Definition:Variation of Function/One-Sided Shift of Finite Type

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

Let $f : X_\mathbf A ^+ \to \C$ be a continuous function.

Let $n \in \N$.

The $n$th variation of $f$ is defined as:
 * $\map {\mathrm{var}_n} f := \sup \set {\size {\map f x - \map f y} : x, y \in X_\mathbf A ^+, \; \forall i \in \hointr 0 n : x_i = y_i}$

Also see

 * Characterization of Lipschitz Continuity on One-Sided Shift of Finite Type by Variations