Entropy and exact Devaney chaos on totally regular continua

TL;DR

This paper investigates the topological entropy of exactly Devaney chaotic maps on totally regular continua, providing bounds and conditions for the existence of such maps with arbitrarily small entropy.

Contribution

It introduces P-Lipschitz maps and establishes new bounds for entropy, also identifying conditions under which these maps exhibit Devaney chaos with minimal entropy.

Findings

Upper bounds for topological entropy of P-Lipschitz maps.

Existence of Devaney chaotic maps with arbitrarily small entropy on certain continua.

Conditions involving free arcs and generalized stars for chaos existence.

Abstract

We study topological entropy of exactly Devaney chaotic maps on totally regular continua, i.e. on (topologically) rectifiable curves. After introducing the so-called P-Lipschitz maps (where P is a finite invariant set) we give an upper bound for their topological entropy. We prove that if a non-degenerate totally regular continuum X contains a free arc which does not disconnect X or if X contains arbitrarily large generalized stars then X admits an exactly Devaney chaotic map with arbitrarily small entropy. A possible application for further study of the best lower bounds of topological entropies of transitive/Devaney chaotic maps is indicated.