Definition:Metric/Shift of Finite Type
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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
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