Shift space

Remarks

The space $\mathcal{A}^{\Z}$ is compact, with the metric $d(x, y) = 2^{-\min\set{\abs{i} : x_i \neq y_i}}$ for $x \neq y$, and $\sigma$ is a homeomorphism; so $\sigma$ restricts to a homeomorphism of each shift space. The whole space $\mathcal{A}^{\Z}$ is the full shift. Equivalent form: $X$ is a shift space if and only if it is defined by a set $\mathcal{F}$ of forbidden words. Here $\mathcal{F}$ is a set of finite words, elements of the free monoid on $\mathcal{A}$. Then $X$ is the set of sequences in which no word of $\mathcal{F}$ occurs as a block of consecutive symbols. One-sided shift spaces are defined in the same way in $\mathcal{A}^{\N}$, with $\sigma(X) \subseteq X$. Example: the golden mean shift $X \subseteq \set{0, 1}^{\Z}$, the sequences with no block $11$. A limit of such sequences has no block $11$, since a block $11$ at positions $i, i+1$ would appear in the sequences close to it. So $X$ is closed, and shifting does not create or remove blocks, so $\sigma(X) = X$. Non-example: the set of sequences in $\set{0, 1}^{\Z}$ with exactly one symbol $1$ satisfies clause 2, but not clause 1. Its elements with the $1$ at position $n$ tend to the all-$0$ sequence as $n \to \infty$, which is not in the set.