The two-sign bilateral shift space, its coding map, and the ball of bounded bilateral real sequences

Write ${\mathrm{\Sigma }}_{\pm }:=\{-1,+1\}$ and

$X_{\pm }:={\mathrm{\Sigma }}_{\pm }^{\mathbb{Z}}=\{s:s\text{ is a function }\mathbb{Z}\to {\mathrm{\Sigma }}_{\pm }\},$

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 $s\in X_{\pm }$ write $s_k:=s(k)$. The elements of $X_{\pm }$ are called two-sign bilateral sequences.

Coding correspondence with the two-symbol space. Let

$\phi :X_{\pm }⟶\{0,1{\}}^{\mathbb{Z}},\phi (s{)}_k:=\frac{s_k+1}{2}\in \{0,1\},$

and let $X_2:=\{0,1{\}}^{\mathbb{Z}}$, $d_2$ and ${\sigma }_2$ be the set, the metric expression and the shift map of The bilateral two-symbol shift space, its metric, its shift map and its cylinder sets. The formulas $\phi (s{)}_k=(s_k+1)\mathrm{/}2$ and ${\phi }^{-1}(t{)}_k=2t_k-1$ define maps $X_{\pm }\to X_2$ and $X_2\to X_{\pm }$ that are inverse to each other, coordinate by coordinate.

Metric expression on $X_{\pm }$. Define

$D_{\pm }(s,t):=d_2(\phi (s),\phi (t))(s,t\in X_{\pm })\mathrm{.}$

Unwinding the series of The bilateral two-symbol shift space, its metric, its shift map and its cylinder sets and using $\mathrm{\mid }\phi (s{)}_k-\phi (t{)}_k\mathrm{\mid }=\frac{1}{2}\mathrm{\mid }{s}_k-t_k\mathrm{\mid }$ for every $k\in \mathbb{Z}$ gives the explicit expression

$D_{\pm }(s,t)={\sum }_{k\ge 0}{2}^{-(k+1)}\mathrm{\mid }{s}_k-t_k\mathrm{\mid }+{\sum }_{k\ge 1}{2}^{-(k+1)}\mathrm{\mid }{s}_{-k}-t_{-k}\mathrm{\mid }\mathrm{.}$

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), because $\mathrm{\mid }{s}_k-t_k\mathrm{\mid }\in \{0,2\}$ for every $k$; hence $D_{\pm }(s,t)$ is a well-defined nonnegative real number. Whether $D_{\pm }$ is a metric is not asserted here; it is proved in the companion lemma on this page, together with the statement that $\phi$ is an isometry of $(X_{\pm },D_{\pm })$ onto $(X_2,d_2)$.

The shift. The (bilateral) shift of $X_{\pm }$ is

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

For $m\in \mathbb{Z}$ its iterate satisfies $S_{\pm }^m(s{)}_k=s_{k+m}$; in particular $S_{\pm }$ is a bijection with inverse $S_{\pm }^{-1}(s{)}_k=s_{k-1}$, and $S_{\pm }={\phi }^{-1}\circ {\sigma }_2\circ \phi$.

Bounded bilateral sequences. A function $x:\mathbb{Z}\to \mathbb{R}$ is bounded when its range $\{x_k:k\in \mathbb{Z}\}$ is a bounded subset of $\mathbb{R}$ (Lower bound, bounded below, bounded set). Write

${\mathrm{\ell }}^{\mathrm{\infty }}(\mathbb{Z}):=\{x:\mathbb{Z}\to \mathbb{R}:x\text{ is bounded}\},\mathrm{\parallel }x{\mathrm{\parallel }}_{\mathrm{\infty }}:=\sup \{\mathrm{\mid }{x}_k\mathrm{\mid }:k\in \mathbb{Z}\}\mathrm{.}$

The set $\{\mathrm{\mid }{x}_k\mathrm{\mid }:k\in \mathbb{Z}\}$ whose supremum is taken is nonempty and bounded above (Lower bound, bounded below, bounded set), so $\mathrm{\parallel }x{\mathrm{\parallel }}_{\mathrm{\infty }}$ is a well-defined nonnegative real number (Suprema and infima are unique, Complete ordered field (least-upper-bound property)). For $R\ge 0$ put

$B_R:=\{x\in {\mathrm{\ell }}^{\mathrm{\infty }}(\mathbb{Z}):\mathrm{\parallel }x{\mathrm{\parallel }}_{\mathrm{\infty }}\le R\}\mathrm{.}$

Thus $B_R$ is the closed ball of radius $R$ in the supremum norm, and $\mathrm{\parallel }x{\mathrm{\parallel }}_{\mathrm{\infty }}\le R$ is by definition the statement that $\mathrm{\mid }{x}_k\mathrm{\mid }\le R$ for every $k\in \mathbb{Z}$ (Lower bound, bounded below, bounded set, Absolute value in an ordered field).

• The only difference between $(X_{\pm },D_{\pm },S_{\pm })$ and the $(X_2,d_2,{\sigma }_2)$ is the relabelling $0\leftrightarrow -1$, $1\leftrightarrow +1$ of the two symbols: $\phi$ is a bijection satisfying $\phi \circ S_{\pm }={\sigma }_2\circ \phi$. Every metric, compactness or chaos statement about the two-symbol shift therefore transfers through $\phi$ once it is known that $\phi$ is an isometry, which is the content 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 clause 3.

• No measure, probability or invariant measure on $X_{\pm }$ is introduced; all statements on this page are topological or metric.

• The space ${\mathrm{\ell }}^{\mathrm{\infty }}(\mathbb{Z})$ and its closed ball $B_R$ are used to state the square-root recurrence 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; nothing in this definition asserts that $B_R$ is complete, which is a separate lemma.