Definition:Lipschitz Space
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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
 | This article is complete as far as it goes, but it could do with expansion. In particular: Can we have a page at some stage defining a "Lipschitz mapping"? You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Expand}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
 | This article, or a section of it, needs explaining. In particular: The nature of each of the many objects supporting this definition are lost several layers down in definitions and it's not a trivial exercise to identify what they are. For a start, I can't find $d_\theta$. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code. |
Also known as
It is also called the space of Lipschitz functions or space of Lipschitz mappings.
In all cases on $\mathsf{Pr} \infty \mathsf{fWiki}$, 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
- Results about Lipschitz spaces can be found here.
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