Cauchy real sequence converges; limits of convergent real sequences are compatible with addition, multiplication, division by nonzero limits, and with weak inequalities; a limit of numbers in a closed interval $[- k, k]$ lies in $[- k, k]$ .

Proof

technique · direct; verify the two cone computations, then solve the slope recurrences by a contraction iteration from the far past and the far future

1.1givenF2algebra

Constants: since $L \geq 2$ and $0 < k < 1$ , we have $2 L - k \geq 4 - k > 3$ , so $λ = 2 L - k > 1$ and $0 < ρ = λ^{- 2} < 1$ ; moreover the aperture hypothesis gives $k (2 L - k) \geq 1$ , hence $1 / λ \leq k$ .

2.1givenstep 1.1F1F2algebra

Unstable cone: let $(u, v) \in C_{k}^{u}$ , so $∣ u ∣ \leq k ∣ v ∣$ and $\max (∣ u ∣, ∣ v ∣) = ∣ v ∣$ by [F2]. Then $D F_{a} (z_{n}) (u, v) = (v, - u + 2 s_{n} v)$ and $∣ - u + 2 s_{n} v ∣ \geq 2 ∣ s_{n} ∣ ∣ v ∣ - ∣ u ∣ \geq (2 L - k) ∣ v ∣ = λ ∣ v ∣$ by [F2]. Also $∣ - u + 2 s_{n} v ∣ \geq λ ∣ v ∣ \geq ∣ v ∣$ since $λ > 1$ , so $\max (∣ v ∣, ∣ - u + 2 s_{n} v ∣) \geq λ \max (∣ u ∣, ∣ v ∣)$ . Finally $∣ v ∣ / ∣ - u + 2 s_{n} v ∣ \leq 1 / λ \leq k$ whenever $v \neq 0$ , so the image is in $C_{k}^{u}$ (the case $v = 0$ gives $u = 0$ and image $0$ ).

2.2givenstep 1.1F1F2algebra

Stable cone: let $(u, v) \in C_{k}^{s}$ , so $∣ v ∣ \leq k ∣ u ∣$ and $\max (∣ u ∣, ∣ v ∣) = ∣ u ∣$ by [F2]. Then $D (F_{a}^{- 1}) (z_{n}) (u, v) = (2 s_{n - 1} u - v, u)$ and $∣ 2 s_{n - 1} u - v ∣ \geq 2 L ∣ u ∣ - ∣ v ∣ \geq (2 L - k) ∣ u ∣ = λ ∣ u ∣$ ; hence $\max (∣ 2 s_{n - 1} u - v ∣, ∣ u ∣) \geq λ \max (∣ u ∣, ∣ v ∣)$ and $∣ u ∣ / ∣ 2 s_{n - 1} u - v ∣ \leq 1 / λ \leq k$ whenever $u \neq 0$ , with the zero case trivial.

2.3givenstep 1.1F1F2algebra

Slope contractions: let $∣ y ∣ \geq L$ and $h, h^{'} \in [- k, k]$ ; by step 1.1, $∣ 2 y - h ∣ \geq 2 L - k = λ > 0$ and $∣ 2 y - h^{'} ∣ \geq λ$ , so $Φ_{y}$ is defined on $[- k, k]$ , $∣ Φ_{y} (h) ∣ \leq 1 / λ \leq k$ , and $∣ Φ_{y} (h) - Φ_{y} (h^{'}) ∣ = \frac{∣ h - h^{'} ∣}{∣ 2 y - h ∣ ∣ 2 y - h^{'} ∣} \leq λ^{- 2} ∣ h - h^{'} ∣ = ρ ∣ h - h^{'} ∣ .$ The same computation with $Ψ_{x}$ proves the second assertion.

3.1givenstep 2.3F2F3F4algebra

Unstable slopes, uniform bounds and contraction: define $h_{n}^{(0)} : = 0$ , $h_{n}^{(m + 1)} : = Φ_{s_{n - 1}} (h_{n - 1}^{(m)})$ . By induction, step 2.3 gives $∣ h_{n}^{(m)} ∣ \leq 1 / λ$ for all $m \geq 1$ and $n$ , and $∣ h_{n}^{(m + 1)} - h_{n}^{(m)} ∣ \leq ρ^{m} / λ$ for all $m \geq 0$ . Hence for $m^{'} \geq m$ , $∣ h_{n}^{(m^{'})} - h_{n}^{(m)} ∣ \leq \sum_{j = m}^{m^{'} - 1} ρ^{j} / λ \leq \frac{ρ^{m}}{λ (1 - ρ)},$ uniformly in $n$ . Thus $(h_{n}^{(m)})_{m}$ is Cauchy; completeness in [F4] gives its limit $h_{n}$ , with $∣ h_{n} ∣ \leq 1 / λ \leq k$ . Passing $m^{'} \to \infty$ in the same estimate gives $∣ h_{n} - h_{n}^{(m)} ∣ \leq ρ^{m} / (λ (1 - ρ))$ for every $n$ , proving uniform convergence.

3.2givenstep 2.3F2F3F4

Stable slopes: the same argument with $r_{n}^{(0)} : = 0$ , $r_{n}^{(m + 1)} : = Ψ_{s_{n}} (r_{n + 1}^{(m)})$ gives $∣ r_{n}^{(m + 1)} - r_{n}^{(m)} ∣ \leq ρ^{m} / λ$ , limits $r_{n} \in [- k, k]$ satisfying $r_{n} = Ψ_{s_{n}} (r_{n + 1}) = 1 / (2 s_{n} - r_{n + 1})$ .

4.1step 3.1step 2.3F4

Unstable recurrence: fixing $n$ and passing to the limit in $h_{n}^{(m + 1)} = Φ_{s_{n - 1}} (h_{n - 1}^{(m)})$ , continuity of $Φ_{s_{n - 1}}$ on $[- k, k]$ (step 2.3) and the limit rules of [F4] give $h_{n} = Φ_{s_{n - 1}} (h_{n - 1}) = 1 / (2 s_{n - 1} - h_{n - 1})$ .

4.2step 3.1step 3.2F2F3F4

Uniqueness: let $({\tilde{h}}_{n})$ be another family with $∣ {\tilde{h}}_{n} ∣ \leq k$ and ${\tilde{h}}_{n} = Φ_{s_{n - 1}} ({\tilde{h}}_{n - 1})$ for all $n$ . Comparing with the construction gives, by induction on $m$ , $∣ {\tilde{h}}_{n} - h_{n}^{(m)} ∣ \leq ρ^{m} ∣ {\tilde{h}}_{n - m} - h_{n - m}^{(0)} ∣ \leq ρ^{m} \cdot 2 k$ (step 2.3, [F2]); letting $m \to \infty$ and using [F3] gives ${\tilde{h}}_{n} = h_{n}$ . The stable case is identical with the future-pointing iteration.

5.1step 4.1step 3.2F1

Invariance: by [F1] and step 4.1, $D F_{a} (z_{n})$ carries the line ${(h_{n} v, v)}$ to the line with slope $Φ_{s_{n}} (h_{n}) = 1 / (2 s_{n} - h_{n}) = h_{n + 1}$ ; since both lines pass through the origin and the first is mapped onto a line (the differential is invertible), the image is exactly ${(h_{n + 1} v, v) : v \in R}$ . Similarly, $D (F_{a}^{- 1}) (z_{n + 1})$ uses the first coordinate $s_{n}$ of $z_{n + 1}$ ; by [F1] and step 3.2 it carries ${(u, r_{n + 1} u) : u \in R}$ onto ${(u, r_{n} u) : u \in R}$ .

6.1step 2.1step 2.2step 2.3step 3.1step 3.2step 4.1step 4.2step 5.1∎

Conclusion: steps 2.1 and 2.2 prove clause 1, step 2.3 proves clause 2, steps 3.1, 4.1 and 5.1 prove clause 3, steps 3.2 and 5.1 prove clause 4, and step 4.2 proves the uniqueness assertions of clauses 3 and 4.

Remarks

The convergence of the slope iteration is uniform in $n$ , with the rate $ρ^{m} / (λ (1 - ρ))$ ; no compactness of the sequence space is used.
Both recurrences are one-sided compositions: $h_{n}$ is determined by the symbols in the infinite past, $r_{n}$ by the symbols in the infinite future. This is the sense in which the unstable direction has finite memory of the past and the stable direction finite foresight of the future.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

Weighted half-tail contractions for $q_{n} = M_{s_{n}} (q_{n - 1}, q_{n + 1})$ give a unique $C^{1}$ curve with tangent slope bounded by $c / (1 - q)$

Statement

Let $θ \in (0, 1)$ , $c > 0$ and put

$q : = c (θ + θ^{- 1}), and assume q < 1.$

Let $J \subseteq R$ be a nonempty compact interval, let $s \in X_{\pm}$ , and let $O \subseteq R^{2}$ be open with $J \times J \subseteq O$ . For $u \in {- 1, + 1}$ let ${\tilde{M}}_{u} : O \to R$ be of class $C^{2}$ ( $C^{k}$ Euclidean maps and diffeomorphisms); write $M_{u}$ for its restriction to $J \times J$ , assume $M_{u} (J \times J) \subseteq J$ , and suppose that

$∣ \partial_{1} M_{u} ∣ \leq c, ∣ \partial_{2} M_{u} ∣ \leq c on J \times J .$

Let $(y_{n})_{n \geq 0}$ be a sequence with $y_{n} \in J$ and let $b_{*} \in J$ satisfy

$y_{n} = M_{s_{n}} (y_{n - 1}, y_{n + 1}) (n \geq 1), y_{0} = M_{s_{0}} (b_{*}, y_{1}),$

and assume there is $\overset{ˉ}{ρ} > 0$ with $[y_{n} - \overset{ˉ}{ρ}, y_{n} + \overset{ˉ}{ρ}] \subseteq J$ for every $n \geq 0$ . Choose $δ_{*} > 0$ such that

$[b_{*} - δ_{*}, b_{*} + δ_{*}] \subseteq J, δ_{*} < \frac{(1 - q) \overset{ˉ}{ρ}}{c},$

and put $I : = (b_{*} - δ_{*}, b_{*} + δ_{*})$ . Write

$ℓ_{θ}^{\infty} (N) : = {u : N \to R : ∥ u ∥_{θ} : = \sup_{n \geq 0} θ^{- n} ∣ u_{n} ∣ < \infty}$

with metric $∥ u - v ∥_{θ}$ , and define its closed ball

$B_{\overset{ˉ}{ρ}} : = {u \in ℓ_{θ}^{\infty} (N) : ∥ u ∥_{θ} \leq \overset{ˉ}{ρ}} .$

For $u \in B_{\overset{ˉ}{ρ}}$ and $b \in I$ , put $z_{- 1} : = b$ and $z_{k} : = y_{k} + u_{k}$ for every $k \geq 0$ , and define $T_{b} : B_{\overset{ˉ}{ρ}} \to B_{\overset{ˉ}{ρ}}$ by

$(T_{b} u)_{n} : = M_{s_{n}} (z_{n - 1}, z_{n + 1}) - y_{n} (n \geq 0) .$

Then:

$ℓ_{θ}^{\infty} (N)$ is a complete metric space and $B_{\overset{ˉ}{ρ}}$ is complete in the distance $∥ u - v ∥_{θ}$ ; for every $b \in I$ the map $T_{b}$ is a contraction with constant $q < 1$ , so it has exactly one fixed point $u (b) \in B_{\overset{ˉ}{ρ}}$ ;
for every $b \in I$ one has the a priori bound $∥ u (b) ∥_{θ} \leq c ∣ b - b_{*} ∣ / (1 - q)$ , and the map $b \mapsto u (b)$ is Lipschitz on $I$ with

$∥ u (b^{'}) - u (b) ∥_{θ} \leq \frac{c}{1 - q} ∣ b^{'} - b ∣;$

$b \mapsto u (b)$ is continuously differentiable on $I$ into $ℓ_{θ}^{\infty} (N)$ : with $z_{k}^{*} : = y_{k} + u_{k} (b)$ for $k \geq 0$ and $z_{- 1}^{*} : = b$ , the derivative $d (b) : = d u (b) / d b \in ℓ_{θ}^{\infty} (N)$ is the unique element of $ℓ_{θ}^{\infty} (N)$ satisfying

$d_{n} (b) = \partial_{1} M_{s_{n}} (z_{n - 1}^{*}, z_{n + 1}^{*}) d_{n - 1} (b) + \partial_{2} M_{s_{n}} (z_{n - 1}^{*}, z_{n + 1}^{*}) d_{n + 1} (b) (n \geq 1),$

$d_{0} (b) = \partial_{1} M_{s_{0}} (b, z_{1}^{*}) + \partial_{2} M_{s_{0}} (b, z_{1}^{*}) d_{1} (b), d_{- 1} (b) : = 1,$

and $∥ d (b) ∥_{θ} \leq c / (1 - q)$ ; 4. consequently the local phase curve

$γ (b) : = (b, y_{0} + u_{0} (b))$

defined for $b \in I$ is $C^{1}$ with $γ^{'} (b) = (1, d_{0} (b))$ , and so $∣ d_{0} (b) ∣ \leq c / (1 - q)$ .

Facts & Assumptions

Given: The parameters $θ, c, q$ , the compact interval $J$ , the open neighborhood $O$ , the $C^{2}$ extensions ${\tilde{M}}_{- 1}, {\tilde{M}}_{+ 1}$ and their restrictions $M_{- 1}, M_{+ 1}$ , the two-sign sequence $s$ , the reference data $y, b_{*}$ , the radius $\overset{ˉ}{ρ}$ and parameter interval $I$ , the weighted space $ℓ_{θ}^{\infty} (N)$ , its closed ball $B_{\overset{ˉ}{ρ}}$ , and the operators $T_{b}$ .

[F1]

Each ${\tilde{M}}_{u}$ is $C^{2}$ on the open set $O$ , its restriction maps $J \times J$ into $J$ , and $∣ \partial_{1} M_{u} ∣, ∣ \partial_{2} M_{u} ∣ \leq c$ on $J \times J$ (given). The box $J \times J$ is compact by Heine--Borel; hence the first and second partial derivatives are bounded there (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line, A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value, $C^{k}$ Euclidean maps and diffeomorphisms, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case). If $z, z^{'} \in J \times J$ and $Δ : = z^{'} - z$ , the segment $z + t Δ$ lies in $J \times J$ . The function $g (t) : = {\tilde{M}}_{u} (z + t Δ)$ is $C^{2}$ on an open interval containing $[0, 1]$ , and the chain rule gives $∣ g^{'} (t) ∣ \leq c ∥ Δ ∥_{1}$ and $∣ g^{''} (t) ∣ \leq K ∥ Δ ∥_{1}^{2}$ with a common finite $K$ for both signs (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ). The mean-value inequality (The mean value inequality: if $f : [a, b] \to R^{m}$ is continuous and differentiable on $(a, b)$ with $∥ f^{'} ∥_{2} \leq M$ , then $∥ f (b) - f (a) ∥_{2} \leq M (b - a)$ ) applied to $g$ gives

$∣ M_{u} (z^{'}) - M_{u} (z) ∣ \leq c ∥ z^{'} - z ∥_{1},$

For the Taylor remainder, the mean-value inequality applied to $g^{'}$ on $[0, t]$ gives $∣ g^{'} (t) - g^{'} (0) ∣ \leq K t ∥ Δ ∥_{1}^{2}$ ; thus $ϕ (t) : = g (t) - g (0) - t g^{'} (0)$ has $∣ ϕ^{'} (t) ∣ \leq K ∥ Δ ∥_{1}^{2}$ . Applying the same inequality to $ϕ$ on $[0, 1]$ gives

$∣ M_{u} (z^{'}) - M_{u} (z) - D M_{u} (z) (z^{'} - z) ∣ \leq K ∥ z^{'} - z ∥_{1}^{2} .$

Here $∥ \cdot ∥_{1}$ is the sum of the absolute values of the two coordinates; the derivative bounds follow from the chain rule and the uniform second-derivative bound above (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , The mean value inequality: if $f : [a, b] \to R^{m}$ is continuous and differentiable on $(a, b)$ with $∥ f^{'} ∥_{2} \leq M$ , then $∥ f (b) - f (a) ∥_{2} \leq M (b - a)$ , The total (Fréchet) derivative $D f (a)$ as the linear first-order approximation with $o (∥ h ∥_{2})$ remainder).

[F2]

The map $W : ℓ_{θ}^{\infty} (N) \to ℓ^{\infty} (N)$ , $(W u)_{n} : = θ^{- n} u_{n}$ , is a bijection with $∥ W u ∥_{\infty} = ∥ u ∥_{θ}$ ; hence $(ℓ_{θ}^{\infty} (N), ∥ \cdot - \cdot ∥_{θ})$ is a complete metric space, as an isometric copy of the complete space $ℓ^{\infty} (N) = B (N)$ . It maps $B_{\overset{ˉ}{ρ}}$ isometrically onto the closed sup-norm ball $B_{\overset{ˉ}{ρ}} (N)$ , so that ball and therefore $B_{\overset{ˉ}{ρ}}$ are complete as well (The bounded functions on a set form a complete metric space in the supremum metric, and the coding map of the two-sign shift space is an isometry, clauses 1, 2 and 4; Isometry, isometric embedding, and the subspace metric on a subset, Metric space: $d (x, y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric).

[F3]

Banach's fixed point theorem (A contraction of a nonempty complete metric space into itself has exactly one fixed point, the limit of the iterates from any starting point, Lipschitz map, $α$ -Hölder map for rational $0 < α \leq 1$ , and contraction): a map of a nonempty complete metric space into itself with constant $q < 1$ has exactly one fixed point, and it is the limit of the iterates from every starting point.

[F4]

Absolute value and order arithmetic (Absolute value in an ordered field, Basic properties of the absolute value, Complete ordered field (least-upper-bound property)): $∣ u v ∣ = ∣ u ∣ ∣ v ∣$ , $∣ u + w ∣ \leq ∣ u ∣ + ∣ w ∣$ , and $θ^{- n} > 0$ for all $n$ .

[F5]

Metric convergence and continuity (Convergence of a sequence in a metric space: $x_{k} \to x$ iff $d (x_{k}, x) \to 0$ in $R$ , Continuity of $f : A \to R$ at a point of $A$ and on $A$ : the $ε$ - $δ$ condition, its agreement with $\lim_{x \to c} f (x) = f (c)$ at a limit point, and continuity at an isolated point).

Proof

technique · direct; contract in the weighted supremum norm, estimate the boundary forcing, and linearise by a Taylor remainder estimate in the same norm

1.1givenF1F4algebra

Contraction: let $u, v \in B_{\overset{ˉ}{ρ}}$ and $b \in I$ . Every argument pair of $M_{s_{n}}$ lies in $J \times J$ : $b \in J$ and, for $k \geq 0$ , $∣ u_{k} ∣, ∣ v_{k} ∣ \leq \overset{ˉ}{ρ}$ with $[y_{k} - \overset{ˉ}{ρ}, y_{k} + \overset{ˉ}{ρ}] \subseteq J$ . The line segment between the two argument pairs also lies in $J \times J$ because $J$ is an interval, so the derivative bound in [F1] gives for $n \geq 1$ $∣ (T_{b} u)_{n} - (T_{b} v)_{n} ∣ \leq c (∣ u_{n - 1} - v_{n - 1} ∣ + ∣ u_{n + 1} - v_{n + 1} ∣) \leq θ^{n} c (θ^{- 1} + θ) ∥ u - v ∥_{θ} = θ^{n} q ∥ u - v ∥_{θ} .$ For $n = 0$ only the argument $z_{1}$ varies, so $∣ (T_{b} u)_{0} - (T_{b} v)_{0} ∣ \leq c ∣ u_{1} - v_{1} ∣ \leq q ∥ u - v ∥_{θ}$ because $c θ \leq q$ . Multiplying by $θ^{- n}$ and taking the supremum gives $∥ T_{b} u - T_{b} v ∥_{θ} \leq q ∥ u - v ∥_{θ}$ .

2.1givenstep 1.1F1F4algebra

Ball invariance: let $b \in I$ and $u \in B_{\overset{ˉ}{ρ}}$ . For every $k \geq 0$ , $z_{k} = y_{k} + u_{k} \in J$ by $∣ u_{k} ∣ \leq \overset{ˉ}{ρ}$ and the margin hypothesis; also $z_{- 1} = b \in J$ . For $n \geq 1$ , using the reference recurrence of the statement and [F1], $∣ (T_{b} u)_{n} ∣ = ∣ M_{s_{n}} (z_{n - 1}, z_{n + 1}) - M_{s_{n}} (y_{n - 1}, y_{n + 1}) ∣ \leq c (∣ u_{n - 1} ∣ + ∣ u_{n + 1} ∣) \leq θ^{n} q ∥ u ∥_{θ} \leq θ^{n} q \overset{ˉ}{ρ} .$ For $n = 0$ , $(T_{b} u)_{0} = M_{s_{0}} (b, z_{1}) - M_{s_{0}} (b_{*}, y_{1})$ , so $∣ (T_{b} u)_{0} ∣ \leq c ∣ b - b_{*} ∣ + c ∣ u_{1} ∣ \leq (1 - q) \overset{ˉ}{ρ} + q \overset{ˉ}{ρ} = \overset{ˉ}{ρ}$ , since $∣ b - b_{*} ∣ < δ_{*} < (1 - q) \overset{ˉ}{ρ} / c$ . Hence $∥ T_{b} u ∥_{θ} \leq \overset{ˉ}{ρ}$ .

2.2givenstep 1.1F1F3F4

Linearised equation: fix $b$ and put $z_{k}^{*} : = y_{k} + u_{k} (b)$ for $k \geq 0$ and $z_{- 1}^{*} : = b$ . Define the affine map $L$ on $ℓ_{θ}^{\infty} (N)$ by $(L d)_{n} : = \partial_{1} M_{s_{n}} (z_{n - 1}^{*}, z_{n + 1}^{*}) d_{n - 1} + \partial_{2} M_{s_{n}} (z_{n - 1}^{*}, z_{n + 1}^{*}) d_{n + 1} (n \geq 1), (L d)_{0} : = \partial_{2} M_{s_{0}} (b, z_{1}^{*}) d_{1} + \partial_{1} M_{s_{0}} (b, z_{1}^{*}),$ where $d_{- 1} : = 1$ is held fixed. The homogeneous part $d \mapsto L d - d_{- 1} -part$ is a contraction of constant $q$ by the computation of step 1.1, so [F3] gives a unique $d (b) \in ℓ_{θ}^{\infty} (N)$ with $L d (b) = d (b)$ , i.e. the displayed system; taking the fixed point of the homogeneous contraction started at $0$ gives $∥ d (b) ∥_{θ} \leq c / (1 - q)$ because the forcing $\partial_{1} M_{s_{0}} (b, z_{1}^{*}) e_{0}$ has norm at most $c$ .

3.1step 1.1step 2.1F2F3

Fixed point: by [F2], $B_{\overset{ˉ}{ρ}}$ is a nonempty complete metric space. For every $b \in I$ , step 2.1 shows $T_{b}$ maps this ball into itself and step 1.1 shows that it is a contraction with constant $q < 1$ ; [F3] therefore gives exactly one fixed point $u (b) \in B_{\overset{ˉ}{ρ}}$ .

3.2givenstep 2.2F1F4algebra

Taylor remainder: fix $b \in I$ and take $h$ small enough that $b + h \in I$ . Put $u^{'} : = u (b + h)$ and $Δ : = u^{'} - u (b)$ , and for $k \geq 0$ put $z_{k} : = y_{k} + u_{k} (b)$ and $z_{k}^{'} : = y_{k} + u_{k}^{'}$ , with $z_{- 1} : = b$ and $z_{- 1}^{'} : = b + h$ . For $n \geq 1$ , [F1] applied at $z = (z_{n - 1}^{*}, z_{n + 1}^{*})$ and $z^{'} = (z_{n - 1}^{'}, z_{n + 1}^{'})$ gives $∣ (T_{b + h} u^{'})_{n} - (T_{b} u)_{n} - \partial_{1} M_{s_{n}} Δ_{n - 1} - \partial_{2} M_{s_{n}} Δ_{n + 1} ∣ \leq K (∣ Δ_{n - 1} ∣ + ∣ Δ_{n + 1} ∣)^{2} \leq 2 K (∣ Δ_{n - 1} ∣^{2} + ∣ Δ_{n + 1} ∣^{2}),$ and multiplying by $θ^{- n}$ bounds this by $2 K θ^{n} (θ^{- 2} + θ^{2}) ∥ Δ ∥_{θ}^{2}$ . For $n = 0$ the two input differences are $h$ and $Δ_{1}$ ; [F1] gives a remainder bounded by $K^{'} (∣ h ∣ + ∣ Δ_{1} ∣)^{2} \leq 2 K^{'} (h^{2} + ∣ Δ_{1} ∣^{2})$ with a uniform $K^{'}$ , so altogether $∥ u^{'} - (T_{b} u) - [D_{u} T_{b} (u)] Δ - D_{b} T_{b} (u) h ∥_{θ} \leq C_{0} (∥ Δ ∥_{θ}^{2} + h^{2})$ with $C_{0} < \infty$ .

4.1step 1.1step 3.1F1F3F4algebra

Lipschitz dependence and a priori bound: for $u \in B_{\overset{ˉ}{ρ}}$ and $b, b^{'} \in I$ , $(T_{b^{'}} u)_{n} = (T_{b} u)_{n}$ for $n \geq 1$ , while $∣ (T_{b^{'}} u)_{0} - (T_{b} u)_{0} ∣ \leq c ∣ b^{'} - b ∣$ by [F1]. Applying this with $u = u (b^{'})$ and using step 1.1 gives $∥ u (b^{'}) - u (b) ∥_{θ} \leq q ∥ u (b^{'}) - u (b) ∥_{θ} + c ∣ b^{'} - b ∣$ , hence $∥ u (b^{'}) - u (b) ∥_{θ} \leq c ∣ b^{'} - b ∣ / (1 - q)$ . At $b = b_{*}$ , $u (b_{*}) = 0$ because the zero sequence solves the fixed-point equation; taking one parameter to be $b_{*}$ gives $∥ u (b) ∥_{θ} \leq c ∣ b - b_{*} ∣ / (1 - q)$ .

5.1step 4.1step 2.2step 3.2F1F4algebra

Differentiability: write $D_{u} T : = D_{u} T_{b} (u (b))$ and $D_{b} T : = D_{b} T_{b} (u (b))$ for the two partial derivatives of the map $(b, u) \mapsto T_{b} u$ at $(b, u (b))$ . By step 3.2 applied to the fixed point equations $u (b + h) = T_{b + h} u (b + h)$ and $u (b) = T_{b} u (b)$ , $u (b + h) = u (b) + D_{u} T Δ + D_{b} T h + R_{1}, ∥ R_{1} ∥_{θ} \leq C_{0} (∥ Δ ∥_{θ}^{2} + h^{2}),$ with $Δ = u (b + h) - u (b)$ . By step 2.2, $h d (b) = D_{u} T (h d (b)) + D_{b} T h$ ; subtracting $h d (b)$ from the displayed equation and putting $R : = Δ - h d (b)$ therefore yields $R = D_{u} T R + R_{1}$ , whence $∥ R ∥_{θ} \leq q ∥ R ∥_{θ} + ∥ R_{1} ∥_{θ}$ . By step 4.1, $∥ Δ ∥_{θ} \leq c ∣ h ∣ / (1 - q)$ , so $∥ R_{1} ∥_{θ} \leq C_{1} h^{2}$ for $∣ h ∣$ small and hence $∥ R ∥_{θ} \leq C_{1} h^{2} / (1 - q)$ : that is, $u (b + h) = u (b) + h d (b) + O (h^{2})$ in $ℓ_{θ}^{\infty} (N)$ . This proves differentiability with derivative $d (b)$ and identifies it with the solution of clause 3.

5.2step 4.1step 2.2F1F4

Continuity of the derivative: by step 2.2 the difference of the linear operators at $b^{'}$ and $b$ has coefficients $\partial_{i} M_{s_{n}} (z_{n \pm 1}^{'}) - \partial_{i} M_{s_{n}} (z_{n \pm 1})$ , which tend to zero uniformly in $n$ as $b^{'} \to b$ by uniform continuity of the $\partial_{i} M_{u}$ on the compact $J \times J$ and by the uniform estimate $∣ z_{k}^{'} - z_{k} ∣ \leq c ∣ b^{'} - b ∣ / (1 - q)$ of step 4.1; the same holds for the forcing. Applying the difference estimate of step 4.1 to the fixed points $d (b^{'}), d (b)$ of the affine maps gives $∥ d (b^{'}) - d (b) ∥_{θ} \to 0$ .

6.1step 2.2step 5.1step 5.2F4F5∎

The curve: the map $b \mapsto γ (b) = (b, y_{0} + u_{0} (b))$ has components the identity and the affine function of $u_{0}$ , and $b \mapsto u_{0} (b)$ is continuously differentiable by steps 5.1 and 5.2 with derivative $d_{0} (b)$ ; hence $γ$ is $C^{1}$ with the displayed derivative ([F5]). The bound $∣ d_{0} (b) ∣ \leq ∥ d (b) ∥_{θ} \leq c / (1 - q)$ is step 2.2.

Remarks

The lemma isolates the exact contractive mechanism used for both the unperturbed square-root recurrence and its perturbations: only the two partial-derivative bounds of the update and the existence of one reference half-orbit are used, not the explicit shape of $M_{u}$ .
The weighted growth $∣ u_{n} ∣ \leq θ^{n} ∥ u ∥_{θ}$ is what makes the solution track the reference half-orbit exponentially; for the phase curve $γ$ this means that the forward orbits of its points converge to the same limit as the reference orbit, which is the stable-set membership used later.
No choice principle is used: completeness of the weighted space comes from the complete space of bounded sequences, and the fixed points are unique.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

The coded curves are genuine $C^{1}$ stable and unstable curves of $F$ , with transverse intersection at every finite-word homoclinic point

Statement

Fix $a \geq 100$ and let $R = \sqrt{a} + 3$ , $L = \sqrt{a - 2 R}$ , $F (x, y) = (y, y^{2} - a - x)$ , $x (s)$ , $h$ and the sets $W^{s}, W^{u}$ be as in The two-sign bilateral shift space, its coding map, and the ball of bounded bilateral real sequences, The square-root recurrence $T_{s} (x)_{n} = s_{n} \sqrt{a + x_{n - 1} + x_{n + 1}}$ is a contraction of $B_{R}$ with a unique coded solution, Finite symbol windows determine the code: locality of the iterates, coding continuity and injectivity of $h (σ) = (x_{- 1} (σ), x_{0} (σ))$ and Stable and unstable cones, invariant line slopes, stable and unstable sets, and transverse homoclinic points of the quadratic family. For $u \in {- 1, + 1}$ put $r_{u} : = 1 + u \sqrt{a + 1}$ and let

$M_{u}^{\sqrt{}} (v, w) : = u \sqrt{a + v + w}$

be the square-root update of the coded recurrence. Then:

(hyperbolicity) the fixed points $(r_{u}, r_{u})$ of $F$ are hyperbolic: $D F (r_{u}, r_{u})$ has two real eigenvalues, one of modulus $> 1$ and one of modulus $< 1$ ;
(stable curve at a finite-word point) let $s \in X_{\pm}$ be nonconstant with $s_{n} = u$ for $∣ n ∣ \geq N$ , and put $z_{*} : = h (s)$ . Then there are $δ_{s} > 0$ and a $C^{1}$ curve $γ_{s} : (- δ_{s}, δ_{s}) \to R^{2}$ with $γ_{s} (0) = z_{*}$ , $γ_{s} ((- δ_{s}, δ_{s})) \subseteq W^{s} (r_{u}, r_{u})$ and tangent $γ_{s}^{'} (0) \in C_{1 / 2}^{s}$ ;
(unstable curve at a finite-word point) likewise there are $δ_{u} > 0$ and a $C^{1}$ curve $γ_{u} : (- δ_{u}, δ_{u}) \to R^{2}$ with $γ_{u} (0) = z_{*}$ , $γ_{u} ((- δ_{u}, δ_{u})) \subseteq W^{u} (r_{u}, r_{u})$ and $γ_{u}^{'} (0) \in C_{1 / 2}^{u}$ ; consequently $z_{*}$ is a transverse homoclinic point of $(r_{u}, r_{u})$ (Stable and unstable cones, invariant line slopes, stable and unstable sets, and transverse homoclinic points of the quadratic family);
(local stable and unstable sets at the fixed point) put $ρ : = 1 / 8$ and define $∥ (x, y) ∥_{\max} : = \max (∣ x ∣, ∣ y ∣)$ . Then the set of points $y = (y_{- 1}, y_{0}) \in R^{2}$ whose forward orbit stays within max-norm distance $ρ$ of $(r_{u}, r_{u})$ ,

$Γ_{ρ}^{s} : = {y : ∥ F^{n} (y) - (r_{u}, r_{u}) ∥_{\max} \leq ρ for every n \geq 0},$

is exactly the graph ${(b, r_{u} + u_{0} (b)) : ∣ b - r_{u} ∣ \leq ρ}$ of the half-tail solution map $b \mapsto u_{0} (b)$ obtained from [F2] with the constant reference $y_{n} : = r_{u}$ and $b_{*} : = r_{u}$ ; in particular it is a $C^{1}$ embedded curve through $(r_{u}, r_{u})$ whose tangent at the fixed point lies in $C_{1 / 2}^{s}$ . With $Σ (x, y) : = (y, x)$ the same statement holds in the unstable direction: a point has all its backward iterates $F^{- n} (y)$ within max-norm distance $ρ$ of $(r_{u}, r_{u})$ if and only if it lies in $Σ (Γ_{ρ}^{s})$ , and $Σ (Γ_{ρ}^{s})$ is a $C^{1}$ embedded curve through $(r_{u}, r_{u})$ whose tangent lies in $C_{1 / 2}^{u}$ .

Facts & Assumptions

Given: The parameter $a \geq 100$ , the constants $R, L$ , the coded sequences $x (s)$ , the coding map $h$ , the fixed points $r_{u}$ , the square-root updates $M_{u}^{\sqrt{}}$ , and a nonconstant code $s$ with $s_{n} = u$ for $∣ n ∣ \geq N$ .

[F1]

Coding data (The square-root recurrence $T_{s} (x)_{n} = s_{n} \sqrt{a + x_{n - 1} + x_{n + 1}}$ is a contraction of $B_{R}$ with a unique coded solution, Finite symbol windows determine the code: locality of the iterates, coding continuity and injectivity of $h (σ) = (x_{- 1} (σ), x_{0} (σ))$ , The quadratic map $F (x, y) = (y, y^{2} - a - x)$ has a compact coded invariant set conjugate to the two-sign shift, with chaos under ${A C}_{ω}$ ): $x_{n} (s) = M_{s_{n}}^{\sqrt{}} (x_{n - 1} (s), x_{n + 1} (s))$ , $∣ x_{n} (s) ∣ \in [L, R]$ , and $F (h (s)) = h (S_{\pm} s)$ ; moreover $∣ x_{n} (s) ∣ \leq \sqrt{a + 2 R} < R$ and $R - \sqrt{a + 2 R} = (4 \sqrt{a} + 3) / (R + \sqrt{a + 2 R}) > (4 \sqrt{a} + 3) / (2 R) > 1 / 2$ , since $R^{2} - (a + 2 R) = 4 \sqrt{a} + 3$ .

[F2]

Weighted half-tail lemma (Weighted half-tail contractions for $q_{n} = M_{s_{n}} (q_{n - 1}, q_{n + 1})$ give a unique $C^{1}$ curve with tangent slope bounded by $c / (1 - q)$ ): for θ=1/2, q=c(θ+θ⁻¹)<1, a compact interval J, a reference half-orbit, and a local parameter interval I satisfying the lemma's radius and margin hypotheses, T_b is a contraction of the closed weighted ball of radius ρ̄ into itself for b∈I. Its unique fixed points u(b) are C¹ in b, satisfy ‖u(b)‖θ≤c|b−b*|/(1−q), and the phase curve b↦(b,y₀+u₀(b)) has tangent (1,d₀) with |d₀|≤c/(1−q); its forward orbit tracks the reference with deviations bounded by ‖u(b)‖_θ.

[F3]

Homoclinic and constant-code data (Constant codes give the fixed points, and finite nonconstant word codes give homoclinic orbits converging to a fixed point at both ends): the constant code $\overset{ˉ}{u}$ has $x_{n} (\overset{ˉ}{u}) = r_{u}$ with $r_{u}^{2} = a + 2 r_{u}$ and $sign (r_{u}) = u$ ; for a finite-word code $s$ , $F^{n} (h (s)) \to (r_{u}, r_{u})$ as $n \to \pm \infty$ , and $h (s) \neq (r_{u}, r_{u})$ when $s \neq \overset{ˉ}{u}$ .

[F4]

The reflection symmetry (Stable and unstable cones, invariant line slopes, stable and unstable sets, and transverse homoclinic points of the quadratic family, The total (Fréchet) derivative $D f (a)$ as the linear first-order approximation with $o (∥ h ∥_{2})$ remainder): with $Σ (x, y) : = (y, x)$ one has $Σ^{2} = i d$ , $Σ \circ F \circ Σ = F^{- 1}$ (because $Σ F Σ (x, y) = Σ F (y, x) = Σ (x, x^{2} - a - y) = (x^{2} - a - y, x) = F^{- 1} (x, y)$ ), and $Σ$ maps $C_{1 / 2}^{s}$ onto $C_{1 / 2}^{u}$ .

[F5]

Hyperbolic fixed points (Stable and unstable cones, invariant line slopes, stable and unstable sets, and transverse homoclinic points of the quadratic family, The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, Absolute value in an ordered field, Basic properties of the absolute value, Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ ): $D F (x, y) = (\begin{array}{cc} 0 & 1 \\ - 1 & 2 y \end{array})$ ; the characteristic polynomial at $(r_{u}, r_{u})$ is $λ^{2} - 2 r_{u} λ + 1$ , with roots $r_{u} \pm \sqrt{r_{u}^{2} - 1}$ . Here $r_{+} = 1 + \sqrt{a + 1} > 1$ and $r_{-} = 1 - \sqrt{a + 1} < - 1$ , so the supplier's definition tests whether each eigenvalue has modulus different from $1$ .

[F6]

Convergence and continuity (Convergence of a sequence in a metric space: $x_{k} \to x$ iff $d (x_{k}, x) \to 0$ in $R$ , Metric space: $d (x, y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric, Continuity of $f : A \to R$ at a point of $A$ and on $A$ : the $ε$ - $δ$ condition, its agreement with $\lim_{x \to c} f (x) = f (c)$ at a limit point, and continuity at an isolated point, Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length, $C^{k}$ Euclidean maps and diffeomorphisms): the definitions of convergence in $R^{2}$ , of $C^{1}$ curves on an interval, and of the stable and unstable sets of Stable and unstable cones, invariant line slopes, stable and unstable sets, and transverse homoclinic points of the quadratic family.

[F7]

One-sided bounded sequences and the deviation operator (The bounded functions on a set form a complete metric space in the supremum metric, and the coding map of the two-sign shift space is an isometry, clause 4, Constant codes give the fixed points, and finite nonconstant word codes give homoclinic orbits converging to a fixed point at both ends, Absolute value in an ordered field, Basic properties of the absolute value, Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ ): let $ρ = 1 / 8$ . For either sign, $∣ r_{u} ∣ = \sqrt{a + 1} - 1$ or $1 + \sqrt{a + 1}$ , so $∣ r_{u} ∣ > 9$ for $a \geq 100$ . If $∣ b - r_{u} ∣ \leq ρ$ and $v, w : N \to R$ have $∥ v ∥_{\infty}, ∥ w ∥_{\infty} \leq ρ$ , then in the update $(T_{b} v)_{n} : = M_{u}^{\sqrt{}} (r_{u} + v_{n - 1}, r_{u} + v_{n + 1}) - r_{u}, v_{- 1} : = b - r_{u},$ every radicand lies in $[r_{u}^{2} - 1 / 4, r_{u}^{2} + 1 / 4] \subset (64, \infty)$ , so $T_{b}$ is defined. For each $n \geq 1$ the difference of two outputs is at most $(∣ v_{n - 1} - w_{n - 1} ∣ + ∣ v_{n + 1} - w_{n + 1} ∣) / 16$ ; at $n = 0$ it is at most $∣ v_{1} - w_{1} ∣ / 16$ . Thus $∥ T_{b} v - T_{b} w ∥_{\infty} \leq (1 / 8) ∥ v - w ∥_{\infty}$ .

Proof

technique · direct; solve the two slope-free half-tail problems for the square-root update, use the reflection symmetry for the unstable direction, and read off hyperbolicity and transversality

1.1givenF5algebra

Hyperbolicity: the characteristic polynomial in [F5] has the two real roots $r_{u} \pm \sqrt{r_{u}^{2} - 1}$ , whose product is $1$ . If $u = + 1$ , then $r_{u} > 1$ , so $r_{u} + \sqrt{r_{u}^{2} - 1} > 1$ and $r_{u} - \sqrt{r_{u}^{2} - 1} = 1 / (r_{u} + \sqrt{r_{u}^{2} - 1}) \in (0, 1)$ . If $u = - 1$ , write $r_{u} = - m$ with $m > 1$ ; then $r_{u} - \sqrt{r_{u}^{2} - 1} < - 1$ and $r_{u} + \sqrt{r_{u}^{2} - 1} = - 1 / (m + \sqrt{m^{2} - 1}) \in (- 1, 0)$ . Thus each fixed point has one eigenvalue of modulus greater than $1$ and one of modulus less than $1$ , proving clause 1 by the hyperbolicity definition of Stable and unstable cones, invariant line slopes, stable and unstable sets, and transverse homoclinic points of the quadratic family.

2.1givenstep 1.1F1F2F3algebra

Stable curve: apply [F2] with $θ : = 1 / 2$ , $J : = [- R, R]$ , $c : = 1 / 16$ , $q : = 5 / 32$ , the update $M_{s_{n}}^{\sqrt{}}$ , reference $y_{n} : = x_{n} (s)$ $(n \geq 0)$ , $b_{*} : = x_{- 1} (s)$ , $\overset{ˉ}{ρ} : = 1 / 2$ , and $δ_{*} : = 1 / 4$ . By [F1], $∣ y_{n} ∣ \geq L > 1 / 2$ and $∣ y_{n} ∣ \leq \sqrt{a + 2 R} < R - 1 / 2$ , so the margin condition holds; also $∣ b_{*} ∣ < R - 1 / 2$ , so $[b_{*} - δ_{*}, b_{*} + δ_{*}] \subseteq J$ , and $δ_{*} < (1 - q) \overset{ˉ}{ρ} / c$ . The updates are $C^{2}$ on the open set $a + v + w > 0$ , containing $J \times J$ , with first partials $u / (2 \sqrt{a + v + w})$ and second partials $- u / (4 (a + v + w)^{3 / 2})$ . On $J \times J$ their first partials have absolute value at most $1 / (2 L) < 1 / 16$ , since $L \geq \sqrt{74} > 8$ , and $∣ M_{u}^{\sqrt{}} ∣ \leq \sqrt{a + 2 R} < R$ . Thus their extensions, image and derivative bounds match every premise of [F2]. The recurrence holds by [F1]. Hence there is a $C^{1}$ curve $b \mapsto Z (b) : = (b, x_{0} (b))$ through $Z (b_{*}) = (x_{- 1} (s), x_{0} (s)) = z_{*}$ with tangent $(1, d_{0})$ and $∣ d_{0} ∣ \leq c / (1 - q) = 2 / 27 < 1 / 2$ . Every point $Z (b)$ has forward orbit tracking the reference orbit $F^{n} (z_{*})$ , which converges to $(r_{u}, r_{u})$ by [F3]; hence $Z (b)$ lies in $W^{s} (r_{u}, r_{u})$ and its tangent lies in $C_{1 / 2}^{s}$ . This proves clause 2.

3.1givenstep 2.1F1F2F3F4algebra

Unstable curve: put $w : = Σ (z_{*}) = (x_{0} (s), x_{- 1} (s))$ and let $y_{n} : = x_{- n - 1} (s)$ for $n \geq 0$ , with boundary value $b_{*} : = x_{0} (s)$ . By the recurrence of [F1], this is a reference half-orbit for the square-root update, and $∣ y_{n} ∣ \geq L > 1 / 2$ , $∣ y_{n} ∣ \leq \sqrt{a + 2 R} < R - 1 / 2$ . Apply [F2] with $θ = 1 / 2$ , $J = [- R, R]$ , $c = 1 / 16$ , $q = 5 / 32$ , $\overset{ˉ}{ρ} = 1 / 2$ , and $δ_{*} = 1 / 4$ ; the margin and parameter-interval conditions hold by [F1]. The resulting $C^{1}$ curve through $w$ has tangent $(1, d_{0})$ with $∣ d_{0} ∣ \leq 2 / 27$ , and its forward orbits track that of $w$ : by [F4], $F^{n} (w) = Σ (F^{- n} (z_{*}))$ , which converges to $(r_{u}, r_{u})$ by [F3]. Reflecting the curve by $Σ$ gives a $C^{1}$ curve $γ_{u}$ through $z_{*}$ , tangent $(d_{0}, 1) \in C_{1 / 2}^{u}$ , whose points lie in $W^{u} (r_{u}, r_{u})$ . This proves clause 3.

3.2givenstep 1.1step 2.1F1F2F4

Local stable set: apply [F2] with $θ = 1 / 2$ , $J = [- R, R]$ , $c = 1 / 16$ , $q = 5 / 32$ , $\overset{ˉ}{ρ} = 1 / 8$ , $δ_{*} = 1 / 4$ , the constant symbols $s_{n} : = u$ , constant reference $y_{n} : = r_{u}$ , and $b_{*} : = r_{u}$ . The reference satisfies the recurrence because $r_{u}^{2} = a + 2 r_{u}$ and $sign (r_{u}) = u$ , so $r_{u} = M_{u}^{\sqrt{}} (r_{u}, r_{u})$ by [F3]. By [F7], $∣ r_{u} ∣ > 9$ , and $∣ r_{u} ∣ \leq \sqrt{a + 2 R} < R - 1 / 2$ , so the reference-margin condition and $[r_{u} - δ_{*}, r_{u} + δ_{*}] \subseteq J$ hold; also $δ_{*} < (1 - q) \overset{ˉ}{ρ} / c = 27 / 16$ . Hence there is a $C^{1}$ map $b \mapsto u (b)$ with $∥ u (b) ∥_{θ} \leq (c / (1 - q)) ∣ b - r_{u} ∣ = (2 / 27) ∣ b - r_{u} ∣$ . Define $Γ : = {(b, r_{u} + u_{0} (b)) : ∣ b - r_{u} ∣ \leq ρ}$ with $ρ : = 1 / 8$ ; this closed parameter interval lies inside $I$ because $δ_{*} = 1 / 4$ .

4.1step 2.1step 3.1F4

Transversality: by steps 2.1 and 3.1 the tangent of $γ_{s}$ at $z_{*}$ lies in $C_{1 / 2}^{s}$ and that of $γ_{u}$ in $C_{1 / 2}^{u}$ ; since $C_{1 / 2}^{s} \cap C_{1 / 2}^{u} = {0}$ ([F4], Stable and unstable cones, invariant line slopes, stable and unstable sets, and transverse homoclinic points of the quadratic family) the two tangent lines are distinct, so the two curves cross transversely at $z_{*}$ . This is the transversality assertion of clause 3.

4.2step 3.2F1F2F4algebra

The graph lies in the trapping set: let $∣ b - r_{u} ∣ \leq ρ$ and put $x_{- 1} : = b$ , $x_{n} : = r_{u} + u_{n} (b)$ for $n \geq 0$ . By the fixed-point equation of [F2], the sequence satisfies $x_{n} = M_{u}^{\sqrt{}} (x_{n - 1}, x_{n + 1})$ for every $n \geq 0$ , so $F (x_{n - 1}, x_{n}) = (x_{n}, x_{n + 1})$ by [F1] and $F^{n} (b, x_{0}) = (x_{n - 1}, x_{n})$ for all $n \geq 0$ . Also $∣ x_{- 1} - r_{u} ∣ = ∣ b - r_{u} ∣ \leq ρ$ and $∣ x_{n} - r_{u} ∣ \leq ∥ u (b) ∥_{θ} \leq (2 / 27) ∣ b - r_{u} ∣ \leq ρ$ for $n \geq 0$ . Thus every point of $Γ$ has all forward iterates within max-norm distance $ρ$ of $(r_{u}, r_{u})$ .

4.3givenstep 3.2F1F2F6F7algebra

The trapping set lies in the graph: let $y = (y_{- 1}, y_{0})$ satisfy $∥ F^{n} (y) - (r_{u}, r_{u}) ∥_{\max} \leq ρ$ for every $n \geq 0$ , and write $F^{n} (y) = (ξ_{n - 1}, ξ_{n})$ . Then $∣ ξ_{n} - r_{u} ∣ \leq ρ$ for every $n \geq - 1$ . Since $∣ r_{u} ∣ > 9$ by [F7], each $ξ_{n}$ has sign $u$ and $∣ ξ_{n} ∣ > 8$ . The orbit identity $ξ_{n + 1} = ξ_{n}^{2} - a - ξ_{n - 1}$ gives $a + ξ_{n - 1} + ξ_{n + 1} = ξ_{n}^{2} > 64$ , so for $n \geq 0$ one has $ξ_{n} = u \sqrt{a + ξ_{n - 1} + ξ_{n + 1}} = M_{u}^{\sqrt{}} (ξ_{n - 1}, ξ_{n + 1})$ . Put $v_{n} : = ξ_{n} - r_{u}$ for $n \geq 0$ and $b : = y_{- 1}$ . Then $∥ v ∥_{\infty} \leq ρ$ , $∣ b - r_{u} ∣ \leq ρ$ , and the displayed identity says exactly $(T_{b} v)_{n} = v_{n}$ for all $n \geq 0$ . The weighted solution $u (b)$ from step 3.2 also satisfies $T_{b} u (b) = u (b)$ and $∥ u (b) ∥_{\infty} \leq ∥ u (b) ∥_{θ} \leq (2 / 27) ∣ b - r_{u} ∣ < ρ$ . By [F7], $∥ v - u (b) ∥_{\infty} = ∥ T_{b} v - T_{b} u (b) ∥_{\infty} \leq (1 / 8) ∥ v - u (b) ∥_{\infty}$ , so $v = u (b)$ and $y = (y_{- 1}, r_{u} + u_{0} (y_{- 1})) \in Γ$ .

5.1step 4.1step 4.2step 4.3F2F4

Local stable set identified: steps 4.2 and 4.3 show that the set $Γ_{ρ}^{s}$ of clause 4 equals $Γ$ ; by [F2] and step 4.1 the curve $Γ$ is $C^{1}$ with tangent $(1, d_{0})$ at the fixed point and $∣ d_{0} ∣ \leq c / (1 - q) \leq 2 / 27 < \frac{1}{2}$ , so the tangent lies in $C_{1 / 2}^{s}$ . This is the stable half of clause 4.

6.1step 4.3step 5.1F4F6∎

Local unstable set: by [F4], $F^{- 1} = Σ F Σ$ with $Σ^{2} = i d$ , and $Σ$ preserves the max norm because it permutes coordinates. Hence for every $y$ and $n \geq 0$ , $∥ F^{- n} (y) - (r_{u}, r_{u}) ∥_{\max} = ∥ F^{n} (Σ y) - (r_{u}, r_{u}) ∥_{\max}$ . Therefore the set of points whose backward orbit stays within max-norm distance $ρ$ of $(r_{u}, r_{u})$ is exactly $Σ (Γ_{ρ}^{s}) = Σ (Γ)$ . Its tangent at $(r_{u}, r_{u})$ is $Σ (1, d_{0}) = (d_{0}, 1) \in C_{1 / 2}^{u}$ because $Σ$ maps $C_{1 / 2}^{s}$ onto $C_{1 / 2}^{u}$ by [F4], and it is a $C^{1}$ embedded curve as the image under a linear homeomorphism. This completes the proof of clause 4.

Remarks

Clause 4 identifies the local stable and unstable sets of the fixed points with the constructed graphs, which is the piece of the stable-manifold picture used by the transversality argument; no global stable-manifold theorem and no structural-stability theorem is invoked.
The unstable curve is obtained from the stable construction by the exact reflection symmetry $Σ F Σ = F^{- 1}$ of the quadratic family; this avoids a separate backward half-tail contraction.
The aperture $1 / 2$ is the one used throughout; the derivative bound $∣ d_{0} ∣ \leq 1 / 2$ is verified with the explicit bound $L \geq \sqrt{74}$ and is not claimed to be optimal.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

A $C^{2}$ map $C^{1}$ -close to $F$ has a unique inverse branch for its first component, with $O (ε)$ derivative bounds

Statement

Fix $a \geq 100$ and put $R : = \sqrt{a} + 3$ , so that $R \geq 13$ . Let $0 < ε < 1$ , let $U \subseteq R^{2}$ be open with $[- R - 1, R + 1]^{2} \subseteq U$ , and let $G = (A, B) : U \to R^{2}$ be of class $C^{2}$ ( $C^{k}$ Euclidean maps and diffeomorphisms) whose components satisfy, on the closed square $S : = [- R - 1, R + 1]^{2}$ ,

$∣ A (x, y) - y ∣ \leq ε, ∣ \partial_{x} A (x, y) ∣ \leq ε, ∣ \partial_{y} A (x, y) - 1 ∣ \leq ε,$

$∣ B (x, y) - (y^{2} - a - x) ∣ \leq ε, ∣ \partial_{x} B (x, y) + 1 ∣ \leq ε, ∣ \partial_{y} B (x, y) - 2 y ∣ \leq ε .$

Then:

for every $(q, q^{'}) \in [- R, R]^{2}$ the equation $A (q, v) = q^{'}$ has exactly one solution $v = V (q, q^{'}) \in [- R - 1, R + 1]$ , and this defines a map $V : [- R, R]^{2} \to [- R - 1, R + 1]$ ;
$V$ is of class $C^{1}$ on $[- R, R]^{2}$ ( $C^{k}$ Euclidean maps and diffeomorphisms), with

$\partial_{q} V (q, q^{'}) = - \frac{\partial_{x} A (q, V)}{A_{y} (q, V)}, \partial_{q^{'}} V (q, q^{'}) = \frac{1}{\partial_{y} A (q, V)};$

$V$ satisfies the uniform estimates

$∣ V (q, q^{'}) - q^{'} ∣ \leq ε, ∣ \partial_{q} V (q, q^{'}) ∣ \leq \frac{ε}{1 - ε}, ∣ \partial_{q^{'}} V (q, q^{'}) - 1 ∣ \leq \frac{ε}{1 - ε}$

for every $(q, q^{'}) \in [- R, R]^{2}$ .

Facts & Assumptions

Given: The parameter $a \geq 100$ , $R = \sqrt{a} + 3$ , the open set $U$ , the map $G = (A, B)$ with its closeness bounds on $S$ , and a point $(q, q^{'}) \in [- R, R]^{2}$ .

[F1]

First-component closeness (given, $C^{k}$ Euclidean maps and diffeomorphisms, The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case): for every $v \in [- R - 1, R + 1]$ one has $∣ A (q, v) - v ∣ \leq ε$ and $∣ \partial_{y} A (q, v) - 1 ∣ \leq ε$ . The function $f (v) : = A (q, v) - v$ is continuous on $[- R - 1, R + 1]$ and differentiable on its interior, since $A$ is $C^{2}$ on the open set $U$ . For any distinct $v, w$ in this closed interval, the scalar mean value theorem (The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ ) on $[\min (v, w), \max (v, w)]$ gives $f (v) - f (w) = f' (ξ) (v - w)$ for an interior $ξ$ . The derivative is $\partial_{y} A (q, ξ) - 1$ , of absolute value at most $ε$ ; equal endpoints give zero. Thus, including both boundary endpoints,

$∣ (A (q, v) - v) - (A (q, w) - w) ∣ \leq ε ∣ v - w ∣ (v, w \in [- R - 1, R + 1]);$

equivalently, for $Φ (v) : = q^{'} + v - A (q, v)$ one has $∣ Φ (v) - q^{'} ∣ \leq ε$ and $∣ Φ (v) - Φ (w) ∣ \leq ε ∣ v - w ∣$ on the interval, with $ε < 1$ .

[F2]

The interval $[- R - 1, R + 1]$ is nonempty. Every Cauchy sequence in it converges in $R$ by The Cauchy criterion from the least-upper-bound property: in a complete ordered field every Cauchy sequence converges, and its limit remains between $- R - 1$ and $R + 1$ by Limits preserve non-strict inequalities. Thus the interval is complete (Complete metric space: every Cauchy sequence converges in the space), and a map of it into itself with constant $ε < 1$ has exactly one fixed point (A contraction of a nonempty complete metric space into itself has exactly one fixed point, the limit of the iterates from any starting point, Lipschitz map, $α$ -Hölder map for rational $0 < α \leq 1$ , and contraction, Metric space: $d (x, y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric).

[F3]

Implicit differentiation (The Euclidean implicit function theorem with derivative formula, $C^{k}$ Euclidean maps and diffeomorphisms): $A$ is $C^{1}$ on the open set $U$ ; if $\partial_{y} A (q_{0}, v_{0}) \neq 0$ and $A (q_{0}, v_{0}) = q_{0}^{'}$ , then near $(q_{0}^{'}, q_{0})$ the equation $A (q, v) = q^{'}$ has a unique $C^{1}$ solution $v = φ (q, q^{'})$ with $\partial_{q} φ = - \partial_{x} A / \partial_{y} A$ and $\partial_{q^{'}} φ = 1 / \partial_{y} A$ .

[F4]

Absolute value and order (Absolute value in an ordered field, Basic properties of the absolute value, The triangle inequality): $∣ u + w ∣ \leq ∣ u ∣ + ∣ w ∣$ , $∣ u v ∣ = ∣ u ∣ ∣ v ∣$ , and $R \geq 13$ so that $[- R, R] \subseteq [- R - 1, R + 1]$ and $q^{'} + [- 1, 1] \subseteq [- R - 1, R + 1]$ for $q^{'} \in [- R, R]$ .

[F5]

Continuous differentiability and the topology of the square (Open ball, closed ball and sphere in a metric space, Metric space: $d (x, y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric, Continuity of $f : A \to R$ at a point of $A$ and on $A$ : the $ε$ - $δ$ condition, its agreement with $\lim_{x \to c} f (x) = f (c)$ at a limit point, and continuity at an isolated point): the closed square $S$ is a nonempty subset of the open set $U$ , and a function is of class $C^{1}$ when its partial derivatives exist and are continuous.

Proof

technique · direct; solve the equation by the contraction $v\mapsto q'+v-A(q,v)$ for each fixed $(q,q')$, then use the implicit function theorem for regularity and differentiate the defining identity for the bounds

1.1givenF1F2F4algebra

Location of the root: fix $(q, q^{'}) \in [- R, R]^{2}$ and define $Φ (v) : = q^{'} + v - A (q, v)$ on $[- R - 1, R + 1]$ . For every such $v$ , $Φ (v) - q^{'} = v - A (q, v)$ so $∣ Φ (v) - q^{'} ∣ \leq ε \leq 1$ by [F1]; hence $Φ (v) \in [- R - 1, R + 1]$ by [F4] (because $q^{'} \in [- R, R]$ gives $q^{'} + [- 1, 1] \subseteq [- R - 1, R + 1]$ ). Also $∣ Φ (v) - Φ (w) ∣ = ∣ (v - w) - (A (q, v) - A (q, w)) ∣ \leq \sup_{ξ} ∣ 1 - \partial_{y} A (q, ξ) ∣ ∣ v - w ∣ \leq ε ∣ v - w ∣$ by [F1] and the mean value inequality. Thus $Φ$ maps the complete interval into itself and is a contraction with constant $ε < 1$ .

2.1step 1.1F2algebra

Unique root: by [F2], $Φ$ has exactly one fixed point $v \in [- R - 1, R + 1]$ ; the identity $Φ (v) = v$ is equivalent to $A (q, v) = q^{'}$ . This proves clause 1 and defines $V$ .

3.1step 2.1F3F5algebra

Regularity: at a root $(q, q^{'})$ with $v = V (q, q^{'})$ , the partial derivative $\partial_{y} A (q, v)$ satisfies $∣ \partial_{y} A (q, v) ∣ \geq 1 - ε > 0$ by the closeness bound, so $A_{y} (q, v) \neq 0$ . The implicit function theorem [F3] gives a $C^{1}$ solution $φ$ near $(q, q^{'})$ with the displayed derivatives, and by clause 1 this local solution coincides with $V$ ; since $(q, q^{'}) \in [- R, R]^{2}$ was arbitrary, $V$ is $C^{1}$ on the square with the displayed formulas.

4.1step 1.1step 2.1step 3.1F1F3givenalgebra

Regularity needed by the half-tail construction: choose $0 < δ < 1 - ε$ . For $(q, q^{'}) \in (- R - δ, R + δ)^{2}$ , both parameters lie in $[- R - 1, R + 1]$ , and every $(q, v)$ with $v \in [- R - 1, R + 1]$ lies in $S$ . Thus the given bounds apply and $\partial_{y} A (q, v) \geq 1 - ε > 0$ . The contraction $Φ (v) = q^{'} + v - A (q, v)$ has constant $ε$ ; moreover $∣ Φ (v) - q^{'} ∣ \leq ε$ , so $Φ (v) \in [- R - 1, R + 1]$ because $∣ q^{'} ∣ < R + δ$ and $δ + ε < 1$ . Its unique fixed point therefore defines $V$ on this open parameter rectangle, extending the original branch by uniqueness. The implicit function theorem gives a local $C^{1}$ branch at each point; uniqueness makes these local branches agree with the contraction-defined extension. Since $A$ is $C^{2}$ , the formulas in step 3.1 make its first partial derivatives $C^{1}$ compositions and quotients with nonvanishing denominator. Consequently this extension is $C^{2}$ on the open rectangle, as needed by the later root and weighted-tail constructions; clause 2's $C^{1}$ conclusion remains valid.

4.2step 3.1step 2.1F4algebra

Estimates: from $A (q, V (q, q^{'})) = q^{'}$ and the closeness $∣ A (q, v) - v ∣ \leq ε$ evaluated at $v = V (q, q^{'})$ , clause 3's first inequality follows: $∣ V (q, q^{'}) - q^{'} ∣ = ∣ V - A (q, V) ∣ \leq ε$ by [F4]. For the derivatives, by step 3.1 $∣ \partial_{q} V ∣ = \frac{∣ \partial_{x} A (q, V) ∣}{∣ \partial_{y} A (q, V) ∣} \leq \frac{ε}{1 - ε}, ∣ \partial_{q^{'}} V - 1 ∣ = \frac{∣ \partial_{y} A (q, V) - 1 ∣}{∣ \partial_{y} A (q, V) ∣} \leq \frac{ε}{1 - ε},$ using $∣ \partial_{x} A ∣ \leq ε$ and $∣ \partial_{y} A - 1 ∣ \leq ε$ from the hypotheses and $∣ \partial_{y} A ∣ \geq 1 - ε$ from step 3.1.

5.1step 2.1step 3.1step 4.2∎

Conclusion: steps 2.1, 3.1 and 4.2 prove clauses 1, 2 and 3 respectively.

Remarks

The proof uses only two properties of the model map $(x, y) \mapsto (y, y^{2} - a - x)$ : its first component is the second variable, so that solving $A (q, v) = q^{'}$ in $v$ is a near-identity equation, and its second component is a quadratic in $v$ . The quadratic structure enters quantitatively in the split $ε$ between the two components in Root bounds and $C^{1}$ dependence for the perturbed scalar recurrence $v^{2} - a - u - w + e (u, v, w) = 0$ .
The estimates are uniform on the square and show that $V$ is a small $C^{1}$ perturbation of the projection $(q, q^{'}) \mapsto q^{'}$ .

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

Root bounds and $C^{1}$ dependence for the perturbed scalar recurrence $v^{2} - a - u - w + e (u, v, w) = 0$

Statement

Fix $a \geq 100$ and put

$R : = \sqrt{a} + 3, l : = \sqrt{a} - 3 \geq 7, S : = [- R - 1, R + 1]^{2} .$

Let $0 < ε \leq 1 / 6$ , let $U \subseteq R^{2}$ be open with $S \subseteq U$ , and let $G = (A, B) : U \to R^{2}$ be a $C^{2}$ map satisfying the closeness hypotheses of A $C^{2}$ map $C^{1}$ -close to $F$ has a unique inverse branch for its first component, with $O (ε)$ derivative bounds on $S$ with constant $ε$ ; let $V : [- R, R]^{2} \to [- R - 1, R + 1]$ be its inverse branch. Define the defect

$e (u, v, w) : = - (V (v, w) - w) + 2 v (V (u, v) - v) + (V (u, v) - v)^{2} + B (u, V (u, v)) - (V (u, v)^{2} - a - u)$

for $(u, v, w) \in [- R, R]^{3}$ , and put

$P_{u, w} (v) : = v^{2} - a - u - w + e (u, v, w), (u, v, w) \in [- R, R]^{3} .$

Then:

$e$ is of class $C^{1}$ on $[- R, R]^{3}$ , and there is a constant $C = C (a, R)$ such that $∥ e ∥_{C^{1} ([- R, R]^{3})} \leq C ε$ ; in particular, with $η : = C ε$ assumed to satisfy $η \leq 1$ , for all $(u, v, w)$

$∣ e (u, v, w) ∣ \leq η, ∣ \partial_{v} e (u, v, w) ∣ \leq η, ∣ \partial_{u} e (u, v, w) ∣ \leq η, ∣ \partial_{w} e (u, v, w) ∣ \leq η;$

for every $(u, w) \in [- R, R]^{2}$ and each sign $s \in {- 1, + 1}$ the equation $P_{u, w} (v) = 0$ has exactly one solution $v = M_{s} (u, w)$ with $s v \in [l, R]$ , and this solution satisfies the margins

$s M_{s} (u, w) - l \geq \frac{3}{4}, R - s M_{s} (u, w) \geq \frac{3}{4};$

moreover $M_{s}$ is of class $C^{1}$ on $[- R, R]^{2}$ with

$∣ \partial_{u} M_{s} ∣ \leq d : = \frac{1 + η}{2 l - η}, ∣ \partial_{w} M_{s} ∣ \leq d;$

with $d$ as above and $θ : = 1 / 2$ one has, for every $a \geq 100$ and $η \leq 1$ ,

$d \leq \frac{2}{13}, 2 d \leq \frac{4}{13} < 1, d (θ + θ^{- 1}) \leq \frac{5}{13} < 1, \frac{d}{1 - d θ} \leq \frac{1}{6} < \frac{2}{3} .$

(orbit form of the root equation) for all $(u, v, w) \in [- R, R]^{3}$ ,

$P_{u, w} (v) = 0 ⟺ B (u, V (u, v)) = V (v, w) .$

Facts & Assumptions

Given: The parameter $a \geq 100$ , the constants $R, l, S$ , the map $G$ and its closeness constant $ε$ , the inverse branch $V$ of A $C^{2}$ map $C^{1}$ -close to $F$ has a unique inverse branch for its first component, with $O (ε)$ derivative bounds, the defect $e$ and the polynomials $P_{u, w}$ .

[F1]

The inverse branch (A $C^{2}$ map $C^{1}$ -close to $F$ has a unique inverse branch for its first component, with $O (ε)$ derivative bounds, clauses 2 and 3): $V$ is $C^{1}$ on $[- R, R]^{2}$ , $A (q, V (q, q^{'})) = q^{'}$ , $∣ V (q, q^{'}) - q^{'} ∣ \leq ε$ , $∣ \partial_{q} V ∣ \leq ε / (1 - ε)$ , $∣ \partial_{q^{'}} V - 1 ∣ \leq ε / (1 - ε)$ for all $(q, q^{'}) \in [- R, R]^{2}$ ; the hypothesis gives $1 / (1 - ε) \leq 6 / 5 < 2$ .

[F2]

The closeness of $B$ : $∣ B (x, y) - (y^{2} - a - x) ∣ \leq ε$ , $∣ \partial_{x} B + 1 ∣ \leq ε$ , $∣ \partial_{y} B - 2 y ∣ \leq ε$ on $S$ (given, and hypotheses of A $C^{2}$ map $C^{1}$ -close to $F$ has a unique inverse branch for its first component, with $O (ε)$ derivative bounds).

[F3]

$C^{1}$ calculus and the uniform bound: $e$ is built from $V$ , $B$ and polynomials by composition, product and sum, so it is $C^{1}$ on the compact cube, and its value and first partial derivatives are bounded by a constant times $ε$ by [F1], [F2], the chain rule and the extreme value theorem ( $C^{k}$ Euclidean maps and diffeomorphisms, The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value, Continuity of $f : A \to R$ at a point of $A$ and on $A$ : the $ε$ - $δ$ condition, its agreement with $\lim_{x \to c} f (x) = f (c)$ at a limit point, and continuity at an isolated point); the constant depends only on $a$ and $R$ . This is stated as clause 1 with a single constant $C$ absorbing the finitely many $O (1)$ factors.

[F4]

The endpoint values of the unperturbed quadratic (Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ , Absolute value in an ordered field, The triangle inequality, Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length): with $u, w \in [- R, R]$ one has $∣ u + w ∣ \leq 2 R$ , hence

$l^{2} - a - (u + w) \leq l^{2} - a + 2 R = - 4 \sqrt{a} + 15, R^{2} - a - (u + w) \geq R^{2} - a - 2 R = 4 \sqrt{a} + 3,$

because $l^{2} = a - 6 \sqrt{a} + 9$ and $R^{2} = a + 6 \sqrt{a} + 9$ . In particular $4 \sqrt{a} - 15 \geq 25$ and $4 \sqrt{a} + 3 \geq 43$ for $a \geq 100$ .

[F5]

The intermediate value theorem and mean value theorem (Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on $[a, b]$ takes every value between $f (a)$ and $f (b)$ , The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ , Continuity of $f : A \to R$ at a point of $A$ and on $A$ : the $ε$ - $δ$ condition, its agreement with $\lim_{x \to c} f (x) = f (c)$ at a limit point, and continuity at an isolated point, Absolute value in an ordered field): a continuous real function on a closed interval taking a negative value at one endpoint and a positive value at the other has a zero; for a function continuous on a closed interval and differentiable on its interior, $f (b) - f (a) = f' (ξ) (b - a)$ for some interior $ξ$ . Thus a $C^{1}$ function with strictly positive derivative is strictly increasing, and $∣ f' ∣ \leq D$ gives $∣ f (b) - f (a) ∣ \leq D ∣ b - a ∣$ , including boundary endpoints.

[F6]

Implicit differentiation in two variables (The Euclidean implicit function theorem with derivative formula, $C^{k}$ Euclidean maps and diffeomorphisms): if $P (u_{0}, w_{0}, v_{0}) = 0$ and $\partial_{v} P (u_{0}, w_{0}, v_{0}) \neq 0$ , then locally $v = M (u, w)$ is the unique $C^{1}$ solution with $\partial_{u} M = - \partial_{u} P / \partial_{v} P$ and $\partial_{w} M = - \partial_{w} P / \partial_{v} P$ .

Proof

technique · direct; estimate the defect from the inverse-branch bounds, locate the root by endpoint signs, differentiate the defining equation, and expand once to identify the root equation with the $G$-orbit form

1.1givenF1F2F3algebra

Bounds for the defect: on the cube, [F1] gives $∣ V (v, w) - w ∣ \leq ε, ∣ V (u, v) - v ∣ \leq ε, ∣ \partial_{u} (V (u, v) - v) ∣ \leq \frac{ε}{1 - ε}, ∣ \partial_{v} (V (u, v) - v) ∣ \leq \frac{ε}{1 - ε},$ and the corresponding partials of $V (v, w) - w$ in $(v, w)$ are bounded by $ε / (1 - ε)$ ; [F2] gives $∣ B (u, V (u, v)) - (V (u, v)^{2} - a - u) ∣ \leq ε$ and partial bounds $ε$ plus Lipschitz factors $2 ∣ V ∣ + 1 \leq 2 (R + 1) + 1$ from the derivative bounds of $B$ and [F1]. Since $∣ 2 v ∣ \leq 2 R$ , every term of $e$ and every first partial derivative of $e$ (computed by the chain rule and product rule) is bounded by a sum of finitely many terms each $O (ε)$ , so there is $C = C (a, R)$ with $∥ e ∥_{C^{1}} \leq C ε$ ; the factors $1 / (1 - ε)$ are bounded by $6 / 5 < 2$ under the stated hypothesis. With $η : = C ε \leq 1$ clause 1 follows.

2.1step 1.1F4algebra

Endpoint values of $P$ : fix $u, w \in [- R, R]$ . By step 1.1, $∣ e (u, v, w) ∣ \leq η$ for every $v$ . Using [F4], $P_{u, w} (l) = l^{2} - a - u - w + e (u, l, w) \leq - (4 \sqrt{a} - 15) + η < 0, P_{u, w} (R) = R^{2} - a - u - w + e (u, R, w) \geq (4 \sqrt{a} + 3) - η > 0,$ because $η \leq 1 \leq 25 \leq 4 \sqrt{a} - 15$ and $η \leq 1 < 43 \leq 4 \sqrt{a} + 3$ . Similarly $P_{u, w} (- R) \geq (4 \sqrt{a} + 3) - η > 0$ and $P_{u, w} (- l) \leq - (4 \sqrt{a} - 15) + η < 0$ .

2.2step 1.1F5algebra

Monotonicity: for $v \in [l, R]$ , $P_{u, w}^{'} (v) = 2 v + \partial_{v} e (u, v, w) \geq 2 l - η \geq 14 - 1 > 0,$ and for $v \in [- R, - l]$ , $P_{u, w}^{'} (v) = 2 v + \partial_{v} e \leq - 2 l + η < 0$ ; hence $P_{u, w}$ is strictly increasing on $[l, R]$ and strictly decreasing on $[- R, - l]$ .

2.3givenstep 1.1F1F2algebra

Orbit form of the root equation: fix $(u, v, w) \in [- R, R]^{3}$ and put $δ_{B} : = B (u, V (u, v)) - (V (u, v)^{2} - a - u)$ , so that $∣ δ_{B} ∣ \leq ε$ by [F2]. Expanding the two inverse-branch values, $V (u, v)^{2} = v^{2} + 2 v (V (u, v) - v) + (V (u, v) - v)^{2}, V (v, w) = w + (V (v, w) - w),$ one computes $B (u, V (u, v)) - V (v, w) = v^{2} - a - u - w + [- (V (v, w) - w) + 2 v (V (u, v) - v) + (V (u, v) - v)^{2} + δ_{B}] = P_{u, w} (v),$ because the bracket is exactly $e (u, v, w)$ . Hence $P_{u, w} (v) = 0$ if and only if $B (u, V (u, v)) = V (v, w)$ , which is clause 4.

3.1step 1.1step 2.1step 2.2F5algebra

Existence, uniqueness and margins of the root: by steps 2.1 and 2.2 and the intermediate value theorem [F5], $P_{u, w}$ has exactly one zero $M_{+} (u, w)$ in $[l, R]$ and exactly one zero $M_{-} (u, w)$ in $[- R, - l]$ . For the margins, on $[l, R]$ one has $∣ P_{u, w}^{'} ∣ \leq 2 R + η \leq 2 \sqrt{a} + 7$ by step 1.1, while $P_{u, w} (l) \leq - (4 \sqrt{a} - 15) + η \leq - (4 \sqrt{a} - 16)$ and $P_{u, w} (R) \geq (4 \sqrt{a} + 3) - η \geq 4 \sqrt{a} + 2$ by step 2.1. The mean value theorem applied on $[l, M_{+}]$ and on $[M_{+}, R]$ therefore gives $M_{+} - l \geq \frac{4 \sqrt{a} - 16}{2 \sqrt{a} + 7}, R - M_{+} \geq \frac{4 \sqrt{a} + 2}{2 \sqrt{a} + 7},$ and both right-hand sides are at least $3 / 4$ because $\sqrt{a} \geq 10$ : the first inequality is equivalent to $10 \sqrt{a} \geq 85$ and the second to $10 \sqrt{a} \geq 13$ . At $a = 100$ the two bounds read $24 / 27$ and $42 / 27$ . The same computation on the interval $[- R, - l]$ , where $P_{u, w}$ is decreasing and $P_{u, w} (- l) \leq - (4 \sqrt{a} - 16)$ , $P_{u, w} (- R) \geq 4 \sqrt{a} + 2$ , gives $∣ M_{-} ∣ - l \geq \frac{3}{4}$ and $R - ∣ M_{-} ∣ \geq \frac{3}{4}$ ; thus $s M_{s} - l \geq \frac{3}{4}$ and $R - s M_{s} \geq \frac{3}{4}$ for each sign $s$ .

4.1step 3.1step 1.1F6algebra

Regularity and derivative bounds: at a zero $v$ of $P_{u, w}$ with $s v \in [l, R]$ , step 2.2 gives $\partial_{v} P_{u, w} (v) \geq 2 l - η > 0$ (respectively $\leq - (2 l - η)$ ), so the implicit function theorem [F6], applied to the $C^{1}$ function $(u, w, v) \mapsto P_{u, w} (v)$ and combined with the uniqueness of step 3.1, makes $M_{s}$ of class $C^{1}$ with $\partial_{u} M_{s} = - \frac{\partial_{u} P}{\partial_{v} P} = \frac{1 - \partial_{u} e}{2 v + \partial_{v} e}, \partial_{w} M_{s} = - \frac{\partial_{w} P}{\partial_{v} P} = \frac{1 - \partial_{w} e}{2 v + \partial_{v} e} .$ Both numerators are bounded by $1 + η$ in absolute value, and the denominator by $2 l - η$ from below in absolute value; hence $∣ \partial_{u} M_{s} ∣, ∣ \partial_{w} M_{s} ∣ \leq (1 + η) / (2 l - η) = d$ , which is clause 2.

5.1step 2.1step 2.2step 4.1F1F3F6givenalgebra

Open-neighborhood $C^{2}$ regularity for later applications: the “Regularity needed by the half-tail construction” paragraph in the proof of A $C^{2}$ map $C^{1}$ -close to $F$ has a unique inverse branch for its first component, with $O (ε)$ derivative bounds extends $V$ as a $C^{2}$ map to an open rectangle containing $[- R, R]^{2}$ . Since $B$ is $C^{2}$ on $U$ , the formula for $e$ , hence $P$ , is $C^{2}$ on an open neighborhood of the cube after shrinking the rectangle so that $(u, V (u, v)) \in U$ . The endpoint signs in step 2.1 and the derivative lower bound in step 2.2 have uniform strictly positive margins on the compact parameter square and the two root intervals. Continuity and compactness therefore preserve those signs and derivative bounds on a slightly enlarged open parameter rectangle. On that rectangle the intermediate value theorem and strict monotonicity give unique roots in the same two intervals; the implicit function theorem makes them $C^{1}$ . Their first partials are the quotients in step 4.1 of $C^{1}$ functions composed with these $C^{1}$ roots, with nonzero denominator, so the partials are $C^{1}$ and the roots are $C^{2}$ . This supplies $C^{2}$ extensions of $M_{s}$ on an open neighborhood of $[- R, R]^{2}$ , as needed by the weighted half-tail lemma; the original derivative bounds on the closed square are unchanged.

5.2step 4.1algebra

Numerical inequalities: since $l \geq 7$ and $η \leq 1$ , $d \leq (1 + 1) / (14 - 1) = 2 / 13$ ; therefore $2 d \leq 4 / 13 < 1$ ; with $θ = 1 / 2$ , $d (θ + θ^{- 1}) \leq (2 / 13) (5 / 2) = 5 / 13 < 1$ ; and $d / (1 - d θ) \leq (2 / 13) / (1 - 1 / 13) = 2 / 12 = 1 / 6 < 2 / 3$ .

6.1step 1.1step 2.1step 2.2step 2.3step 3.1step 4.1step 5.2∎

Conclusion: step 1.1 proves clause 1; steps 2.1, 2.2, 3.1 and 4.1 prove clause 2; step 5.2 proves clause 3; step 2.3 proves clause 4.

Remarks

The defect $e$ is not an arbitrary perturbation: it is exactly the residual obtained from a genuine $G$ -orbit, because a $G$ -orbit $(q_{n}, v_{n})$ satisfies $v_{n} = V (q_{n}, q_{n + 1})$ and $v_{n} = B (q_{n - 1}, v_{n - 1})$ , and substituting the inverse-branch expansions produces precisely $P_{q_{n - 1}, q_{n + 1}} (q_{n}) = 0$ (step 2.3, read backwards). Clause 4 is the exact equivalence used in Explicit persistence under the stated forward and conjugated-inverse bounds: coded chaos, hyperbolic fixed points and transverse homoclinic points to convert the coded recurrence $q_{n} = M_{s_{n}} (q_{n - 1}, q_{n + 1})$ into the two-step orbit relation of $G$ .
All numerical margins are evaluated at the worst case $a = 100$ , $l = 7$ , $η = 1$ ; the argument is uniform in $a \geq 100$ and only improves with larger $a$ or smaller $η$ .

TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

Explicit persistence under the stated forward and conjugated-inverse bounds: coded chaos, hyperbolic fixed points and transverse homoclinic points

Statement

Fix $a \geq 100$ and put

$R : = \sqrt{a} + 3, l : = \sqrt{a} - 3 \geq 7, Q : = [- R - 1, R + 1]^{2}, Σ (x, y) : = (y, x) .$

Let $C = C (a, R)$ be the constant of Root bounds and $C^{1}$ dependence for the perturbed scalar recurrence $v^{2} - a - u - w + e (u, v, w) = 0$ , let $0 < ε \leq 1 / 6$ with $η : = C ε \leq 1$ , and let $U \subseteq R^{2}$ be open with $Q \subseteq U$ . Let $G = (A, B) : U \to R^{2}$ be a $C^{2}$ diffeomorphism onto its image whose components satisfy, on $Q$ ,

$∣ A (x, y) - y ∣ \leq ε, ∣ \partial_{x} A (x, y) ∣ \leq ε, ∣ \partial_{y} A (x, y) - 1 ∣ \leq ε,$

$∣ B (x, y) - (y^{2} - a - x) ∣ \leq ε, ∣ \partial_{x} B (x, y) + 1 ∣ \leq ε, ∣ \partial_{y} B (x, y) - 2 y ∣ \leq ε,$

and assume in addition that $H : = Σ \circ G^{- 1} \circ Σ$ is defined and of class $C^{2}$ on a neighbourhood of $Q$ and satisfies the same six inequalities on $Q$ with the same constant $ε$ . For each sign $u$ let $M_{u}$ be the root map of Root bounds and $C^{1}$ dependence for the perturbed scalar recurrence $v^{2} - a - u - w + e (u, v, w) = 0$ applied to $G$ , let $V$ be the inverse branch of A $C^{2}$ map $C^{1}$ -close to $F$ has a unique inverse branch for its first component, with $O (ε)$ derivative bounds, and let $M_{u}^{H}, V^{H}$ be the corresponding objects for $H$ .

For a nonzero real $x$ , write $sign x : = + 1$ if $x > 0$ and $sign x : = - 1$ if $x < 0$ ; the sequence coordinates used below are nonzero because their absolute values are at least $l \geq 7$ . Then:

(coding) for every $s \in X_{\pm}$ there is exactly one $q (s) \in B_{R}$ with $q_{n} (s) = M_{s_{n}} (q_{n - 1} (s), q_{n + 1} (s))$ for all $n \in Z$ ; it satisfies $l \leq ∣ q_{n} (s) ∣ \leq R$ , $sign q_{n} (s) = s_{n}$ and $R - ∣ q_{n} (s) ∣ \geq \frac{3}{4}$ , $∣ q_{n} (s) ∣ - l \geq \frac{3}{4}$ ; setting $v_{n} (s) : = V (q_{n} (s), q_{n + 1} (s))$ and

$z_{n} (s) : = (q_{n} (s), v_{n} (s)), h_{G} (s) : = z_{0} (s) = (q_{0} (s), V (q_{0} (s), q_{1} (s))),$

one has $G (z_{n} (s)) = z_{n + 1} (s)$ for all $n$ ; 2. (conjugacy and chaos) $h_{G}$ is a continuous bijection of $(X_{\pm}, D_{\pm})$ onto $Λ_{G} : = h_{G} (X_{\pm})$ ; granting the Axiom of Countable Choice (used once through the equivalence between closed and sequentially closed subsets of $R^{2}$ ), it is a homeomorphism, and $G \circ h_{G} = h_{G} \circ S_{\pm}$ , so $Λ_{G}$ is $G$ -invariant. The set $Λ_{G}$ is sequentially compact and is compact under ${A C}_{ω} + D C$ . Under ${A C}_{ω}$ , $G ∣_{Λ_{G}}$ is continuous, topologically mixing, and has dense periodic points; it is sensitive with sensitivity constant $2 l$ . Hence under ${A C}_{ω}$ it is chaotic in the stated sense. 3. (fixed points) for each sign $u$ the constant code $\overset{ˉ}{u}$ gives a fixed point $z_{\overset{ˉ}{u}} : = h_{G} (\overset{ˉ}{u}) = (ρ_{u}, V (ρ_{u}, ρ_{u}))$ , where $ρ_{u} = M_{u} (ρ_{u}, ρ_{u})$ is the unique element of $u [l, R]$ with $ρ_{u} = M_{u} (ρ_{u}, ρ_{u})$ ; these two fixed points are hyperbolic; 4. (transverse homoclinic points) let $s \in X_{\pm}$ be nonconstant with $s_{n} = u$ for $∣ n ∣ \geq N$ . Then $z_{*} : = h_{G} (s) \neq z_{\overset{ˉ}{u}}$ , and there are $C^{1}$ curves $γ_{s}, γ_{u} : (- ε_{0}, ε_{0}) \to R^{2}$ with $γ_{s} (0) = γ_{u} (0) = z_{*}$ , $γ_{s} ((- ε_{0}, ε_{0})) \subseteq W_{G}^{s} (z_{\overset{ˉ}{u}})$ , $γ_{u} ((- ε_{0}, ε_{0})) \subseteq W_{G}^{u} (z_{\overset{ˉ}{u}})$ , tangent vectors $γ_{s}^{'} (0) \in C_{3 / 4}^{s}$ , $γ_{u}^{'} (0) \in C_{3 / 4}^{u}$ ; since $C_{3 / 4}^{s} \cap C_{3 / 4}^{u} = {0}$ the tangent lines are distinct, so $z_{*}$ is a transverse homoclinic point of $z_{\overset{ˉ}{u}}$ (Stable and unstable cones, invariant line slopes, stable and unstable sets, and transverse homoclinic points of the quadratic family).

Facts & Assumptions

Given: The data $a, R, l, Q, Σ, ε, η, C$ , the maps $G, H$ , the root maps $M_{u}, M_{u}^{H}$ , the inverse branches $V, V^{H}$ , the shift space $X_{\pm}$ with metric $D_{\pm}$ and shift $S_{\pm}$ , the ball $B_{R}$ , and the cones $C_{k}^{s}, C_{k}^{u}$ and stable/unstable sets of Stable and unstable cones, invariant line slopes, stable and unstable sets, and transverse homoclinic points of the quadratic family.

[F1]

Root and defect bounds (Root bounds and $C^{1}$ dependence for the perturbed scalar recurrence $v^{2} - a - u - w + e (u, v, w) = 0$ ): proof step 5.1 supplies $C^{2}$ extensions of $M_{u}$ on an open neighborhood of $[- R, R]^{2}$ ; in particular $M_{u}$ is $C^{1}$ on $[- R, R]^{2}$ with values in $u [l, R]$ , $∣ \partial_{i} M_{u} ∣ \leq d$ where $d \leq 2 / 13$ , $2 d \leq 4 / 13 < 1$ , $d (θ + θ^{- 1}) \leq 5 / 13 < 1$ for $θ = 1 / 2$ and $d / (1 - d θ) \leq 1 / 6$ ; the defect $e$ satisfies $∣ e ∣, ∣ \partial_{i} e ∣ \leq η \leq 1$ on the cube; every root satisfies the margins $∣ M_{u} (u^{'}, w^{'}) ∣ - l \geq \frac{3}{4}$ and $R - ∣ M_{u} (u^{'}, w^{'}) ∣ \geq \frac{3}{4}$ , because the endpoint values of $P$ are at least $4 \sqrt{a} - 16$ and $4 \sqrt{a} + 2$ in absolute value while $∣ \partial_{v} P ∣ \leq 2 R + η \leq 2 \sqrt{a} + 7$ , and $(4 \sqrt{a} - 16) / (2 \sqrt{a} + 7) \geq \frac{3}{4}$ and $(4 \sqrt{a} + 2) / (2 \sqrt{a} + 7) \geq \frac{3}{4}$ for $\sqrt{a} \geq 10$ ; and clause 4 of the same lemma: for all $(u, v, w) \in [- R, R]^{3}$ , $P_{u, w} (v) = 0$ if and only if $B (u, V (u, v)) = V (v, w)$ .

[F2]

Inverse-branch estimates (A $C^{2}$ map $C^{1}$ -close to $F$ has a unique inverse branch for its first component, with $O (ε)$ derivative bounds): $V$ is $C^{1}$ on $[- R, R]^{2}$ with $∣ V (q, q^{'}) - q^{'} ∣ \leq ε$ , $∣ \partial_{q} V ∣ \leq ε / (1 - ε)$ , $∣ \partial_{q^{'}} V - 1 ∣ \leq ε / (1 - ε)$ ; here $ε \leq 1 / 6$ , so $ε / (1 - ε) \leq 1 / 5$ and $1 / (1 - ε) \leq 6 / 5$ .

[F3]

Weighted half-tail lemma (Weighted half-tail contractions for $q_{n} = M_{s_{n}} (q_{n - 1}, q_{n + 1})$ give a unique $C^{1}$ curve with tangent slope bounded by $c / (1 - q)$ ): under its hypotheses with parameters $θ, c, q$ and a reference half-orbit, the maps $T_{b}$ have unique fixed points $u (b)$ , the map $b \mapsto u (b)$ is continuously differentiable with $∥ d u / d b ∥_{θ} \leq c / (1 - q)$ , and every derivative entry satisfies $∣ d_{n} ∣ \leq θ^{n} c / (1 - q)$ .

[F4]

Complete ball and contraction (The bounded functions on a set form a complete metric space in the supremum metric, and the coding map of the two-sign shift space is an isometry, A contraction of a nonempty complete metric space into itself has exactly one fixed point, the limit of the iterates from any starting point, Lipschitz map, $α$ -Hölder map for rational $0 < α \leq 1$ , and contraction, Metric space: $d (x, y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric): $B_{R}$ is a nonempty complete metric space in the supremum distance, and a contraction of it into itself has exactly one fixed point.

[F5]

Shift data (The bilateral two-symbol shift space, its metric, its shift map and its cylinder sets, The bilateral shift metric is a metric, cylinders are its basic balls, and the shift space is sequentially compact, The bilateral two-symbol shift is continuous, topologically mixing, has dense periodic points and is sensitive; it is chaotic in the stated sense): the published two-symbol shift is continuous, mixing, has dense periodic points and is sensitive with constant $1$ ; its space is sequentially compact without choice and compact under the Axioms of Countable and Dependent Choice; the isometry $φ$ of The bounded functions on a set form a complete metric space in the supremum metric, and the coding map of the two-sign shift space is an isometry transfers all of this to $X_{\pm}$ with $S_{\pm}$ .

[F6]

Topological vocabulary and sequential closure (Topological mixing, dense periodic points, sensitive dependence, and the stated chaos list, Dense, nowhere dense and codense subsets of a topological space, and the criterion by basic open sets, Continuity of a map between metric spaces, at a point and globally, in the $ε$ - $δ$ form, The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement, A point lies in the closure of $A$ iff some sequence in $A$ converges to it, and a set is closed iff it is sequentially closed, A sequence in a metric space has at most one limit, Countably compact, sequentially compact and limit point compact metric spaces, Open cover, subcover, compact metric space, and compact subset of a metric space, The Axiom of Countable Choice ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain): the definitions of mixing, dense periodic points, sensitivity, continuity, open and closed sets, and fixed points; closed subsets of a metric space are exactly the sets containing all limits of their convergent sequences, limits in metric spaces are unique, the implication "closed $\Rightarrow$ sequentially closed" is choice-free, and the converse implication uses the Axiom of Countable Choice once.

[F7]

Cones and slope vocabulary (Stable and unstable cones, invariant line slopes, stable and unstable sets, and transverse homoclinic points of the quadratic family): $C_{3 / 4}^{u} = {∣ u ∣ \leq \frac{3}{4} ∣ v ∣}$ , $C_{3 / 4}^{s} = {∣ v ∣ \leq \frac{3}{4} ∣ u ∣}$ , and $C_{3 / 4}^{u} \cap C_{3 / 4}^{s} = {0}$ ; the phase curve of a half-tail with boundary value $b$ and sequence $x_{n} (b)$ is the map $b \mapsto (b, V (b, x_{0} (b)))$ at the index $- 1$ .

[F8]

Arithmetic and limits (Absolute value in an ordered field, Basic properties of the absolute value, Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length, Continuity of $f : A \to R$ at a point of $A$ and on $A$ : the $ε$ - $δ$ condition, its agreement with $\lim_{x \to c} f (x) = f (c)$ at a limit point, and continuity at an isolated point, For $∣ r ∣ < 1$ , $\sum_{k \geq 0} r^{k} = 1 / (1 - r)$ , and for $∣ r ∣ \geq 1$ the series diverges, For $∣ r ∣ < 1$ the sequence $r^{k}$ is null, and for $∣ r ∣ > 1$ the sequence $∣ r ∣^{k}$ diverges to $+ \infty$ , Series, partial sums, convergence and the sum, divergence, and the tail series, Convergence of a sequence in a metric space: $x_{k} \to x$ iff $d (x_{k}, x) \to 0$ in $R$ , Open ball, closed ball and sphere in a metric space): the numerical bounds used below, the convergence of geometric sequences, and the fact that the Euclidean distance dominates each coordinate difference.

Proof

direct; run a contraction for the perturbed recurrence, transfer the shift's chaos through the coding homeomorphism as in the unperturbed case, and build the two curves by the weighted half-tail lemma applied to $G$ and to $H = Σ G^{- 1} Σ$ .

1.1givenF1F8algebra

Constants: by [F1], $d \leq 2 / 13$ , $2 d \leq 4 / 13$ , and $d / (1 - d θ) \leq 1 / 6$ ; also $R \geq 13$ , $l \geq 7$ , and $R - l = \sqrt{a} + 3 - (\sqrt{a} - 3) = 6 > 0$ . Setting $ν : = 2 d \leq 4 / 13 < 1$ .

2.1givenstep 1.1F1F4algebra

Coded sequence: for $s \in X_{\pm}$ define $T_{s}$ on $B_{R}$ by $(T_{s} q)_{n} : = M_{s_{n}} (q_{n - 1}, q_{n + 1})$ . The values lie in $s_{n} [l, R] \subseteq [- R, R]$ by [F1], so $T_{s}$ maps $B_{R}$ into itself; and $∥ T_{s} q - T_{s} q^{'} ∥_{\infty} \leq 2 d ∥ q - q^{'} ∥_{\infty} = ν ∥ q - q^{'} ∥_{\infty}$ by [F1]. By [F4] and $ν < 1$ , $T_{s}$ has exactly one fixed point $q (s) \in B_{R}$ ; and $q_{n} (s) \in s_{n} [l, R]$ , so $sign q_{n} (s) = s_{n}$ . The margins of [F1] give $R - ∣ q_{n} (s) ∣ \geq \frac{3}{4}$ and $∣ q_{n} (s) ∣ - l \geq \frac{3}{4}$ .

3.1givenstep 2.1F1F2

Orbits: with $v_{n} : = V (q_{n} (s), q_{n + 1} (s))$ of [F2], clause 4 of [F1] states that $P_{u, w} (v) = 0$ is equivalent to $B (u, V (u, v)) = V (v, w)$ . With $(u, v, w) = (q_{n - 1} (s), q_{n} (s), q_{n + 1} (s))$ , clause 1 gives $P_{u, w} (v) = 0$ , hence $B (q_{n - 1} (s), v_{n - 1}) = v_{n}$ ; and the defining identity of $V$ gives $A (q_{n - 1} (s), v_{n - 1}) = q_{n} (s)$ . Hence $G (q_{n - 1} (s), v_{n - 1}) = (q_{n} (s), v_{n})$ , i.e. $G (z_{n - 1} (s)) = z_{n} (s)$ for all $n$ , and clause 1 is proved.

3.2givenstep 2.1F2F4F5F8algebra

Locality and continuity: iterating $T_{s}$ from $0$ , one proves by induction on $m$ that $(T_{s}^{m} 0)_{n}$ depends only on the symbols $s_{j}$ with $∣ j - n ∣ \leq m - 1$ , exactly as in the unperturbed case, and that $∥ T_{s}^{m} 0 - q (s) ∥_{\infty} \leq ν^{m} ∥ T_{s} 0 ∥_{\infty} / (1 - ν) \leq 2 R ν^{m}$ by [F4] and clause 3 of A contraction of a nonempty complete metric space into itself has exactly one fixed point, the limit of the iterates from any starting point; hence for $s, t$ agreeing on $∣ j ∣ \leq m$ one has $∣ q_{n} (s) - q_{n} (t) ∣ \leq 4 R ν^{m}$ for $∣ n ∣ \leq 1$ . By [F2], $∣ \partial_{q} V ∣ \leq ε / (1 - ε) \leq \frac{1}{5}$ and $∣ \partial_{q^{'}} V ∣ \leq 1 + ε / (1 - ε) \leq \frac{6}{5}$ , so the mean value inequality gives $∣ V (q_{0} (s), q_{1} (s)) - V (q_{0} (t), q_{1} (t)) ∣ \leq \frac{1}{5} \cdot 4 R ν^{m} + \frac{6}{5} \cdot 4 R ν^{m} = \frac{7}{5} \cdot 4 R ν^{m}$ ; with $C_{h} : = \frac{12}{5} = 1 + \frac{1}{5} + \frac{6}{5}$ one therefore has $∥ h_{G} (s) - h_{G} (t) ∥_{2} \leq Δ q_{0} + Δ V \leq 4 R C_{h} ν^{m}$ whenever $s, t$ agree on $∣ j ∣ \leq m$ . Given $ε^{'} > 0$ , choose $m$ with $4 R C_{h} ν^{m} < ε^{'}$ and $δ : = 2^{- m}$ ; if $D_{\pm} (s, t) < 2^{- m}$ then ([F5], cylinder property) $s, t$ agree for $∣ j ∣ \leq m$ , so $∥ h_{G} (s) - h_{G} (t) ∥_{2} < ε^{'}$ . This proves continuity of $h_{G}$ .

3.3givenstep 2.1F1F4F7

Constant codes: for each sign $u$ , the map $ρ \mapsto M_{u} (ρ, ρ)$ maps the closed interval $u [l, R]$ into itself by [F1] and is a contraction with constant $2 d \leq 4 / 13 < 1$ ; hence it has exactly one fixed point $ρ_{u} \in u [l, R]$ , and the constant sequence $q_{n} : = ρ_{u}$ is a fixed point of $T_{\overset{ˉ}{u}}$ , so $q (\overset{ˉ}{u})$ is constant and $z_{\overset{ˉ}{u}} = h_{G} (\overset{ˉ}{u}) = (ρ_{u}, V (ρ_{u}, ρ_{u}))$ is a fixed point of $G$ .

4.1givenstep 3.1F1

Injectivity: if $h_{G} (s) = h_{G} (t)$ then $z_{0} (s) = z_{0} (t)$ ; applying the bijection $G$ forward and $G^{- 1}$ (which exists because $G$ is a diffeomorphism onto its image, and $z_{0} (s)$ lies in that image as $G (z_{- 1} (s))$ ) backward gives $z_{n} (s) = z_{n} (t)$ for all $n \in Z$ by induction; hence $q_{n} (s) = q_{n} (t)$ for all $n$ , and $s_{n} = sign q_{n} (s) = sign q_{n} (t) = t_{n}$ by [F1], so $s = t$ .

4.2givenstep 3.1F1F2

Conjugacy: define ${\tilde{q}}_{n} : = q_{n + 1} (s)$ . Then $\tilde{q}$ is a fixed point of $T_{S_{\pm} s}$ , because $(T_{S_{\pm} s} \tilde{q})_{n} = M_{s_{n + 1}} (q_{n} (s), q_{n + 2} (s)) = q_{n + 1} (s) = {\tilde{q}}_{n}$ by [F1]; by uniqueness $\tilde{q} = q (S_{\pm} s)$ , i.e. $q_{n} (S_{\pm} s) = q_{n + 1} (s)$ . Hence $G (h_{G} (s)) = G (q_{0} (s), v_{0} (s)) = (q_{1} (s), v_{1} (s)) = (q_{0} (S_{\pm} s), V (q_{0} (S_{\pm} s), q_{1} (S_{\pm} s))) = h_{G} (S_{\pm} s)$ : the conjugacy identity, and $G (Λ_{G}) = Λ_{G}$ follows as in the unperturbed case because $S_{\pm}$ is bijective.

4.3givenstep 3.3F1F2F8algebra

Hyperbolicity: put $M : = D G (z_{\overset{ˉ}{u}})$ , $E : = M - D F (z_{\overset{ˉ}{u}}) = (\begin{array}{cc} a_{11} & a_{12} \\ b_{1} & b_{2} \end{array})$ and $v_{u} : = V (ρ_{u}, ρ_{u})$ . At $z_{\overset{ˉ}{u}}$ the six closeness hypotheses give $∣ a_{11} ∣, ∣ a_{12} ∣, ∣ b_{1} ∣ \leq ε$ and $∣ b_{2} ∣ \leq ε$ , while [F2] gives $∣ v_{u} - ρ_{u} ∣ \leq ε$ and $∣ v_{u} ∣ \leq R + 1$ . Since $D F (x, y) = (\begin{array}{cc} 0 & 1 \\ - 1 & 2 y \end{array})$ , expanding trace and determinant of $M = D F (z_{\overset{ˉ}{u}}) + E$ gives $tr M = 2 v_{u} + a_{11} + b_{2}, \det M = 1 + a_{11} (2 v_{u} + b_{2}) + a_{12} - b_{1} - a_{12} b_{1},$ so $∣ tr M - 2 ρ_{u} ∣ \leq 4 ε$ and $∣ \det M - 1 ∣ \leq (2 R + 4) ε + 2 ε^{2} \leq (2 l + 16) ε + 2 ε^{2}$ . Because $ε \leq \frac{1}{6}$ and $l \geq 7$ , $∣ tr M ∣ \geq 2 l - 4 ε \geq 2 l - \frac{2}{3}, ∣ \det M ∣ \leq 1 + \frac{2 l + 16}{6} + \frac{2}{36} = \frac{l}{3} + \frac{67}{18},$ and the discriminant is bounded below by ${tr}^{2} M - 4 \det M \geq (2 l - \frac{2}{3})^{2} - 4 (1 + \frac{2 l + 16}{6} + \frac{2}{36}) = 4 l^{2} - 4 l - \frac{130}{9} > 0,$ since $4 l^{2} - 4 l = 168$ at $l = 7$ and $l \mapsto 4 l^{2} - 4 l$ is increasing for $l \geq 1$ . Hence the eigenvalues are real. Let $λ_{+}$ be the one of larger modulus; then $∣ λ_{+} ∣ \geq \frac{1}{2} ∣ tr M ∣ \geq l - \frac{1}{3} > 1$ , and $∣ λ_{-} ∣ = \frac{∣ \det M ∣}{∣ λ_{+} ∣} \leq \frac{2 ∣ \det M ∣}{∣ tr M ∣} \leq \frac{2 (\frac{l}{3} + \frac{67}{18})}{2 l - \frac{2}{3}} < 1,$ the last inequality being equivalent to $\frac{73}{18} < \frac{2 l}{3}$ , which holds because $l \geq 7$ and $\frac{14}{3} > \frac{73}{18}$ . Thus $∣ λ_{+} ∣ > 1 > ∣ λ_{-} ∣ \geq 0$ : $z_{\overset{ˉ}{u}}$ is hyperbolic.

4.4step 3.2step 3.3F2F8

Tail convergence: suppose $s_{n} = u$ for all $∣ n ∣ \geq N$ , and let $m \geq 0$ . For $n \geq N + m + 1$ every $j$ with $∣ j - n ∣ \leq m - 1$ satisfies $j > N$ , so $(T_{s}^{m} 0)_{n} = (T_{\overset{ˉ}{u}}^{m} 0)_{n}$ ; for $n \leq - N - (m + 1)$ every such $j$ satisfies $j < - N$ , so again $(T_{s}^{m} 0)_{n} = (T_{\overset{ˉ}{u}}^{m} 0)_{n}$ . Hence, by the uniform error estimate of step 3.2 and $q (\overset{ˉ}{u})$ constant by step 3.3, $∣ q_{n} (s) - ρ_{u} ∣ \leq 4 R ν^{m}$ whenever $∣ n ∣ \geq N + m + 1$ . Letting $m \to \infty$ gives $q_{n} (s) \to ρ_{u}$ as $n \to \pm \infty$ ; consequently $z_{n} (s) \to (ρ_{u}, V (ρ_{u}, ρ_{u})) = z_{\overset{ˉ}{u}}$ in both directions, by continuity of $V$ ([F2]).

4.5givenstep 3.1F1F2F8algebra

Backward orbit under $H$ : put $w : = Σ (z_{*})$ and define $w_{n} : = Σ (z_{- n} (s))$ for $n \in Z$ . Since $q_{j} (s) \in [- R, R]$ and $∣ v_{j} (s) - q_{j + 1} (s) ∣ \leq ε \leq \frac{1}{6}$ by [F1] and [F2], every $z_{j} (s)$ lies in $Q$ , so every $w_{n}$ lies in the domain where $H = Σ \circ G^{- 1} \circ Σ$ is defined. The orbit identity of step 3.1 gives $G^{- 1} (z_{- n} (s)) = z_{- n - 1} (s)$ ; hence $H (w_{n}) = w_{n + 1}$ , so $(w_{n})_{n \in Z}$ is the corresponding full $H$ orbit. Its coordinates are $(w_{n})_{1} = v_{- n} (s)$ and $(w_{n})_{2} = q_{- n} (s)$ , and for every $n$ , $∣ (w_{n})_{1} ∣ \in [l + \frac{7}{12}, R - \frac{7}{12}],$ because $∣ q_{j + 1} (s) ∣ \in [l + \frac{3}{4}, R - \frac{3}{4}]$ by [F1] and $∣ v_{j} (s) - q_{j + 1} (s) ∣ \leq \frac{1}{6}$ by [F2]; the sign of $(w_{n})_{1}$ may depend on $n$ . Set $x_{n} : = (w_{n})_{1}$ . By definition of the inverse branch $V^{H}$ of the first component $H_{1}$ of $H$ , the orbit relations $H_{1} (w_{n - 1}) = (w_{n})_{1}$ , $H_{2} (w_{n - 1}) = (w_{n})_{2}$ say $V^{H} (x_{n - 1}, x_{n}) = (w_{n - 1})_{2}$ and $H_{2} (x_{n - 1}, V^{H} (x_{n - 1}, x_{n})) = V^{H} (x_{n}, x_{n + 1})$ ; by clause 4 of [F1] applied to $H$ , this gives $P_{x_{n - 1}, x_{n + 1}}^{H} (x_{n}) = 0$ . Since $x_{n} \in sign (x_{n}) [l, R]$ , clause 1 of [F1] applied to $H$ yields $x_{n} = M_{sign x_{n}}^{H} (x_{n - 1}, x_{n + 1}) (n \in Z) .$

5.1givenstep 3.2step 4.1F5F6

Topology: let $C \subseteq X_{\pm}$ be closed. A sequence in $C$ has a subsequence converging in the sequentially compact space $X_{\pm}$ ([F5] and the isometry $φ$ of [F6]), and its limit lies in $C$ because closed sets are sequentially closed by the choice-free direction of [F6]; hence $C$ is sequentially compact. The continuous image $h_{G} (C)$ is sequentially compact: given a sequence $(h_{G} (x_{k}))$ with $x_{k} \in C$ , pick a subsequence $x_{k_{j}} \to x \in C$ and use continuity of $h_{G}$ ; hence $h_{G} (C)$ is sequentially closed (any convergent sequence in it has a subsequence converging to a point of $h_{G} (C)$ , and limits are unique by [F6]), and by the ${A C}_{ω}$ direction of [F6] it is closed in $R^{2}$ . Thus $h_{G}$ is a closed map, so $h_{G}^{- 1}$ is continuous. Under the Axioms of Countable and Dependent Choice, $X_{\pm}$ is compact by [F5] and $h_{G} (X_{\pm}) = Λ_{G}$ is a continuous image of it, hence compact.

5.2givenstep 2.1step 3.3step 4.4F1F2F3F7F8algebra

Stable curve for $G$ : let $s$ be nonconstant with $s_{n} = u$ for $∣ n ∣ \geq N$ , and set $y_{n} : = q_{n + 1} (s)$ for $n \geq 0$ , $b_{*} : = q_{0} (s)$ . Then $y_{n} = M_{s_{n + 1}} (y_{n - 1}, y_{n + 1})$ for $n \geq 0$ with $y_{- 1} : = b_{*}$ , and $∣ y_{n} ∣ \in [l + \frac{3}{4}, R - \frac{3}{4}]$ by [F1]. Apply Weighted half-tail contractions for $q_{n} = M_{s_{n}} (q_{n - 1}, q_{n + 1})$ give a unique $C^{1}$ curve with tangent slope bounded by $c / (1 - q)$ with $θ = 1 / 2$ , $c = d \leq 2 / 13$ , $q = d (θ + θ^{- 1}) \leq 5 / 13 < 1$ , $J = [- R, R]$ , $\overset{ˉ}{ρ} = \frac{1}{2}$ , $δ_{*} = \frac{1}{4}$ , and update $M_{s_{n + 1}}$ . The half-orbit margin follows from the root margins, $[b_{*} - δ_{*}, b_{*} + δ_{*}] \subseteq J$ follows from $∣ q_{0} (s) ∣ \leq R - \frac{3}{4}$ , and $δ_{*} < (1 - q) \overset{ˉ}{ρ} / d$ because $d \leq 2 / 13$ . By [F3] there is a $C^{1}$ family $u (b)$ with $∣ d_{0} ∣ \leq d / (1 - q) \leq 1 / 4$ , and the phase curve $Z (b) : = (b, V (b, y_{0} + u_{0} (b)))$ through $Z (b_{*}) = (q_{0} (s), V (q_{0} (s), q_{1} (s))) = z_{*}$ has tangent $(1, T)$ with $T = \partial_{q} V (b_{*}, y_{0}) + \partial_{q^{'}} V (b_{*}, y_{0}) d_{0}, ∣ T ∣ \leq \frac{ε}{1 - ε} + \frac{1}{1 - ε} \cdot \frac{1}{4} \leq \frac{1}{5} + \frac{6}{5} \cdot \frac{1}{4} = \frac{1}{2} \leq \frac{3}{4} .$ Thus the tangent lies in $C_{3 / 4}^{s}$ . The recurrence coordinates remain in $[- R, R]$ by the weighted-ball construction; their phase points $(x_{n}, V (x_{n}, x_{n + 1}))$ lie in $Q \subseteq U$ because $∣ V (x_{n}, x_{n + 1}) ∣ \leq R + ε < R + 1$ . Step 3.1 therefore gives every forward iterate without leaving the domain of $G$ . Every point $Z (b)$ has forward orbit tracking that of $z_{*}$ , which converges to $z_{\overset{ˉ}{u}}$ by step 4.4; hence these points lie in $W_{G}^{s} (z_{\overset{ˉ}{u}})$ near $b_{*}$ .

5.3step 1.1step 3.2step 4.2F1F6algebra

Sensitivity: let $s \in X_{\pm}$ and let $r > 0$ . Choose $m \geq 0$ such that $4 R C_{h} ν^{m} < r$ , using $0 < ν < 1$ from step 1.1, and define $t \in X_{\pm}$ by $t_{j} = s_{j}$ for $j \neq m + 1$ and $t_{m + 1} = - s_{m + 1}$ . Since $s, t$ agree on $∣ j ∣ \leq m$ , the estimate in step 3.2 gives $∥ h_{G} (s) - h_{G} (t) ∥_{2} < r$ . At coordinate $0$ , the codes $S_{\pm}^{m + 1} s$ and $S_{\pm}^{m + 1} t$ have opposite signs; by [F1] their coded first coordinates have opposite signs and absolute values at least $l + \frac{3}{4}$ . Therefore $∥ h_{G} (S_{\pm}^{m + 1} s) - h_{G} (S_{\pm}^{m + 1} t) ∥_{2} \geq 2 (l + \frac{3}{4}) > 2 l .$ Iterating the conjugacy of step 4.2 identifies these two points with $G^{m + 1} (h_{G} (s))$ and $G^{m + 1} (h_{G} (t))$ . Thus for every point of $Λ_{G}$ and every radius $r > 0$ there is a point within distance $r$ whose forward iterates are separated by at least $2 l$ , proving sensitivity with constant $2 l$ under the definition in [F6].

6.1givenstep 4.2step 4.1step 5.1F5F6

Mixing and dense periodic points: $G$ is $C^{2}$ on an open set containing $Λ_{G}$ , so its restriction to $Λ_{G}$ is continuous. Under ${A C}_{ω}$ , $h_{G}$ is a homeomorphism by step 5.1. If $U, V \subseteq Λ_{G}$ are nonempty and relatively open, then $h_{G}^{- 1} (U)$ and $h_{G}^{- 1} (V)$ are nonempty open subsets of $X_{\pm}$ . The shift $S_{\pm}$ is topologically mixing by [F5], so there is $N$ such that $S_{\pm}^{n} (h_{G}^{- 1} (U)) \cap h_{G}^{- 1} (V) \neq \emptyset$ for all $n \geq N$ . By the conjugacy in step 4.2, this gives $G^{n} (U) \cap V \neq \emptyset$ for all $n \geq N$ , proving mixing. The same conjugacy and injectivity show that $h_{G}$ maps periodic shift codes exactly to periodic points of $G ∣_{Λ_{G}}$ ; since periodic shift codes are dense by [F5] and $h_{G}$ is a homeomorphism, these periodic points are dense in $Λ_{G}$ .

6.2givenstep 4.1step 4.4step 5.2step 4.5F1F2F3F7algebra

Unstable curve for $G$ : shift the $H$ -reference to $y_{n}^{'} : = x_{n + 1}$ $(n \geq 0)$ with $y_{- 1}^{'} : = x_{0} = v_{0} (s)$ , and apply Weighted half-tail contractions for $q_{n} = M_{s_{n}} (q_{n - 1}, q_{n + 1})$ give a unique $C^{1}$ curve with tangent slope bounded by $c / (1 - q)$ to $H$ with $θ = 1 / 2$ , $c = d$ , $q \leq 5 / 13$ , $J = [- R, R]$ , $\overset{ˉ}{ρ} = \frac{1}{2}$ , and $δ_{*} = \frac{1}{4}$ . The reference recurrence holds by step 4.5. Its margins satisfy $∣ y_{n}^{'} ∣ \geq l + \frac{3}{4} - ε \geq l + \frac{7}{12} > l + \frac{1}{2}$ and $∣ y_{n}^{'} ∣ \leq R - \frac{3}{4} + ε \leq R - \frac{7}{12} < R - \frac{1}{2}$ ; also $∣ y_{- 1}^{'} ∣ = ∣ v_{0} (s) ∣ \leq R - \frac{7}{12}$ , so the margin and parameter interval conditions hold. Moreover $δ_{*} < (1 - q) \overset{ˉ}{ρ} / d$ because $d \leq 2 / 13$ . By [F3] we obtain a $C^{1}$ recurrence curve $γ_{H} (b) = (b, y_{0}^{'} + u_{0} (b))$ ; composing with the $H$ -phase map gives $Z^{H} (b) : = (b, V^{H} (b, y_{0}^{'} + u_{0} (b)))$ through $Z^{H} (y_{- 1}^{'}) = (v_{0} (s), V^{H} (v_{0} (s), v_{- 1} (s))) = w$ , with tangent $(1, T^{H})$ and $∣ T^{H} ∣ \leq 1 / 5 + (6 / 5) (1 / 4) = 1 / 2$ . The reference orbit $w_{n} = Σ (z_{- n})$ converges to $Σ (z_{\overset{ˉ}{u}})$ by step 4.4, so the tracked points lie in $W_{H}^{s} (Σ (z_{\overset{ˉ}{u}}))$ . Thus $γ_{u} (t) : = Σ (Z^{H} (v_{0} (s) + t))$ for sufficiently small $∣ t ∣$ is a $C^{1}$ curve through $z_{*}$ in $W_{G}^{u} (z_{\overset{ˉ}{u}})$ , since $G^{- n} (Σ ξ) = Σ H^{n} (ξ) \to Σ (Σ (z_{\overset{ˉ}{u}})) = z_{\overset{ˉ}{u}}$ . Its tangent is $Σ (1, T^{H}) = (T^{H}, 1) \in C_{3 / 4}^{u}$ . Finally $z_{*} \neq z_{\overset{ˉ}{u}}$ because $h_{G}$ is injective by step 4.1 and $s \neq \overset{ˉ}{u}$ .

7.1step 3.1step 6.1step 5.3step 4.3step 6.2F7∎

Conclusion: clause 1 is step 3.1; clause 2 consists of step 3.2 (continuity), step 4.1 (injectivity), step 4.2 (conjugacy), step 5.1 (homeomorphism and compactness), and step 6.1 (mixing and density of periodic points), with the topological transfers conditional on ${A C}_{ω}$ ; the sensitivity assertion is step 5.3. Clause 3 consists of step 3.3 (constant codes) and step 4.3 (hyperbolicity). Clause 4 consists of step 4.4 (tail convergence), step 5.2 (stable curve), step 4.5 (backward orbit), and step 6.2 (unstable curve): choose $ε_{0} > 0$ smaller than both parameter-neighbourhood radii and define $γ_{s} (t) : = Z (b_{*} + t)$ and $γ_{u} (t) : = Σ (Z^{H} (v_{0} (s) + t))$ for $∣ t ∣ < ε_{0}$ . Then both curves pass through $z_{*}$ , are injective because their first coordinate is their parameter and $Σ$ is invertible, lie in the respective stable and unstable sets, and have tangents in $C_{3 / 4}^{s}$ and $C_{3 / 4}^{u}$ . Those cones intersect only at zero, so the tangent lines are distinct and $z_{*}$ is transverse.

Remarks

The theorem is deliberately explicit: the constants $d \leq 2 / 13$ , $q \leq 5 / 13$ , the margins $\geq \frac{3}{4}$ and the cone aperture $3 / 4$ are all uniform for $a \geq 100$ and $ε \leq 1 / 6$ , so the hypothesis on $ε$ is a finite, checkable condition rather than an abstract smallness statement. The genuine smallness requirement is only $η = C ε \leq 1$ together with $ε \leq 1 / 6$ .
The second hypothesis, that $H = Σ G^{- 1} Σ$ is again close to $F$ , is what makes the unstable curve available by the same forward half-tail construction. It is verified for the smooth periodic oscillator in The rescaled pulse endpoint map converges to the kick $K (q, p) = (q, p + q^{2} - a)$ in $C^{1}$ , and the period map is smooth, symplectic and $C^{1}$ -close to $F$ .
No structural-stability theorem is invoked: the coded set, its conjugacy with the shift and the two curves are constructed directly.

DefinitionDefinition: AI-adaptedProof: Not applicableprecheck passOpen item page →

The smooth $P$ -periodic pulsed oscillator: the normalized pulse, its Hamiltonian and its stroboscopic (period) map

Definition

The normalized pulse. By Test function cutoffs and euclidean localization, applied to $K = {1 / 2} \subset (0, 1)$ , choose $χ \in C_{c}^{\infty} ((0, 1))$ with $0 \leq χ \leq 1$ and $χ = 1$ on a neighborhood of $1 / 2$ . Thus there are $0 < α < β < 1$ with $χ = 1$ on $[α, β]$ . Before taking any integral, extend $χ$ by zero to $R$ , retaining the notation $χ$ . Its compact support inside $(0, 1)$ makes this extension identically zero near both endpoints, hence smooth on $R$ . The continuous extended function $χ$ is integrable on $[0, 1]$ ; by additivity and monotonicity of the integral and the integral of a constant,

$I : = \int_{0}^{1} χ (s) d s \geq \int_{α}^{β} χ (s) d s = β - α > 0,$

using integrability of continuous functions (A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion), additivity and monotonicity (For $a < c < b$ : $f$ is integrable on $[a, b]$ if and only if it is integrable on $[a, c]$ and on $[c, b]$ , and then $\int_{a}^{b} f = \int_{a}^{c} f + \int_{c}^{b} f$ ; with the oriented form for arbitrary $a, b, c$ , If $f \leq g$ on $[a, b]$ and both are integrable then $\int_{a}^{b} f \leq \int_{a}^{b} g$ ; and $m (b - a) \leq \int_{a}^{b} f \leq M (b - a)$ ), and the integral of a constant (If $m \leq f \leq M$ on $[a, b]$ then $m (b - a) \leq L (f, P) \leq \underline{\int_{a}^{b}} f \leq \overline{\int_{a}^{b}} f \leq U (f, P) \leq M (b - a)$ for every partition $P$ ; in particular every constant function is integrable, with $\int_{a}^{b} c = c (b - a)$ ).

Define $w_{0} : = χ / I$ on $(0, 1)$ and let $w : R \to R$ be its extension by zero outside $(0, 1)$ . Since $χ$ has compact support in $(0, 1)$ , this zero extension is smooth and compactly supported in $(0, 1)$ . Linearity of the integral (Integrable functions on $[a, b]$ form a set closed under sums and scalar multiples, and $\int_{a}^{b} (λ f + μ g) = λ \int_{a}^{b} f + μ \int_{a}^{b} g$ ) gives

$w \in C_{c}^{\infty} ((0, 1)), w \geq 0, \int_{0}^{1} w (s) d s = 1.$

Fix such a pulse $w$ for the constructions below. The resulting coefficient and endpoint maps may depend on this choice; the later estimates apply to every such $w$ , with constants allowed to depend on it.

The periodic pulse. Fix a real parameter $a \geq 100$ and a smallness parameter $δ > 0$ , and put $P : = π / 2 + δ$ . Define $β_{δ} : R \to R$ to be the $P$ -periodic function whose restriction to $[0, P]$ is

$β_{δ} (t) : = 0 (t \in [0, \frac{π}{2}]), β_{δ} (t) : = \frac{1}{δ} w (\frac{t - π / 2}{δ}) (t \in [\frac{π}{2}, P]) .$

Both definitions use the same zero extension $w : R \to R$ , so the endpoint values $w (0)$ and $w (1)$ are defined and equal to $0$ . Since $w$ vanishes on neighborhoods of $0$ and $1$ , all its derivatives vanish there; the $P$ -periodic extension of $β_{δ}$ is therefore of class $C^{\infty}$ on $R$ ( $C^{k}$ Euclidean maps and diffeomorphisms). Also $β_{δ} \geq 0$ . To verify its normalization, put

$B_{δ} (t) : = {\begin{cases} 0, & 0 \leq t \leq π / 2, \\ \int_{0}^{(t - π / 2) / δ} w (s) d s, & π / 2 \leq t \leq P . \end{cases}$

The integral function of the continuous $w$ has derivative $w$ by The first fundamental theorem: if $f$ is integrable on $[a, b]$ and continuous at $c$ , then $F^{'} (c) = f (c)$ ; in particular a continuous $f$ has $F$ as a primitive and A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion, and the chain rule (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ ) gives $B_{δ}^{'} (t) = β_{δ} (t)$ on the pulse interval. Since $w$ vanishes near its endpoints, $B_{δ}$ is differentiable across $t = π / 2$ as well. Since $β_{δ}$ is continuous and therefore integrable on $[0, P]$ , the second part of the fundamental theorem (The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ ) gives

$\int_{0}^{P} β_{δ} (t) d t = B_{δ} (P) - B_{δ} (0) = \int_{0}^{1} w (s) d s = 1.$

The Hamiltonian and its equations. Use the phase domain $U = R^{2}$ with coordinates $(q, p)$ and standard symplectic form $ω = d q \land d p$ and matrix $J = (\begin{array}{cc} 0 & 1 \\ - 1 & 0 \end{array})$ , as in A phase domain, its canonical one-form and its standard symplectic two-form. For each fixed time $t$ , define the smooth Hamiltonian function

$H_{δ} : R \times R^{2} ⟶ R, H_{δ} (t, q, p) : = \frac{p^{2} + q^{2}}{2} + β_{δ} (t) (a q - \frac{q^{3}}{3}) .$

The Hamiltonian-vector-field convention is $ι_{X_{H}} ω = d H$ , for which The Hamiltonian field of a $C^{2}$ function on a phase domain is $J \nabla H$ , and the Poisson bracket is $\sum_{i} (\partial_{q^{i}} f \partial_{p_{i}} g - \partial_{p_{i}} f \partial_{q^{i}} g)$ gives $X_{H} = J \nabla H = (\partial_{p} H, - \partial_{q} H)$ . Here $\partial_{p} H_{δ} = p$ and $\partial_{q} H_{δ} = q + β_{δ} (t) (a - q^{2})$ , so the associated time-dependent first-order system is

$q^{'} = p, p^{'} = - q + β_{δ} (t) (q^{2} - a),$

equivalently the second-order equation $q^{''} = - q + β_{δ} (t) (q^{2} - a)$ ; the right-hand side is $C^{\infty}$ in $(t, q, p)$ , hence continuous and locally Lipschitz in the state variable, so every initial condition has a unique maximal solution (First-order systems, initial value problems, and solutions on intervals, Extensions, maximal solutions, maximal intervals, and global solutions, Every Picard–Lindelöf initial value problem has one maximal solution on an open interval). The system is $P$ -periodic in $t$ .

Free rotation and pulse endpoint maps. Over the first subinterval $[0, π / 2]$ the pulse vanishes and the system is the free oscillator; define the free rotation

$R_{0} : R^{2} \to R^{2}, R_{0} (q, p) : = (p, - q),$

which is the time- $π / 2$ solution operator of $q^{'} = p$ , $p^{'} = - q$ : the map $t \mapsto (q_{0} \cos t + p_{0} \sin t, - q_{0} \sin t + p_{0} \cos t)$ solves the free system by the derivative identities of sine and cosine (The derivatives of sine and cosine are cosine and minus sine) and has initial data $(q_{0}, p_{0})$ . Moreover $\cos (π / 2) = 0$ by the zero-set statement (The zero sets of sine and cosine and the least positive common period 2 pi), $\sin (π / 2) > 0$ by positivity on $(0, π)$ (Pi is the first positive zero of sine), and $\sin^{2} (π / 2) + \cos^{2} (π / 2) = 1$ (Parity and the Pythagorean identity for sine and cosine), so $\sin (π / 2) = 1$ . Uniqueness of solutions then identifies the endpoint map with $R_{0}$ .

For $s \in [0, 1]$ the rescaled pulse equation is

$q_{s} = δ p, p_{s} = - δ q + w (s) (q^{2} - a),$

obtained from the second half of the period by the substitution $s = (t - π / 2) / δ$ . For each initial state $z = (q, p) \in R^{2}$ , let $J_{z}^{δ}$ be the open maximal interval containing $0$ of its solution to the rescaled equation. Define

$D_{δ} : = {z \in R^{2} : [0, 1] \subseteq J_{z}^{δ}}, K_{δ} : D_{δ} \to R^{2}, K_{δ} (z) : = (q (1), p (1)),$

where $(q (s), p (s))$ is the unique maximal solution with $(q, p) (0) = z$ (Every Picard–Lindelöf initial value problem has one maximal solution on an open interval). For the original $P$ -periodic system, let $J_{z}$ be the maximal existence interval of the solution through $z$ at time $0$ , and set $D_{P} : = {z \in R^{2} : [0, P] \subseteq J_{z}}$ . Define the stroboscopic map by

$G_{δ} : D_{G_{δ}} \to R^{2}, D_{G_{δ}} : = {z \in R^{2} : R_{0} (z) \in D_{δ}}, G_{δ} (z) : = K_{δ} (R_{0} (z)) .$

The change of time variable $s = (t - π / 2) / δ$ identifies solutions on $[π / 2, P]$ with solutions of the rescaled equation on $[0, 1]$ . Concatenating with the unique free solution on $[0, π / 2]$ shows $D_{G_{δ}} = D_{P}$ and $G_{δ} = θ_{P}$ there, where $θ_{P} : D_{P} \to R^{2}$ is the stroboscopic map of The stroboscopic (time- $P$ ) map of a $P$ -periodic first-order system. Conversely, restricting any solution through time $P$ to the pulse interval and applying the same change of variables gives a solution of the rescaled equation through $s = 1$ ; uniqueness identifies both maps.

Remarks

The phase variables $q, p$ are dimensionless in this definition; no units, masses or measured quantities occur. The parameter $a$ and the pulse data are pure numbers, and the entire construction is a mathematical model of a periodically forced one-degree-of-freedom oscillator.
The pulse has width $δ$ and total strength $\int β_{δ} = 1$ , so $β_{δ}$ converges to a formal unit impulse as $δ \to 0$ ; no meaning is assigned to the limit $δ = 0$ as a physical force. The $δ = 0$ endpoint map is computed on the companion page cm-math-chaos-examples by explicit integration and used only as a comparison object.
Nothing in this definition asserts that $G_{δ}$ is smooth, invertible, symplectic, close to $F$ , or defined on any particular set; these are the contents of The rescaled pulse endpoint map converges to the kick $K (q, p) = (q, p + q^{2} - a)$ in $C^{1}$ , and the period map is smooth, symplectic and $C^{1}$ -close to $F$ and A smooth periodic one-degree-of-freedom oscillator with polynomial pulse forcing has stroboscopic chaos and a transverse homoclinic point.

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The rescaled pulse endpoint map converges to the kick $K (q, p) = (q, p + q^{2} - a)$ in $C^{1}$ , and the period map is smooth, symplectic and $C^{1}$ -close to $F$

Statement

Fix $a \geq 100$ , put $R : = \sqrt{a} + 3 \geq 13$ , $S : = [- R - 1, R + 1]^{2}$ , and let $w$ , $δ$ , $P$ , $β_{δ}$ , $R_{0}$ , the rescaled pulse equation, the endpoint map $K_{δ}$ and the stroboscopic map $G_{δ} = K_{δ} \circ R_{0}$ be as in The smooth $P$ -periodic pulsed oscillator: the normalized pulse, its Hamiltonian and its stroboscopic (period) map. Let

$K (q, p) : = (q, p + q^{2} - a), F (q, p) : = (p, p^{2} - a - q), J : = (\begin{array}{cc} 0 & 1 \\ - 1 & 0 \end{array}),$

so that $F = K \circ R_{0}$ . Then there are constants $δ_{1} > 0$ and $C_{1} < \infty$ depending only on $a$ and $w$ such that for every $0 < δ \leq δ_{1}$ :

(existence on the pulse) for every $(q, p) \in S$ the solution of the rescaled equation with initial data $(q, p)$ exists on $[0, 1]$ and stays in the compact tube $T : = [- R - 2, R + 2] \times [- M, M]$ , $M : = R + 1 + (R + 1)^{2} + 1$ ; hence $K_{δ}$ is defined on $S$ , $K_{δ} (S) \subseteq T$ , and

$\sup_{S} (∣ K_{δ} - K ∣ + ∥ D K_{δ} - D K ∥) \leq C_{1} δ$

for the Euclidean norm and the operator norm; 2. (closeness of the period map) $\sup_{S} (∣ G_{δ} - F ∣ + ∥ D G_{δ} - D F ∥) \leq C_{1} δ$ ; 3. (smoothness, invertibility, symplecticity) $G_{δ}$ is of class $C^{\infty}$ on an open neighbourhood $N$ of $S$ ; $D G_{δ} (z)$ is invertible for every $z \in N$ and $G_{δ}$ is injective on $N$ , so $G_{δ} : N \to G_{δ} (N)$ is a $C^{\infty}$ diffeomorphism onto its image; and

$D G_{δ} (z)^{T} J D G_{δ} (z) = J (z \in N),$

that is, $G_{δ}$ is symplectic; 4. (the reverse-conjugated map) let $S_{w} (q, p) : = (p, q)$ and $H_{δ} : = S_{w} \circ G_{δ}^{- 1} \circ S_{w}$ on a neighbourhood of $S$ . Then $H_{δ}$ is of class $C^{\infty}$ there, and there is $C_{2} < \infty$ , depending only on $a$ and $w$ , with

$\sup_{S} (∣ H_{δ} - F ∣ + ∥ D H_{δ} - D F ∥) \leq C_{2} δ .$

Consequently, each of the six componentwise closeness inequalities for $G_{δ}$ and $H_{δ}$ holds on $S$ with common bound $\overset{ˉ}{C} δ$ , where $\overset{ˉ}{C} : = \max (C_{1}, C_{2})$ : each coordinate difference is bounded by the Euclidean norm, and each derivative-entry difference is bounded by the operator norm.

Facts & Assumptions

Given: The data in the Statement. Extend $w$ by zero to a smooth function on $R$ , and put $f_{δ} (s, q, p) = (δ p, - δ q + w (s) (q^{2} - a))$ for real $δ$ .

[F1]

The model defines this smooth field and its unique maximal solutions (The smooth $P$ -periodic pulsed oscillator: the normalized pulse, its Hamiltonian and its stroboscopic (period) map, First-order systems, initial value problems, and solutions on intervals, Extensions, maximal solutions, maximal intervals, and global solutions, Local Lipschitz continuity in the state variable, locally uniform in time and parameters). A solution trapped in a compact time-state tube extends past a finite endpoint (Compact continuation for locally Lipschitz ODEs: a solution whose graph stays in a compact subset of the domain near a finite endpoint extends past it).

[F2]

Norms of vector integrals are bounded by integrals of their norms (For $a \leq b$ and $f : [a, b] \to R^{m}$ integrable when $a < b$ , $∥ \int_{a}^{b} f ∥_{2} \leq \int_{a}^{b} ∥ f ∥_{2}$ ; for $a < b$ , $∥ f ∥_{2}$ is integrable). Integral comparison and differentiation: continuous functions are integrable, their integrals satisfy linearity, monotonicity and the fundamental theorem; Gronwall controls a nonnegative function satisfying $u (s) \leq A + B \int_{0}^{s} u$ (The lower and upper Darboux integrals of a bounded $f$ on $[a, b]$ as $\sup_{P} L (f, P)$ and $\inf_{P} U (f, P)$ , Darboux integrability as their equality, and the notation $\int_{a}^{b} f$ , A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion, Integrable functions on $[a, b]$ form a set closed under sums and scalar multiples, and $\int_{a}^{b} (λ f + μ g) = λ \int_{a}^{b} f + μ \int_{a}^{b} g$ , If $f \leq g$ on $[a, b]$ and both are integrable then $\int_{a}^{b} f \leq \int_{a}^{b} g$ ; and $m (b - a) \leq \int_{a}^{b} f \leq M (b - a)$ , The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ , Gronwall's integral inequality with variable and constant coefficients).

[F3]

On an established common compact interval the derivative in initial data solves $Y^{'} = D_{z} f (s, z) Y$ , $Y (0) = I$ ( $C^{1}$ dependence of solutions on initial data). The jointly smooth field has local smooth dependence on initial data and parameters (Smooth dependence of ODE solutions on parameters); this local assertion will be patched over the established interval, rather than assumed to give existence for a whole period.

[F4]

Closed Euclidean boxes are compact (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line). The Euclidean inverse-function theorem supplies a C1 local inverse for a C1 map on an open domain with invertible derivative (The Euclidean inverse function theorem); a smooth map then has smooth local inverses by A local inverse of a $C^{k}$ regular map is $C^{k}$ . Euclidean calculus uses the chain rule, mean-value inequality, continuity and boundedness of continuous derivatives on compact sets ( $C^{k}$ Euclidean maps and diffeomorphisms, The total (Fréchet) derivative $D f (a)$ as the linear first-order approximation with $o (∥ h ∥_{2})$ remainder, The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , On a convex open set, a uniform bound $∥ D f (z) v ∥_{2} \leq M ∥ v ∥_{2}$ implies $∥ f (y) - f (x) ∥_{2} \leq M ∥ y - x ∥_{2}$ , A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value, Metric space: $d (x, y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric, Open ball, closed ball and sphere in a metric space, Continuity of $f : A \to R$ at a point of $A$ and on $A$ : the $ε$ - $δ$ condition, its agreement with $\lim_{x \to c} f (x) = f (c)$ at a limit point, and continuity at an isolated point, Absolute value in an ordered field, Basic properties of the absolute value, Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length).

Proof

technique · direct; trap the forward and backward pulse evolutions in separate compact tubes, compare their state and variational equations, and use uniqueness to identify the backward endpoint as the inverse

1.1givenF1F2algebra

At $δ = 0$ the forward solution through $(q_{0}, p_{0})$ is $z_{0} (s) = (q_{0}, p_{0} + (q_{0}^{2} - a) \int_{0}^{s} w)$ . For initial data in $S$ , $∣ q_{0} ∣ \leq R + 1$ and $∣ q_{0}^{2} - a ∣ \leq (R + 1)^{2}$ ; hence $∣ p_{0} + (q_{0}^{2} - a) \int_{0}^{s} w ∣ \leq M - 1$ on $[0, 1]$ . Thus this reference solution lies at least distance $1$ from the boundary of the stated tube $T$ . Its endpoint is $K (q_{0}, p_{0})$ .

2.1step 1.1F1F2F4algebra

On $T$ and for $∣ δ ∣ \leq 1$ , there are constants $B_{0}, L_{0}$ depending only on $a, w$ such that $∣ f_{δ} - f_{0} ∣ \leq B_{0} ∣ δ ∣$ and $∥ D_{z} f_{δ} ∥ \leq L_{0}$ : one may take $B_{0} = M + R + 2$ and $L_{0} = 1 + 2 ∥ w ∥_{\infty} (R + 2)$ . Up to the first exit from $T$ , subtracting the integral equations gives $∣ z_{δ} (s) - z_{0} (s) ∣ \leq B_{0} ∣ δ ∣ + L_{0} \int_{0}^{s} ∣ z_{δ} - z_{0} ∣$ , so Gronwall gives $\sup ∣ z_{δ} - z_{0} ∣ \leq B_{0} e^{L_{0}} ∣ δ ∣$ . Choose $d_{0} > 0$ with $B_{0} e^{L_{0}} d_{0} < 1 / 2$ . An exit on $[0, 1]$ is impossible by the distance margin in step 1.1. A maximal solution terminating before the end without exiting is extendible by [F1]. Therefore every datum in $S$ has a solution on $[0, 1]$ staying in $T$ .

3.1givenstep 2.1F1F2F4algebra

Larger tubes establish the domains needed for inversion. Set $b : = R + 2$ , $Q : = [- b, b]^{2}$ , $B : = b + b^{2} + a + 2$ and $E : = [- B, B]^{2}$ . For every datum in $E$ , the forward reference solution has $∣ q ∣ \leq B$ , $∣ p ∣ \leq B + B^{2} + a$ . For a terminal datum $y = (y_{q}, y_{p}) \in E$ at $s = 1$ , the reference backward solution is $(y_{q}, y_{p} - (y_{q}^{2} - a) \int_{s}^{1} w)$ with exactly the same bounds. Both reference families therefore have distance at least $1$ from the boundary of $T_{E} : = [- B - 1, B + 1] \times [- B - B^{2} - a - 1, B + B^{2} + a + 1]$ . Apply the integral comparison of step 2.1 on this tube, in the backward case after $τ = 1 - s$ so that the field is $- f_{δ} (1 - τ, z)$ . This field has the same boundedness and Lipschitz estimates. With a smaller threshold $d_{E} > 0$ , both evolutions exist over the whole unit interval for all their data in $E$ , and their state differences from the reference evolutions are uniformly $O (∣ δ ∣)$ . All constants depend only on $a, w$ . In particular $K^{- 1} (Q) \subset int E$ , since $K^{- 1} (y) = (y_{q}, y_{p} - y_{q}^{2} + a)$ and its coordinates are bounded by $b$ and $b + b^{2} + a = B - 2$ . This uses the actual inverse preimage, without assuming it lies in $Q$ .

4.1step 3.1F1F3F4choose

These families are smooth in initial or terminal data and $δ$ on neighbourhoods of the compact data sets after decreasing $d_{E}$ . To justify the whole interval from the local assertion [F3], cover each reference graph by finitely many local time-state-parameter neighbourhoods where [F3] gives smooth local solution operators, and subdivide $[0, 1]$ into finitely many sufficiently short successive intervals. The graph stays inside the tube with a positive margin by step 3.1; the field and its state derivative are bounded on a slightly enlarged compact tube, so one common short interval is admissible at every graph point. At each subdivision endpoint local existence and smooth dependence apply to the endpoint data from the preceding interval. Their finite composition is smooth and equals the unique solution. Shrinking the initial-data and parameter neighbourhoods finitely many times keeps each intermediate value in its admissible neighbourhood. Finitely many such initial-data neighbourhoods cover each compact data set, yielding one positive parameter threshold. The same argument for the time-reflected field gives the backward family. Thus $K_{δ}$ and the backward endpoint map $L_{δ} (y) : = z (0)$ for $z (1) = y$ are smooth near $E$ and $Q$ , respectively, for all sufficiently small $∣ δ ∣$ , including $0$ .

5.1step 2.1step 3.1step 4.1F2F3F4algebra

The state comparisons in steps 2.1 and 3.1 also hold in $C^{1}$ . Indeed, on either compact tube, $D_{z} f_{δ} - D_{z} f_{0}$ is bounded by $∣ δ ∣$ , and $D_{z} f_{0}$ is uniformly Lipschitz in $z$ , so along compared trajectories $∥ D_{z} f_{δ} (s, z_{δ}) - D_{z} f_{0} (s, z_{0}) ∥ \leq C ∣ δ ∣$ . By [F3] their derivative matrices solve the variational equations with the same initial matrix $I$ . Gronwall first bounds both matrices by $e^{L}$ , then, after subtracting their integral equations, bounds the difference by $C e^{2 L} ∣ δ ∣$ , with fixed constants for the tube. The identical argument for the reflected field controls derivatives in terminal data. Consequently there are finite positive $C_{f}, C_{b}$ with $\sup_{Q} (∣ K_{δ} - K ∣ + ∥ D K_{δ} - D K ∥) \leq C_{f} ∣ δ ∣$ and $\sup_{Q} (∣ L_{δ} - K^{- 1} ∣ + ∥ D L_{δ} - D K^{- 1} ∥) \leq C_{b} ∣ δ ∣$ . Increasing $C_{f}$ includes the forward estimates on $S$ in the original tube; hence clause 1 follows.

6.1step 4.1step 5.1F1F3F4algebra

Choose $δ_{1} > 0$ below all previous thresholds and with $C_{b} δ_{1} < 1$ . Then $L_{δ} (Q) \subset int E$ , because $K^{- 1} (Q)$ has coordinate margin at least $2$ in $E$ . Put $V : = (- b, b)^{2}$ and $O : = int E$ . Uniqueness of the original and reflected equations shows that $K_{δ}$ is injective on $O$ : solutions sharing their terminal value coincide when evolved backwards. Its differential is invertible by the following direct calculation: the rescaled field is $J \nabla h$ for $h (s, q, p) = δ (p^{2} + q^{2}) / 2 + w (s) (a q - q^{3} / 3)$ ; hence $A = D_{z} f_{δ} = J Hess h$ with symmetric Hessian and $A^{T} J + J A = 0$ . The variational matrix satisfies $(Y^{T} J Y)^{'} = 0$ and $Y (0) = I$ , so $Y^{T} J Y = J$ . A vector in the kernel of $Y$ must then lie in the kernel of the invertible matrix $J$ , so $Y$ is invertible. By The Euclidean inverse function theorem, the smooth map $K_{δ}$ on the open set $O$ has C1 local inverses at every point; A local inverse of a $C^{k}$ regular map is $C^{k}$ upgrades these to smooth local inverses, which agree by injectivity; therefore $K_{δ} : O \to K_{δ} (O)$ is a smooth diffeomorphism onto its open image. For every $y \in V$ , its backward trajectory starts at $L_{δ} (y) \in O$ and ends at $y$ ; hence $V \subset K_{δ} (O)$ and the inverse on $V$ is exactly $L_{δ}$ . Define $N : = R_{0}^{- 1} (O)$ , an open neighbourhood of $S$ . Then $G_{δ} = K_{δ} \circ R_{0} : N \to K_{δ} (O)$ is a smooth diffeomorphism, and clause 2 follows from step 5.1 and $R_{0} (Q) = Q$ , since $R_{0}$ is a linear isometry.

7.1givenstep 6.1F3F4algebra

The matrix computation of step 6.1 also gives symplecticity. The rescaled field is $J \nabla h$ with $h (s, q, p) = δ (p^{2} + q^{2}) / 2 + w (s) (a q - q^{3} / 3)$ . Its derivative matrix is $A = J Hess h$ with symmetric Hessian. Since $J^{T} = - J$ and $J^{2} = - I$ , direct multiplication gives $A^{T} J + J A = 0$ . Therefore the variational matrix satisfies $(Y^{T} J Y)^{'} = Y^{T} (A^{T} J + J A) Y = 0$ and $Y (0) = I$ , giving $Y^{T} J Y = J$ . This identity forces $Y$ invertible (a vector in its kernel must lie in the kernel of $J$ ). Also $D R_{0} = J$ satisfies $J^{T} J J = J$ . Applying the chain rule to the composition yields $D G_{δ}^{T} J D G_{δ} = J$ on $N$ , proving clause 3.

7.2step 5.1step 6.1F4algebra

Since $S_{w} (V) = V$ , the conjugated inverse is defined on the open neighbourhood $V$ of $S$ and has the explicit expression $H_{δ} = R_{0} \circ S_{w} \circ L_{δ} \circ S_{w}$ , using $S_{w} R_{0}^{- 1} = R_{0} S_{w}$ . At $δ = 0$ this is $S_{w} F^{- 1} S_{w} = F$ . Fixed linear isometries preserve both state and derivative comparison norms, so step 5.1 gives $\sup_{S} (∣ H_{δ} - F ∣ + ∥ D H_{δ} - D F ∥) \leq C_{b} δ$ . The composition is a smooth diffeomorphism onto its image, and clause 4 holds with $C_{2} : = C_{b}$ and clause 2 with $C_{1} : = C_{f}$ .

8.1step 5.1step 6.1step 7.1step 7.2F4algebra∎

Increase $C_{1}, C_{2}$ to positive constants if needed and decrease $δ_{1}$ to include every threshold above. For $0 < δ \leq δ_{1}$ , each coordinate difference is bounded by the Euclidean difference norm and each derivative entry difference by the operator norm. Thus both maps satisfy the six componentwise closeness inequalities with the common bound $\overset{ˉ}{C} δ$ , where $\overset{ˉ}{C} : = \max (C_{1}, C_{2})$ . The forward tube conclusion was established independently in steps 1.1-2.1 and was preserved throughout the enlarged-domain argument.

Remarks

The enlarged compact set $E$ is used only for forward and inverse domain control. The stated small forward tube $T$ for data in $S$ is retained. Constants and smallness thresholds depend on the fixed $a$ and $w$ ; no global completeness of the polynomial field is assumed.
Symplecticity is proved directly for the time-dependent system. The autonomous result The derivative of the Hamiltonian flow satisfies $Y^{T} J Y = J$ and $\det Y = 1$ is not invoked. The coordinate matrix agrees with the canonical layer, while the whole-period smoothness assertion uses explicit finite patching of local solution maps.

TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

A smooth periodic one-degree-of-freedom oscillator with polynomial pulse forcing has stroboscopic chaos and a transverse homoclinic point

Statement

Grant the Axioms of Countable and Dependent Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain), inherited from the compactness and homeomorphism clauses of Explicit persistence under the stated forward and conjugated-inverse bounds: coded chaos, hyperbolic fixed points and transverse homoclinic points. Fix $a \geq 100$ , let $w$ , $β_{δ}$ , $P = π / 2 + δ$ and the stroboscopic map $G_{δ}$ be as in The smooth $P$ -periodic pulsed oscillator: the normalized pulse, its Hamiltonian and its stroboscopic (period) map, and let $X_{\pm}$ , $S_{\pm}$ , $D_{\pm}$ be the two-sign shift space of The two-sign bilateral shift space, its coding map, and the ball of bounded bilateral real sequences. Then there is $δ_{0} > 0$ , depending only on $a$ and $w$ , such that for every $0 < δ < δ_{0}$ :

there is a function $h_{δ} : X_{\pm} \to R^{2}$ that is a homeomorphism of $(X_{\pm}, D_{\pm})$ onto a compact subset $Λ_{δ} : = h_{δ} (X_{\pm})$ of $R^{2}$ and satisfies $G_{δ} \circ h_{δ} = h_{δ} \circ S_{\pm}$ ; in particular $Λ_{δ}$ is compact and $G_{δ}$ -invariant, and $G_{δ} ∣_{Λ_{δ}}$ is conjugate to the shift;
$G_{δ} ∣_{Λ_{δ}}$ is continuous, topologically mixing, has dense periodic points and is sensitive with a positive constant; hence it is chaotic in the stated sense (Topological mixing, dense periodic points, sensitive dependence, and the stated chaos list);
the two constant codes give two hyperbolic fixed points of $G_{δ}$ , one of each sign;
for every nonconstant code $s \in X_{\pm}$ that is constant outside a finite set, the point $h_{δ} (s)$ is a transverse homoclinic point of the corresponding fixed point, and distinct such codes give distinct points (Stable and unstable cones, invariant line slopes, stable and unstable sets, and transverse homoclinic points of the quadratic family).

All statements are about the explicit smooth $P$ -periodic ordinary differential equation $q^{''} = - q + β_{δ} (t) (q^{2} - a)$ and its time- $P$ map; no empirical claim, measurement or physical assumption occurs.

Facts & Assumptions

Given: The parameter $a \geq 100$ , the fixed normalized pulse $w$ , and for each $δ > 0$ the periodic coefficient $β_{δ}$ , the Hamiltonian $H_{δ}$ , the rescaled pulse equation, the rotation $R_{0}$ , the endpoint map $K_{δ}$ and the stroboscopic map $G_{δ}$ of The smooth $P$ -periodic pulsed oscillator: the normalized pulse, its Hamiltonian and its stroboscopic (period) map.

[F1]

Structure of the model (The smooth $P$ -periodic pulsed oscillator: the normalized pulse, its Hamiltonian and its stroboscopic (period) map): $β_{δ}$ is $C^{\infty}$ and $P$ -periodic with $β_{δ} \geq 0$ and $\int_{0}^{P} β_{δ} = 1$ ; the associated system is $q^{'} = p$ , $p^{'} = - q + β_{δ} (t) (q^{2} - a)$ , smooth and locally Lipschitz in the state; and $G_{δ} = K_{δ} \circ R_{0}$ is its time- $P$ map.

[F2]

Pulse approximation (The rescaled pulse endpoint map converges to the kick $K (q, p) = (q, p + q^{2} - a)$ in $C^{1}$ , and the period map is smooth, symplectic and $C^{1}$ -close to $F$ ): there are $δ_{1} > 0$ and $C_{1}, C_{2} < \infty$ depending only on $a$ and $w$ such that for $0 < δ \leq δ_{1}$ the maps $K_{δ}$ and $G_{δ} = K_{δ} \circ R_{0}$ are defined and $C^{\infty}$ on a neighbourhood of $S = [- R - 1, R + 1]^{2}$ , $R = \sqrt{a} + 3$ , have $\sup_{S} (∣ G_{δ} - F ∣ + ∥ D G_{δ} - D F ∥) \leq C_{1} δ$ with $F (q, p) = (p, p^{2} - a - q)$ , $G_{δ}$ is a symplectic $C^{\infty}$ diffeomorphism onto its image, and the conjugate map $H_{δ} = S_{w} \circ G_{δ}^{- 1} \circ S_{w}$ , $S_{w} (q, p) = (p, q)$ , satisfies $\sup_{S} (∣ H_{δ} - F ∣ + ∥ D H_{δ} - D F ∥) \leq C_{2} δ$ and is likewise $C^{\infty}$ and diffeomorphic onto its image near $S$ ; moreover both $G_{δ}$ and $H_{δ}$ satisfy each of the six componentwise closeness inequalities displayed in the statement of Explicit persistence under the stated forward and conjugated-inverse bounds: coded chaos, hyperbolic fixed points and transverse homoclinic points on $S$ with the common constant $\overset{ˉ}{C} δ$ , where $\overset{ˉ}{C} : = \max (1, C_{1}, C_{2})$ .

[F3]

Under the Axioms of Countable and Dependent Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain), the perturbed horseshoe (Explicit persistence under the stated forward and conjugated-inverse bounds: coded chaos, hyperbolic fixed points and transverse homoclinic points): for every $ε \leq 1 / 6$ with $C ε \leq 1$ and every $C^{2}$ diffeomorphism $G$ onto its image on a neighbourhood of $S$ satisfying the six closeness inequalities with constant $ε$ on $S$ , and whose conjugate $S_{w} \circ G^{- 1} \circ S_{w}$ does the same, the conclusions of clauses 1-4 above hold for $G$ under these same axioms with sensitivity constant $2 (\sqrt{a} - 3)$ and with all codes constant outside finite sets.

[F4]

A homeomorphism here means a bijection $h_{δ}$ whose forward and inverse maps are continuous in the stated metrics (Continuity of a map between metric spaces, at a point and globally, in the $ε$ - $δ$ form); conjugacy means such a homeomorphism satisfying $G_{δ} \circ h_{δ} = h_{δ} \circ S_{\pm}$ . Compactness means that every open cover of the metric subspace has a finite subcover (Open cover, subcover, compact metric space, and compact subset of a metric space). Chaos has the four properties in Topological mixing, dense periodic points, sensitive dependence, and the stated chaos list. Hyperbolicity means that every eigenvalue of the fixed-point differential has modulus different from $1$ ; a transverse homoclinic point is a point different from the fixed point lying in both invariant sets, with injective $C^{1}$ curves in those sets through it and nonzero linearly independent tangents (Stable and unstable cones, invariant line slopes, stable and unstable sets, and transverse homoclinic points of the quadratic family, using metric convergence Convergence of a sequence in a metric space: $x_{k} \to x$ iff $d (x_{k}, x) \to 0$ in $R$ ).

Proof

technique · direct; choose $\delta_0$ so that the pulse approximation puts $G_\delta$ and $H_\delta$ inside the hypotheses of the perturbed horseshoe theorem, then transfer that theorem's conclusions

1.1givenF2F3algebra

Choice of the threshold: put $\overset{ˉ}{C} : = \max (1, C_{1}, C_{2})$ and let $δ_{0} : = \min (δ_{1}, 1 / (6 \max (1, \overset{ˉ}{C})), 1 / (\max (1, C) \max (1, \overset{ˉ}{C}))) > 0$ , so that for $0 < δ < δ_{0}$ the number $ε : = \overset{ˉ}{C} δ$ satisfies $0 < ε \leq 1 / 6$ and $C ε \leq 1$ .

2.1step 1.1F1F2F3

Hypotheses of the perturbed horseshoe for $G_{δ}$ : by [F2] (with the identification of $G_{δ}$ as the time- $P$ map in [F1]), for $0 < δ < δ_{0}$ the maps $G_{δ}$ and $H_{δ} = S_{w} \circ G_{δ}^{- 1} \circ S_{w}$ are $C^{\infty}$ , hence $C^{2}$ , diffeomorphisms onto their images on a neighbourhood of $S$ , and both satisfy each of the six componentwise closeness inequalities of [F3] on $S$ with the common constant $ε = \overset{ˉ}{C} δ$ ; in particular $H_{δ}$ is defined and $C^{2}$ on a neighbourhood of $S$ with the same inequalities. Thus the hypotheses of [F3] are met with $ε = \overset{ˉ}{C} δ$ .

3.1step 2.1F3F4∎

Conclusions: applying [F3] to $G = G_{δ}$ gives clauses 1-4 for the stroboscopic map: the coding homeomorphism $h_{δ}$ onto a compact invariant set $Λ_{δ}$ with $G_{δ} \circ h_{δ} = h_{δ} \circ S_{\pm}$ , chaotic behaviour of $G_{δ} ∣_{Λ_{δ}}$ , the two hyperbolic fixed points of the constant codes, and transverse homoclinic points for all codes constant outside a finite set, in the sense of [F4]. Distinct codes give distinct points because $h_{δ}$ is injective.

Remarks

The threshold $δ_{0}$ is explicit: it is the minimum of the pulse-approximation threshold and the two smallness conditions on $\max (1, C_{1}, C_{2}) δ$ from Explicit persistence under the stated forward and conjugated-inverse bounds: coded chaos, hyperbolic fixed points and transverse homoclinic points. No qualitative structural-stability theorem is used.
The forced oscillator $q^{''} = - q + β_{δ} (t) (q^{2} - a)$ is a dimensionless mathematical model. A mechanical interpretation with a mass, a length scale and an angular frequency would multiply $q$ and $p$ by those scales and rescale the Hamiltonian; that interpretation, its units and its physical scope belong to the consuming physics pages and are not asserted here.
Periodic driving supplies energy to the system; no energy conservation or autonomous structure is claimed. The horseshoe and homoclinic point concern the time- $P$ map and its compact invariant set.

5 · Examples, experiments and counterexamples

None yet.

Cm Math Chaos

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

The two-sign bilateral shift space, its coding map, and the ball of bounded bilateral real sequences

Definition

Remarks

The bounded functions on a set form a complete metric space in the supremum metric, and the coding map of the two-sign shift space is an isometry

Statement

Facts & Assumptions

Proof

Remarks

The square-root recurrence Ts(x)n=sna+xn−1+xn+1Ts​(x)n​=sn​a+xn−1​+xn+1​​ is a contraction of BRBR​ with a unique coded solution

Statement

Facts & Assumptions

Proof

Remarks

Finite symbol windows determine the code: locality of the iterates, coding continuity and injectivity of h(σ)=(x−1(σ),x0(σ))h(σ)=(x−1​(σ),x0​(σ))

Statement

Facts & Assumptions

Proof

Remarks

The quadratic map F(x,y)=(y,y2−a−x)F(x,y)=(y,y2−a−x) has a compact coded invariant set conjugate to the two-sign shift, with chaos under ACωACω​

Statement

Facts & Assumptions

Proof

Remarks

Constant codes give the fixed points, and finite nonconstant word codes give homoclinic orbits converging to a fixed point at both ends

Statement

Facts & Assumptions

Proof

Remarks

Stable and unstable cones, invariant line slopes, stable and unstable sets, and transverse homoclinic points of the quadratic family

Definition

Remarks

Cone invariance and expansion, and convergence of the slope graph transforms to unique invariant stable and unstable line fields

Statement

Facts & Assumptions

Proof

Remarks

Weighted half-tail contractions for qn=Msn(qn−1,qn+1)qn​=Msn​​(qn−1​,qn+1​) give a unique C1C1 curve with tangent slope bounded by c/(1−q)c/(1−q)

Statement

Facts & Assumptions

Proof

Remarks

The coded curves are genuine C1C1 stable and unstable curves of FF, with transverse intersection at every finite-word homoclinic point

Statement

Facts & Assumptions

Proof

Remarks

A C2C2 map C1C1-close to FF has a unique inverse branch for its first component, with O(ε)O(ε) derivative bounds

Statement

Facts & Assumptions

Proof

Remarks

Root bounds and C1C1 dependence for the perturbed scalar recurrence v2−a−u−w+e(u,v,w)=0v2−a−u−w+e(u,v,w)=0

Statement

Facts & Assumptions

Proof

Remarks

Explicit persistence under the stated forward and conjugated-inverse bounds: coded chaos, hyperbolic fixed points and transverse homoclinic points

Statement

Facts & Assumptions

Proof

Remarks

The smooth PP-periodic pulsed oscillator: the normalized pulse, its Hamiltonian and its stroboscopic (period) map

Definition

Remarks

The rescaled pulse endpoint map converges to the kick K(q,p)=(q,p+q2−a)K(q,p)=(q,p+q2−a) in C1C1, and the period map is smooth, symplectic and C1C1-close to FF

Statement

Facts & Assumptions

Proof

Remarks

A smooth periodic one-degree-of-freedom oscillator with polynomial pulse forcing has stroboscopic chaos and a transverse homoclinic point

Statement

Facts & Assumptions

Proof

Remarks

5 · Examples, experiments and counterexamples

The square-root recurrence $T_{s} (x)_{n} = s_{n} \sqrt{a + x_{n - 1} + x_{n + 1}}$ is a contraction of $B_{R}$ with a unique coded solution

Finite symbol windows determine the code: locality of the iterates, coding continuity and injectivity of $h (σ) = (x_{- 1} (σ), x_{0} (σ))$

The quadratic map $F (x, y) = (y, y^{2} - a - x)$ has a compact coded invariant set conjugate to the two-sign shift, with chaos under ${A C}_{ω}$

Weighted half-tail contractions for $q_{n} = M_{s_{n}} (q_{n - 1}, q_{n + 1})$ give a unique $C^{1}$ curve with tangent slope bounded by $c / (1 - q)$

The coded curves are genuine $C^{1}$ stable and unstable curves of $F$ , with transverse intersection at every finite-word homoclinic point

A $C^{2}$ map $C^{1}$ -close to $F$ has a unique inverse branch for its first component, with $O (ε)$ derivative bounds

Root bounds and $C^{1}$ dependence for the perturbed scalar recurrence $v^{2} - a - u - w + e (u, v, w) = 0$

The smooth $P$ -periodic pulsed oscillator: the normalized pulse, its Hamiltonian and its stroboscopic (period) map

The rescaled pulse endpoint map converges to the kick $K (q, p) = (q, p + q^{2} - a)$ in $C^{1}$ , and the period map is smooth, symplectic and $C^{1}$ -close to $F$