Equivalence of Collet-Eckmann conditions for slowly recurrent rational maps

TL;DR

This paper demonstrates that for slowly recurrent rational maps on the Riemann sphere, key dynamical conditions are equivalent and preserved under topological conjugacy, simplifying the understanding of their complex dynamics.

Contribution

It establishes the equivalence and invariance of Collet-Eckmann conditions within a specific family of rational maps, clarifying their dynamical properties.

Findings

Collet-Eckmann, second Collet-Eckmann, and topological Collet-Eckmann conditions are equivalent

These conditions are invariant under topological conjugacy

Simplifies classification of slowly recurrent rational maps

Abstract

In this short note we observe that within the family of slowly recurrent rational maps on the Riemann sphere, the Collet-Eckmann, second Collet-Eckmann, and topological Collet-Eckmann conditions are equivalent and also invariant under topological conjugacy.