Jumps of entropy for $C^r$ interval maps

Abstract

We study the jumps 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.