Topological invariance of the Collet–Eckmann condition for one-dimensional maps

Abstract

This paper is devoted to studying the topological invariance of several non-uniform hyperbolicity conditions of one-dimensional maps. In contrast with the case of maps with only one critical point, it is known that for maps with several critical points the Collet–Eckmann condition is not in itself invariant under topological conjugacy. We show that the Collet–Eckmann condition together with any of several slow recurrence conditions is invariant under topological conjugacy. This extends and gives a new proof of a result by Luzzatto and Wang that also applies to the complex setting.