Statistical properties of unimodal maps: smooth families with negative Schwarzian derivative

TL;DR

This paper proves that in smooth families of unimodal maps with negative Schwarzian derivative, most parameters exhibit Collet-Eckmann dynamics with predictable statistical properties, confirming a conjecture in this area.

Contribution

It establishes that a residual set of such families have almost all non-regular parameters with robust statistical behavior, confirming the Palis conjecture for this class.

Findings

Almost every non-regular parameter is Collet-Eckmann.

Critical orbit exhibits subexponential recurrence.

Results confirm the Palis conjecture in this setting.

Abstract

We prove that there is a residual set of families of smooth or analytic unimodal maps with quadratic critical point and negative Schwarzian derivative such that almost every non-regular parameter is Collet-Eckmann with subexponential recurrence of the critical orbit. Those conditions lead to a detailed and robust statistical description of the dynamics. This proves the Palis conjecture in this setting.