Vivification

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 $𝐱=(x_n{)}_{n\in M}$, where $M=\mathbb{N}$ (resp. $M=\mathbb{Z}$) and $x_n$ is in A for any integer n. The shift operator $\sigma$ acts on an infinite or bi-infinite word by shifting all symbols to the left, i.e.,

$(\sigma (𝐱))(n)=x_{n+1}$ 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 $({𝐱}_k{)}_{k\ge 0}$ of elements of S, the limit $\underset{k\to \mathrm{\infty }}{lim}{𝐱}_k$ 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 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 Template:Harvtxt, based on the Hebrew word סופי meaning "finite", to refer to the fact that this is a generalization of a finiteness property.

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).

Further reading

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