Characterization of Lipschitz Continuity on Shift of Finite Type by Variations

Theorem
Let $\struct {X_\mathbf A, \sigma_\mathbf A}$ be a shift of finite type.

Let $f : X_\mathbf A \to \C$ be a continuous mapping.

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

Let $C > 0$.

Let $x, y \in X_\mathbf A$.

Then:
 * $\size {\map f x - \map f y} \le C \map {d_\theta} {x, y}$


 * $\forall n \in \N : \map {\mathrm {var}_n} f \le C \theta ^n$
 * $\forall n \in \N : \map {\mathrm {var}_n} f \le C \theta ^n$

where:
 * $d_\theta$ denotes the metric
 * $\mathrm {var}_n$ denotes the $n$th variation.