Definition:Space of Lipschitz Functions/One-Sided Shift of Finite Type

Definition

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

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

The space of Lipschitz functions 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

Also written as $F_\theta ^+$.

Also see

• Characterization of Lipschitz Continuity on One-Sided Shift of Finite Type by Variations
• Norm on Space of Lipschitz Functions

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