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Expansivity implies existence of Hölder continuous Lyapunov function

  • * Corresponding author: Łukasz Struski

    * Corresponding author: Łukasz Struski 
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  • 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.

    Mathematics Subject Classification: Primary:37D20.


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

    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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