Expansiveness, Lyapunov exponents and entropy for set valued maps

TL;DR

This paper extends the concepts of expansiveness, Lyapunov exponents, and entropy to set valued maps on topological spaces, establishing new links between stability, chaos, and topological complexity.

Contribution

It introduces a new notion of expansiveness for set valued maps and relates Lyapunov exponents to topological entropy, providing novel theoretical insights.

Findings

Expansiveness for set valued maps implies positive topological entropy on Peano spaces.

Positivity of Lyapunov exponents leads to positive topological entropy.

Stable points are either empty or constitute the entire space on Peano spaces.

Abstract

In this paper we introduce a notion of expansiveness for a set valued map defined on a topological space different from that given by Richard Williams at \cite{Wi, Wi2} and prove that the topological entropy of an expansive set valued map defined on a Peano space of positive dimension is greater than zero. We define Lyapunov exponent for set valued maps and prove that positiveness of its Lyapunov exponent implies positiveness for the topological entropy. Finally we introduce the definition of (Lyapunov) stable points for set valued maps and prove a dichotomy for the set of stable points for set valued maps defined on Peano spaces: either it is empty or the whole space.