Definition:Metric/Shift of Finite Type

Definition
Let $\struct {X _\mathbf A, \sigma_\mathbf A}$ be a shift of finite type.

Let $\theta \in \openint 0 1$.

Then the metric $d_\theta$ on $X _\mathbf A$ is defined by:
 * $\forall x, y \in X_\mathbf A : \map {d_\theta} {x, y} = \theta ^N$

where:
 * $ N := \sup \set {n \in \N \cup \set \infty: x_i = y_i \text { for all } i \in \openint {-n} n}$
 * $\theta ^\infty := 0$

Here we consider the suprema related to the extended natural numbers $\struct {\N \cup \set \infty, \le}$.

Also see

 * Metric on Shift of Finite Type is Metric
 * Metric on Shift of Finite Type is Non-Archimedean