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The bilateral two-symbol shift is continuous, topologically mixing, has dense periodic points and is sensitive; it is chaotic in the stated sense

Statement

Let $X = {0, 1}^{Z}$ , let $d$ be the metric of The bilateral shift metric is a metric, cylinders are its basic balls, and the shift space is sequentially compact and let $σ : X \to X$ , $σ (s)_{k} = s_{k + 1}$ , be the bilateral shift of The bilateral two-symbol shift space, its metric, its shift map and its cylinder sets. Then:

$σ$ is continuous (Continuity of a map between metric spaces, at a point and globally, in the $ε$ - $δ$ form);
$σ$ is topologically mixing;
the periodic points of $σ$ are dense in $X$ ;
$σ$ has sensitive dependence on initial conditions with sensitivity constant $δ = 1$ ;

consequently $σ$ is chaotic in the stated sense (Topological mixing, dense periodic points, sensitive dependence, and the stated chaos list).

Facts & Assumptions

Given: The shift space $(X, d)$ , the shift map $σ$ and the cylinders $[w]_{p, q} = {s : s_{k} = w_{k} for p \leq k \leq q}$ .

[F1]

$d$ is a metric; every cylinder is open, and every nonempty open subset of $X$ contains a cylinder (The bilateral shift metric is a metric, cylinders are its basic balls, and the shift space is sequentially compact, 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).

[F2]

$σ$ is a bijection with $σ (s)_{k} = s_{k + 1}$ and inverse $σ^{- 1} (s)_{k} = s_{k - 1}$ ; for $m \geq 1$ , $σ^{m} (s)_{k} = s_{k + m}$ (The bilateral two-symbol shift space, its metric, its shift map and its cylinder sets).

[F3]

For $s, t \in X$ , the defining series of $d$ are nonnegative and summable with $\sum_{k \geq 0} 2^{- k} = 2$ and $\sum_{k \geq m + 1} 2^{- k} = 2^{- m}$ (The bilateral two-symbol shift space, its metric, its shift map and its cylinder sets, For $∣ r ∣ < 1$ , $\sum_{k \geq 0} r^{k} = 1 / (1 - r)$ , and for $∣ r ∣ \geq 1$ the series diverges, A series of nonnegative terms converges iff its partial sums are bounded, and then the sum is their supremum, Series, partial sums, convergence and the sum, divergence, and the tail series).

[F4]

A map $g : X \to X$ is continuous at $s$ when for every $ε > 0$ there is $δ > 0$ with $g (B (s, δ)) \subseteq B (g (s), ε)$ (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).

[F5]

A subset $D \subseteq X$ is dense when every nonempty open subset meets $D$ (Dense, nowhere dense and codense subsets of a topological space, and the criterion by basic open sets).

[F6]

The list of properties defining "chaotic in the stated sense" is in Topological mixing, dense periodic points, sensitive dependence, and the stated chaos list: continuity, topological mixing, dense periodic points and sensitive dependence.

Proof

technique · direct; use cylinders as a basis and prescribe finitely many coordinates

1.1givenF1F2F3F4algebra

For $s, t \in X$ and any $m \geq 0$ , if $s_{j} = t_{j}$ for $∣ j ∣ \leq m + 1$ , then $σ (s)_{j} = σ (t)_{j}$ for $∣ j ∣ \leq m$ ; hence $σ ([s]_{F_{m + 1}}) \subseteq [σ (s)]_{F_{m}}$ . By clause 2 of [F1], $B (s, 2^{- (m + 1)}) \subseteq [s]_{F_{m + 1}}$ and $[σ (s)]_{F_{m}} \subseteq \overline{B} (σ (s), 2^{1 - m})$ , so $d (s, t) < 2^{- (m + 1)}$ implies $d (σ s, σ t) \leq 2^{1 - m}$ ; given $ε > 0$ choose $m$ with $2^{1 - m} < ε$ and $δ = 2^{- (m + 1)}$ . This proves continuity at every $s$ , which is clause 1.

1.2givenF1F2algebra

Let $U, V \subseteq X$ be nonempty and open. By [F1] there are cylinders $[w]_{p, q} \subseteq U$ and $[w^{'}]_{p^{'}, q^{'}} \subseteq V$ . Let $n$ be an integer with $(p, \dots, q) \cap (p^{'} + n, \dots, q^{'} + n) = \emptyset$ ; this holds for every $n > q - p^{'}$ . Define $s \in X$ by $s_{k} : = w_{k}$ for $p \leq k \leq q$ , $s_{k} : = w_{k - n}^{'}$ for $p^{'} + n \leq k \leq q^{'} + n$ , and $s_{k} : = 0$ for all remaining $k$ ; the two prescriptions are compatible because their index sets are disjoint. Then $s \in [w]_{p, q}$ , and for $j \in [p^{'}, q^{'}]$ one has $σ^{n} (s)_{j} = s_{j + n} = w_{j}^{'}$ , so $σ^{n} (s) \in [w^{'}]_{p^{'}, q^{'}}$ . Hence $σ^{n} (U) \cap V \neq \emptyset$ for every $n > q - p^{'}$ , which exhibits the integer $N$ required by topological mixing.

1.3givenF1F2algebra

Let $U \subseteq X$ be nonempty and open and choose a cylinder $[w]_{p, q} \subseteq U$ by [F1]; put $L : = q - p + 1 \geq 1$ and define $s \in X$ by $s_{k} : = w_{p + ((k - p) m o d L)}$ , the $L$ -periodic extension of the word $w$ . Then $s_{k} = w_{k}$ for $p \leq k \leq q$ , so $s \in [w]_{p, q} \subseteq U$ , and $s_{k + L} = s_{k}$ for every $k$ ; by [F2], $σ^{L} (s)_{k} = s_{k + L} = s_{k}$ for all $k$ , so $s$ is a periodic point of $σ$ . Thus every nonempty open set meets $Per (σ)$ , and [F5] gives clause 3.

1.4givenF2F3algebra

Let $x \in X$ and $ε > 0$ ; choose an integer $k \geq 1$ with $2^{- k} < ε$ and define $y \in X$ by $y_{j} : = x_{j}$ for $j \neq k$ and $y_{k} : = 1 - x_{k}$ . Then $d (x, y) = 2^{- k} < ε$ because exactly one term of the defining sum is nonzero and equal to $2^{- k}$ . Moreover $σ^{k} (x)_{0} = x_{k} \neq y_{k} = σ^{k} (y)_{0}$ , so in the defining sum for $d (σ^{k} x, σ^{k} y)$ the term with index $0$ equals $2^{0} \cdot 1 = 1$ and all terms are nonnegative; hence $d (σ^{k} x, σ^{k} y) \geq 1$ . Therefore $δ = 1$ is a sensitivity constant, which is clause 4.

2.1step 1.11.21.31.4F6∎

Clauses 1-4 and [F6] give that $σ$ is continuous, topologically mixing, has dense periodic points and is sensitive; by definition, $σ$ is chaotic in the stated sense.

Remarks

Compactness of $(X, d)$ is not used in the proof: mixing, density of periodic points and sensitivity are checked directly from cylinders. The companion lemma The bilateral shift metric is a metric, cylinders are its basic balls, and the shift space is sequentially compact proves that $X$ is sequentially compact, and compact under the Axioms of Countable and Dependent Choice, so the example also matches the usual convention in which a chaotic system is a continuous self-map of a compact metric space.
The integer $k$ in step 1.4 is positive, so the witness sequence $y$ is obtained by flipping a coordinate in the future tail; flipping a coordinate at index $k \leq 0$ would instead give $d (x, y) = 2^{- ∣ k ∣}$ but the displayed iterate argument uses $σ^{k}$ with $k \geq 1$ . The sensitivity constant $1$ is not claimed to be optimal: any $0 < δ \leq 1$ is also a sensitivity constant, and no statement is made about the largest such constant.

The bilateral two-symbol shift is continuous, topologically mixing, has dense periodic points and is sensitive; it is chaotic in the stated sense

Statement

Facts & Assumptions

Proof

Remarks

Depends on

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