Summability implies Collet-Eckmann almost surely

TL;DR

This paper strengthens Jakobson's theorem by showing that for a generic family of interval maps satisfying summability, the parameters for which the Collet-Eckmann and polynomial recurrence conditions hold are densely distributed around the original map.

Contribution

It proves that the Collet-Eckmann and polynomial recurrence conditions are almost surely satisfied in a generic family of maps with summability, extending Jakobson's theorem.

Findings

Parameters satisfying Collet-Eckmann are dense near the original map.

Almost all maps in the family exhibit strong recurrence properties.

The result applies to a broad class of interval maps with summability.

Abstract

We provide a strengthened version of the famous Jakobson's theorem. Consider an interval map $f$ satisfying a summability condition. For a generic one-parameter family $f_t$ of maps with $f_0=f$, we prove that $t=0$ is a Lebesgue density point of the set of parameters for which $f_t$ satisfies both the Collect-Eckmann condition and a strong polynomial recurrence condition.