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 Lipschitz space 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 the space of Lipschitz functions or space of Lipschitz mappings.

In all cases on, the term mapping is preferred over function.

If no confusion can arise, the Lipschitz space can also be denoted by $F_\theta$.

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

Also see

 * Characterization of Lipschitz Continuity on Shift of Finite Type by Variations


 * Definition:Lipschitz Norm