The bilateral two-symbol shift space, its metric, its shift map and its cylinder sets

Let

$X:=\{0,1{\}}^{\mathbb{Z}}=\{s:s\text{ is a function }\mathbb{Z}\to \{0,1\}\}$

be the set of bi-infinite binary sequences, 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 $s\in X$ write $s_k:=s(k)$ for the value at the integer $k$. Define

$d(s,t):={\sum }_{k=0}^{\mathrm{\infty }}{2}^{-k}\mathrm{\mid }{s}_k-t_k\mathrm{\mid }+{\sum }_{k=1}^{\mathrm{\infty }}{2}^{-k}\mathrm{\mid }{s}_{-k}-t_{-k}\mathrm{\mid }(s,t\in X)\mathrm{.}$

Both series converge by comparison with the geometric series ${\sum }_{k\ge 0}{2}^{-k}=2$ (Series, partial sums, convergence and the sum, divergence, and the tail series, For $\mathrm{\mid }r\mathrm{\mid }<1$, ${\sum }_{k\ge 0}{r}^k=1\mathrm{/}(1-r)$, and for $\mathrm{\mid }r\mathrm{\mid }\ge 1$ the series diverges, Integer powers $a^m$, Absolute value in an ordered field), since $\mathrm{\mid }{s}_j-t_j\mathrm{\mid }\in \{0,1\}$ for every $j\in \mathbb{Z}$; hence $d(s,t)$ is a well-defined real number. That $d$ is a metric on $X$ is proved in the companion lemma The bilateral shift metric is a metric, cylinders are its basic balls, and the shift space is sequentially compact ↗ and is not part of this definition.

The (bilateral) shift is the map

$\sigma :X⟶X,\sigma (s{)}_k:=s_{k+1}(k\in \mathbb{Z},s\in X)\mathrm{.}$

For $n\ge 1$ the $n$-fold iterate is written ${\sigma }^n$, defined by ${\sigma }^1:=\sigma$ and ${\sigma }^{n+1}:=\sigma \circ {\sigma }^n$; a point $s\in X$ is periodic when ${\sigma }^n(s)=s$ for some $n\ge 1$.

A cylinder is a set of the form

$[w{]}_{p,q}:=\{s\in X:s_k=w_k\text{ for }p\le k\le q\},$

where $p\le q$ are integers and $w=(w_p,\dots ,w_q)\in \{0,1{\}}^{\{p,\dots ,q\}}$ is a finite binary word; the integers $p,q$ and the word $w$ are part of the data. When $p=q$ this is the set of sequences with prescribed value at the single coordinate $p$.

• The set $X$ is uncountable and the map $\sigma$ is a bijection with inverse ${\sigma }^{-1}(s{)}_k=s_{k-1}$; the definition itself asserts only the displayed formulas, while continuity of $\sigma$ is a theorem (The bilateral two-symbol shift is continuous, topologically mixing, has dense periodic points and is sensitive; it is chaotic in the stated sense).

• No measure, probability or invariant measure on $X$ is introduced by this definition. The example used in this collection is topological; statements about invariant measures for the one-sided fair-coin shift are separate published results and are not consumed here.

• Cylinders form a basis for the metric topology. More precisely, for $m\ge 0$ the centered cylinder $[s{]}_{-m,m}$ satisfies $B(s,2^{-m})\subseteq [s{]}_{-m,m}\subseteq \bar{B}(s,2^{1-m})$. An arbitrary cylinder $[w{]}_{p,q}$ contains $B(s,2^{-\max \{\mathrm{\mid }p\mathrm{\mid },\mathrm{\mid }q\mathrm{\mid }\}})$ about each of its points $s$, but need not have a comparable outer radius (The bilateral shift metric is a metric, cylinders are its basic balls, and the shift space is sequentially compact ↗, clause 2 and proof step 2.1).