Statistical properties of unimodal maps: the quadratic family

Abstract

We prove that almost every nonregular real quadratic map is Collet-Eckmann and has polynomial recurrence of the critical orbit (proving a conjecture by Sinai). It follows that typical quadratic maps have excellent ergodic properties, as exponential decay of correlations (Keller and Nowicki, Young) and stochastic stability in the strong sense (Baladi and Viana). This is an important step in achieving the same results for more general families of unimodal maps.

Authors

Artur Ávila

CNRS UMR 7599
Université Pierre et Marie Curie
75005 Paris
France

Carlos Gustavo Moreira

IMPA
Rio de Janeiro, 22460-320
Brazil