Expansivity and Cone-fields in Metric Spaces

Abstract

Due to the results of Lewowicz and Tolosa expansivity can be characterized with the aid of Lyapunov function. In this paper we study a similar problem for uniform expansivity and show that it can be described using generalized cone-fields on metric spaces. We say that a function f : X → X is uniformly expansive on a set Λ ⊂ X if there exist ε > 0 and α ∈ (0,

Expansivity implies existence of Hölder continuous Lyapunov function

“…In [20] we showed that uniform expansivity implies classical expansivity. Moreover, we give a simple example which demonstrates that the opposite implication does not hold (see [20], Ex. 2).…”

“…In [15] Mazur shows a relationship between hyperbolic and cone-hyperbolic structures (which we introduced in [21]) in metric spaces. In [20] we used the same tool to describe uniform expansivity. We showed that a function f is uniformly expansive if and only if there exists a generalized cone-field such that f is cone-hyperbolic.…”

“…In Section 2 we introduce basic notations and definitions concerning generalization of cone-fields to metric spaces. We recall definitions of cone-fields, cone-hyperbolicity and uniformly expansivity [20,21]. We apply a theorem from [20,21] (see Theorem B) to obtain the main result of the paper, namely the construction of a Lyapunov function for a uniformly expansive system [see Section 5].…”

“…We recall definitions of cone-fields, cone-hyperbolicity and uniformly expansivity [20,21]. We apply a theorem from [20,21] (see Theorem B) to obtain the main result of the paper, namely the construction of a Lyapunov function for a uniformly expansive system [see Section 5]. In Sections 3 and 4 we show, by using a quasiconvex function [23], that the partial functions c u , c s needed to construct this Lyapunov function are Hölder continuous.…”

“…In this section we introduce basic notations and definitions concerning generalization of cone-fields to metric spaces. For the convenience of the reader we recall notation from [20,21].…”