Statistical properties of unimodal maps: the quadratic family

2000/10/06 by Artur Avila, Avila, Artur, Carlos Gustavo Moreira +1
Mathematics · #37E05 #37F10 #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS #msc:37E05 #msc:37F10

paper · pdf · doi:10.48550/arxiv.math/0010062

42 pages, no figures, fifth version, to appear in Annals of Mathematics

arxiv created 2003/06/10 · arxiv updated 2009/11/30

Abstract

We prove that almost every non-regular 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 to get the same results for more general families of unimodal maps.

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