Definition:Lipschitz Seminorm

Definition

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

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

Let $\map {F_\theta} {X_\mathbf A}$ be the Lipschitz space on $X_\mathbf A$.

The Lipschitz seminorm of $f \in \map {F_\theta} {X_\mathbf A}$ is defined as:

$\ds \size f_\theta := \sup_{n \mathop \in \N} \dfrac {\map {\mathrm {var}_n} f} {\theta ^n}$

Also see

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

Source of Name

This entry was named for Rudolf Otto Sigismund Lipschitz.

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