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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.

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

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

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

Depends on

Dependency tree · 2 levels

Sources

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