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

TL;DR

This paper proves that the Collet-Eckmann condition combined with slow recurrence conditions remains invariant under topological conjugacy for one-dimensional maps with multiple critical points, extending previous results.

Contribution

It demonstrates the topological invariance of the Collet-Eckmann condition with slow recurrence conditions for maps with multiple critical points, broadening the understanding of hyperbolicity invariance.

Findings

Collet-Eckmann condition with slow recurrence is topologically invariant

Extends previous invariance results to maps with multiple critical points

Provides a new proof applicable to complex settings

Abstract

This paper is devoted to study 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 invariance 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.