Jumps of entropy for $C^r$ interval maps

TL;DR

This paper investigates the continuity properties of topological entropy for $C^r$ interval maps, establishing conditions under which entropy remains continuous and analyzing related diagram entropy.

Contribution

It provides new results on the continuity of topological entropy for $C^r$ maps, especially at points where entropy exceeds a specific threshold, using Buzzi-Hofbauer diagrams.

Findings

Entropy is continuous at $f$ if $h_{top}(f) > rac{ ext{log}^+ orm{f'}_ty}{r}$.

Continuity of entropy is linked to the properties of associated Buzzi-Hofbauer diagrams.

The study offers conditions ensuring entropy stability for smooth interval maps.

Abstract

We study the jump of topological entropy for $C^r$ interval or circle maps. We prove in particular that the topological entropy is continuous at any $f \in C^r([0; 1])$ with $h_{top}(f) > \frac{\log^+ \|f'\|_\infty}{r}$. To this end we study the continuity of the entropy of the Buzzi-Hofbauer diagrams associated to $C^r$ interval maps.