Statistical properties of unimodal maps: the quadratic family


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.


Artur Ávila

CNRS UMR 7599, Université Pierre et Marie Curie, 75005 Paris, France

Carlos Gustavo Moreira

IMPA, Rio de Janeiro, 22460-320, Brazil