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Explicit mixing, periodic and sensitivity checks in the bilateral shift

Example

Let $X = {0, 1}^{Z}$ with the metric $d$ and the shift $σ$ of The bilateral two-symbol shift space, its metric, its shift map and its cylinder sets, and let cylinders be written $[w]_{p, q}$ .

A metric computation. Let $x \in X$ be arbitrary and let $y$ agree with $x$ on the coordinates $- 2, \dots, 2$ and differ at coordinate $3$ only. Then $y \in [x]_{- 2, 2}$ and $d (x, y) = 2^{- 3} = 1 / 8$ , which is at most $2^{1 - 2} = 1 / 2$ , in accordance with $[x]_{F_{m}} \subseteq \overline{B} (x, 2^{1 - m})$ .
A periodic point in a prescribed cylinder. The word $(1, 0, 1)$ on the coordinates $- 1, 0, 1$ extends to the $3$ -periodic sequence $s$ with $s_{- 1} = 1$ , $s_{0} = 0$ , $s_{1} = 1$ and $s_{k + 3} = s_{k}$ for all $k$ . Then $s \in [1, 0, 1]_{- 1, 1}$ and $σ^{3} (s) = s$ .
A sensitivity witness. For every $x \in X$ and every $k \geq 1$ , let $y$ agree with $x$ except at coordinate $k$ , where $y_{k} = 1 - x_{k}$ . Then $d (x, y) = 2^{- k}$ and $d (σ^{k} (x), σ^{k} (y)) \geq 1$ .
A mixing computation. For the cylinders $U = [1, 0]_{0, 1}$ and $V = [0]_{- 1, - 1}$ , one has $σ^{n} (U) \cap V \neq \emptyset$ for every $n \geq 2$ .

Facts & Assumptions

Given: The shift space $(X, d)$ , the shift $σ$ and the cylinder notation.

[L1]

A cylinder $[w]_{p, q}$ is the set of sequences with prescribed finite block; $d$ is a metric; and $σ^{n} (x)_{k} = x_{k + n}$ (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).

[L2]

The shift is topologically mixing, has dense periodic points and is sensitive with constant $1$ (The bilateral two-symbol shift is continuous, topologically mixing, has dense periodic points and is sensitive; it is chaotic in the stated sense, Topological mixing, dense periodic points, sensitive dependence, and the stated chaos list).

Verification

technique · direct; evaluate the definitions on the explicit sequences

1.1givenL1algebra

Since $y_{j} = x_{j}$ for $∣ j ∣ \leq 2$ , every term of $d (x, y)$ with $∣ j ∣ \leq 2$ vanishes; the only nonzero term is the one with $j = 3$ , equal to $2^{- 3} ∣ x_{3} - y_{3} ∣ = 2^{- 3}$ . Hence $d (x, y) = 1 / 8 \leq 2^{1 - 2}$ , which illustrates the second inclusion of The bilateral shift metric is a metric, cylinders are its basic balls, and the shift space is sequentially compact on a concrete pair.

1.2givenL1L2algebra

The displayed sequence $s$ is well defined because the prescription is periodic, it lies in the cylinder $[1, 0, 1]_{- 1, 1}$ by construction, and $σ^{3} (s)_{k} = s_{k + 3} = s_{k}$ for every $k$ , so $σ^{3} (s) = s$ ; thus $s$ is a periodic point of $σ$ and it lies in the prescribed cylinder, illustrating clause 3 of [L2] on an explicit block.

1.3givenL1L2algebra

For $y$ as in item 3, exactly one term of the defining sum of $d (x, y)$ is nonzero, so $d (x, y) = 2^{- k}$ . Moreover $σ^{k} (x)_{0} = x_{k} \neq y_{k} = σ^{k} (y)_{0}$ , so the term with $j = 0$ in the sum for $d (σ^{k} (x), σ^{k} (y))$ equals $1$ and all terms are nonnegative; hence $d (σ^{k} (x), σ^{k} (y)) \geq 1$ . This realizes the sensitivity constant $1$ of [L2] with the pair $(x, y)$ .

2.1givenL1L2algebra∎

Compute $σ^{n} (U) = {z : z_{- n} = 1, z_{1 - n} = 0}$ and $V = {z : z_{- 1} = 0}$ ; the intersection is nonempty exactly when the three coordinate prescriptions are consistent, which they are as soon as no two of the coordinates $- n$ , $1 - n$ , $- 1$ coincide with conflicting values. For $n = 1$ the prescriptions are $z_{- 1} = 1$ and $z_{- 1} = 0$ , a contradiction, so $σ (U) \cap V = \emptyset$ . For $n \geq 2$ : if $n = 2$ the coordinates are $- 2, - 1, - 1$ and the prescriptions are $z_{- 2} = 1$ , $z_{- 1} = 0$ , consistent; if $n \geq 3$ the three coordinates are distinct, so a sequence with the prescribed values exists. Hence $σ^{n} (U) \cap V \neq \emptyset$ for every $n \geq 2$ , as claimed, and the mixing condition of [L2] holds on this pair of basic open sets with $N = 2$ .

Remarks

Item 4 is a finite check of the "all sufficiently large $n$ " quantifier: the intersection fails at $n = 1$ but succeeds for every $n \geq 2$ . This is exactly the distinction between mixing and the weaker open-set transitivity, and no uniformity over all pairs of cylinders is claimed by the computation.
The example uses no measure on $X$ ; the statements verified are the topological ones of Topological mixing, dense periodic points, sensitive dependence, and the stated chaos list.

Depends on

Dependency tree · 2 levels

Sources

C. Ulcigrai, MATH36206/MATHM6206 Topological and Symbolic Dynamics, Sections 2.6-2.7: cylinders, periodic sequences obtained by repeating blocks, and Theorem 2.7.1