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November  2017, 22(9): 3575-3589. doi: 10.3934/dcdsb.2017180

## Expansivity implies existence of Hölder continuous Lyapunov function

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

* Corresponding author: Łukasz Struski

Received  August 2016 Revised  May 2017 Published  July 2017

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.

Citation: Łukasz Struski, Jacek Tabor. Expansivity implies existence of Hölder continuous Lyapunov function. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3575-3589. doi: 10.3934/dcdsb.2017180
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##### References:
Sequences $\alpha, \alpha'\in\Phi(K,L,\varepsilon)$ restricted to the set $\text{dom}_\varepsilon(\alpha)\cap\text{dom}_\varepsilon(\alpha')=[0,n]_Z$, $n\in{\bar{\mathbb{N}}}$
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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