Definition:Space of Lipschitz Functions/One-Sided Shift of Finite Type
Jump to navigation
Jump to search
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