Shift space

In symbolic dynamics and related branches of mathematics, a shift space or subshift is a set of infinite words that represent the evolution of a discrete system. In fact, shift spaces and symbolic dynamical systems are often considered synonyms. The most widely studied shift spaces are the subshifts of finite type and the sofic shifts.

In the classical framework a shift space is any subset ${\displaystyle \Lambda }$ of ${\displaystyle A^{\mathbb {Z} }:=\{(x_{i})_{i\in \mathbb {Z} }:\ x_{i}\in A\ \forall i\in \mathbb {Z} \}}$, where ${\displaystyle A}$ is a finite set, which is closed for the Tychonov topology and invariant by translations. More generally one can define a shift space as the closed and translation-invariant subsets of ${\displaystyle A^{\mathbb {G} }}$, where ${\displaystyle A}$ is any non-empty set and ${\displaystyle \mathbb {G} }$ is any monoid.