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The single-flip code at $a = 100$ : a nonperiodic coded orbit converging to the fixed point at both ends

Example

Take $a = 100$ and let $s \in X_{\pm}$ be the code with

$s_{0} = - 1, s_{n} = + 1 (n \neq 0),$

the single-flip code. Then $s$ is constant outside a finite set (indeed outside ${0}$ ), is not the constant code $\overset{ˉ}{+ 1}$ , and is not periodic: $S_{\pm}^{m} s \neq s$ for every $m \geq 1$ , because the unique coordinate at which $S_{\pm}^{m} s$ equals $- 1$ is $- m \neq 0$ whereas $s$ equals $- 1$ only at $0$ . Consequently the coded point $z : = h (s)$ satisfies $z \neq (r_{+}, r_{+})$ and $F^{m} (z) \neq z$ for every $m \geq 1$ , while

$F^{n} (z) ⟶ (r_{+}, r_{+}) (n \to + \infty and n \to - \infty) .$

Facts & Assumptions

Given: The parameter $a = 100$ , the single-flip code $s$ , the coding map $h$ and the phase map $F$ of 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 ${A C}_{ω}$ , and the fixed point $(r_{+}, r_{+})$ , $r_{+} = 1 + \sqrt{101}$ , of Constant codes give the fixed points, and finite nonconstant word codes give homoclinic orbits converging to a fixed point at both ends.

[L1]

Finite-word data (Constant codes give the fixed points, and finite nonconstant word codes give homoclinic orbits converging to a fixed point at both ends): a code that equals the constant code $\overset{ˉ}{u}$ outside a finite set has coded orbit converging to $(r_{u}, r_{u})$ at both ends, and $h (s) \neq (r_{u}, r_{u})$ for $s \neq \overset{ˉ}{u}$ ; distinct codes give distinct points.

[L2]

Conjugacy and injectivity (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 ${A C}_{ω}$ , Finite symbol windows determine the code: locality of the iterates, coding continuity and injectivity of $h (σ) = (x_{- 1} (σ), x_{0} (σ))$ ): $F^{n} (h (t)) = h (S_{\pm}^{n} t)$ for all $t$ and $n$ , and $h$ is injective; hence $F^{m} (h (s)) = h (s)$ would force $S_{\pm}^{m} s = s$ , and $h (s) = h (\overset{ˉ}{+ 1})$ would force $s = \overset{ˉ}{+ 1}$ .

[L3]

The constant code (Constant codes give the fixed points, and finite nonconstant word codes give homoclinic orbits converging to a fixed point at both ends, clause 1): $h (\overset{ˉ}{+ 1}) = (r_{+}, r_{+})$ with $r_{+} = 1 + \sqrt{101}$ .

[L4]

Signs (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, clause 4, 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): $sign x_{n} (t) = t_{n}$ for every code $t$ and every $n$ ; in particular $x_{0} (s) < 0 < r_{+}$ because $s_{0} = - 1$ and $r_{+} > 1 > 0$ .

Verification

technique · direct; combine the finite-word convergence with the injectivity of the coding to exclude periodicity

1.1givenalgebra

Nonperiodicity of the code: for $m \geq 1$ , the coordinate $0$ of $S_{\pm}^{m} s$ equals $s_{m} = + 1$ , while $s_{0} = - 1$ ; hence $S_{\pm}^{m} s \neq s$ . Also $s \neq \overset{ˉ}{+ 1}$ because $s_{0} = - 1$ .

2.1givenstep 1.1L2L4

The point differs from the fixed point: by [L4], $x_{0} (s) < 0 < r_{+}$ , so $h (s) \neq (r_{+}, r_{+})$ directly; alternatively this follows from the injectivity and $s \neq \overset{ˉ}{+ 1}$ in [L1], [L2].

2.2step 1.1L2

No return to the initial point: if $F^{m} (h (s)) = h (s)$ for some $m \geq 1$ , then $h (S_{\pm}^{m} s) = h (s)$ by [L2], so $S_{\pm}^{m} s = s$ by injectivity, contradicting step 1.1.

3.1step 2.1step 2.2L1L3∎

Two-sided convergence: the code $s$ equals $\overset{ˉ}{+ 1}$ outside ${0}$ , so [L1] applies with $u = + 1$ and gives $F^{n} (h (s)) \to (r_{+}, r_{+})$ as $n \to \pm \infty$ . Together with steps 2.1 and 2.2 this exhibits $h (s)$ as a nonperiodic homoclinic point of the fixed point $(r_{+}, r_{+})$ .

Remarks

The verification uses only the coding dictionary and the finite-word convergence; it does not compute the coordinates of $h (s)$ , which have no closed form.
The same argument applies to every nonconstant code that is constant outside a finite set: none of these coded orbits is periodic, and each converges to the corresponding fixed point in both time directions.

The single-flip code at $a = 100$ : a nonperiodic coded orbit converging to the fixed point at both ends

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources

The single-flip code at a=100a=100: a nonperiodic coded orbit converging to the fixed point at both ends

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources

The single-flip code at $a = 100$ : a nonperiodic coded orbit converging to the fixed point at both ends