Two counterexamples in rational and interval dynamics

TL;DR

This paper presents two counterexamples in rational and interval dynamics, demonstrating limitations of existing theorems and conjectures regarding Collet-Eckmann properties and their invariance.

Contribution

It constructs specific examples showing that certain Collet-Eckmann conditions do not imply each other or topological invariance, challenging previous assumptions.

Findings

Existence of a polynomial satisfying Topological Collet-Eckmann but with a non-Collet-Eckmann recurrent critical orbit

Collet-Eckmann property is not a topological invariant for real polynomials with negative Schwarzian derivative

Counterexamples contradict previous theorems and conjectures in dynamical systems

Abstract

In rational dynamics, we prove the existence of a polynomial that satisfies the Topological Collet-Eckmann condition, but which has a recurrent critical orbit that is not Collet-Eckmann. This shows that the converse of the main theorem in [11] does not hold. In interval dynamics, we show that the Collet-Eckmann property for recurrent critical orbits is not a topological invariant for real polynomials with negative Schwarzian derivative. This contradicts a conjecture of Swiatek [22].