Entropy-continuity for interval maps with holes

TL;DR

This paper investigates how the topological entropy of piecewise monotonic maps with holes changes under perturbations, demonstrating local Hölder continuity with an exponent depending on the entropy value.

Contribution

It establishes the local Hölder continuity of topological entropy for interval maps with holes under various perturbations, linking the Hölder exponent to the entropy itself.

Findings

Topological entropy varies locally Hölder continuously with respect to perturbations.

The Hölder exponent depends on the current value of the topological entropy.

Continuity results apply to perturbations like sliding or expanding holes.

Abstract

We study the dependence of the topological entropy of piecewise monotonic maps with holes under perturbations, for example sliding a hole of fixed size at uniform speed or expanding a hole with uniform expansion. We show that under suitable conditions the topological entropy varies locally Hoelder continuously with the local Hoelder exponent depending itself on the value of the topological entropy.