Expansivity implies existence of Hölder continuous Lyapunov function

Łukasz Struski; Jacek Tabor

doi:10.3934/dcdsb.2017180

Article Contents

2017, Volume 22, Issue 9: 3575-3589. Doi: 10.3934/dcdsb.2017180

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Expansivity implies existence of Hölder continuous Lyapunov function

Łukasz Struski^, and
Jacek Tabor

Faculty of Mathematics and Computer Science, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland

* Corresponding author: Łukasz Struski
* Corresponding author: Łukasz Struski

Received: July 31, 2016

Revised: April 30, 2017

Published: October 2017

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Abstract

The Lyapunov function is a very useful tool in the theory of dynamical systems, in particular in the study of the stability of an equilibrium point. In this paper we construct a locally Hölder continuous Lyapunov function for a uniformly expansive set for a map f in a metric space X. In the construction a basic role is played by the functions defining the stable and unstable cone-fields. As a tool we also use the approximately quasiconvex functions.

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Mathematics Subject Classification: Primary:37D20.

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Figure 1. Sequences $\alpha, \alpha'\in\Phi(K,L,\varepsilon)$ restricted to the set $\text{dom}_\varepsilon(\alpha)\cap\text{dom}_\varepsilon(\alpha')=[0,n]_Z$, $n\in{\bar{\mathbb{N}}}$

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Figure 2. All possible situations in Case Ⅲ (see Figure 1(c)) depending on the position of $\tilde{n}$, where $\text{dom}_e(\alpha)\cap\text{dom}_e(\alpha')=[0,n]_Z$

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