Definition:Lipschitz Space

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

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

The space of Lipschitz mappings on $X _\mathbf A$ with respect to the metric $d_\theta$ is defined as:
 * $\ds\map {F_\theta} {X_\mathbf A} := \set {f \in \map C {X _\mathbf A, \C} : \sup _{n \mathop \in \N} \dfrac {\map {\mathrm {var}_n} f} {\theta ^n} < \infty }$

where:
 * $\map C {X _\mathbf A, \C}$ denotes the continuous mapping space
 * $\mathrm {var}_n$ denotes the $n$th variation

Also known as
It is also called Space of Lipschitz functions.

If no confusion, it is also denoted by $F_\theta$.

It is also written as $\struct {F_\theta, \norm \cdot_\theta}$ together with the norm.

Also see

 * Characterization of Lipschitz Continuity on Shift of Finite Type by Variations
 * Norm on Space of Lipschitz Functions