Definition:Shift of Finite Type

Definition
Let $\mathbf A = \sqbrk a_k$ be a logical matrix for a $k \in \Z: k \ge 2$.

Let:
 * $X_\mathbf A = \set {x = \sequence {x_n}_{n \mathop \in \Z} : x_n \in \set {1, 2, \ldots, k}, a_{x_n, x_{n + 1} } = 1}$

Let $\sigma_\mathbf A : X_\mathbf A \to X_\mathbf A$ be the forward shift operator, that is:
 * $\map {\sigma _{\mathbf A} } x := y$

where $y_n = x_{n + 1}$ for all $n \in \Z$.

Then the pair $\struct {X _\mathbf A, \sigma_\mathbf A}$ is called a shift of finite type.

Topology
$X _\mathbf A$ is given the subspace topology of the product topological space $\set {1, 2, \ldots, k} ^\Z$, where $\set {1, 2, \ldots, k}$ is given the discrete topology.

Also known as
This is also called two-sided shift of finite type.

Also see

 * Shift of Finite Type is Compact
 * Definition:Metric/Shift of Finite Type
 * Shift of Finite Type is Metrizable