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.

Notation

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

$(\sigma(\mathbf x))(n)=x_{n+1}$ for all n.

In the following we choose $M=\mathbb N$ and thus speak of infinite words, but all definitions are naturally generalizable to the bi-infinite case.

Definition

A set of infinite words over A is a shift space if it is closed with respect to the natural product topology of $A^\mathbb N$ and invariant under the shift operator. Thus a set $S\subseteq A^\mathbb N$ is a subshift if and only if

1. for any (pointwise) convergent sequence $(\mathbf x_k)_{k\geq 0}$ of elements of S, the limit $\lim_{k\to\infty}\mathbf x_k$ also belongs to S; and
2. $\sigma(S)=S$.

A shift space S is sometimes denoted as $(S,\sigma)$ to emphasize the role of the shift operator.

Some authors[1] use the term subshift for a set of infinite words that is just invariant under the shift, and reserve the term shift space for those that are also closed.

Characterization and sofic subshifts

A subset S of $A^\mathbb N$ is a shift space if and only if there exists a set X of finite words such that S coincides with the set of all infinite words over A having no factor in X.

In particular, if X is finite then S is called a subshift of finite type and more generally when X is a regular language, the corresponding subshift is called sofic. The name "sofic" was coined by Weiss (1973), based on the Hebrew word סופי meaning "finite", to refer to the fact that this is a generalization of a finiteness property.[2]

Examples

The first trivial example of shift space (of finite type) is the full shift $A^\mathbb N$.

Let $A=\{a,b\}$. The set of all infinite words over A containing at most one b is a sofic subshift, not of finite type. The set of all infinite words over A whose b form blocks of prime length is not sofic (this can be shown by using the pumping lemma).