# Definition:Shift of Finite Type

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## Definition

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

Let $\set {1, 2, \ldots, k}$ be given the discrete topology.

Let $X _\mathbf A$ be the subspace of the product space $\set {1, 2, \ldots, k} ^\Z$ defined as:

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

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**.

## Also known as

$\struct {X _\mathbf A, \sigma_\mathbf A}$ is also called **two-sided shift of finite type** as well as **topological Markov chain**.

The mapping $\sigma_\mathbf A : X_\mathbf A \to X_\mathbf A$ is also called simply **shift operator** as well as **shift**.

## Also see

- Definition:Variation of Function/Shift of Finite Type
- Definition:Lipschitz Space
- Definition:Lipschitz Seminorm
- Shift of Finite Type is Metrizable

## Sources

- 1990: William Parry and Mark Pollicott:
*Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics*: Chapter $1$: Subshifts of Finite Type and Function Spaces