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.

Let A be a finite set of states. An infinite (respectively bi-infinite) word over A is a sequence ${\displaystyle \mathbf {x} =(x_{n})_{n\in M}}$, where ${\displaystyle M=\mathbb {N} }$ (resp. ${\displaystyle M=\mathbb {Z} }$) and ${\displaystyle x_{n}}$ is in A for any ${\displaystyle n\in M}$. The shift operator ${\displaystyle \sigma }$ acts on an infinite or bi-infinite word by shifting all symbols to the left, i.e.,

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