# Definition:Lipschitz Space

Jump to navigation
Jump to search

## 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