Length-expanding Lipschitz maps on totally regular continua

TL;DR

This paper constructs length-expanding Lipschitz maps on totally regular continua, demonstrating their existence and chaotic properties, which extends the dynamical systems theory on complex topological structures.

Contribution

It introduces length-expanding Lipschitz maps on totally regular continua and proves their existence, along with chaotic dynamics similar to the tent map.

Findings

Existence of LEL maps on all totally regular continua.

Construction of exactly Devaney chaotic maps with finite entropy.

Maps exhibit properties like dense periodicity and length-expansiveness.

Abstract

The tent map is an elementary example of an interval map possessing many interesting properties, such as dense periodicity, exactness, Lipschitzness and a kind of length-expansiveness. It is often used in constructions of dynamical systems on the interval/trees/graphs. The purpose of the present paper is to construct, on totally regular continua (i.e. on topologically rectifiable curves), maps sharing some typical properties with the tent map. These maps will be called length-expanding Lipschitz maps, briefly LEL maps. We show that every totally regular continuum endowed with a suitable metric admits a LEL map. As an application we obtain that every totally regular continuum admits an exactly Devaney chaotic map with finite entropy and the specification property.