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We prove that almost every nonregular real quadratic map is ColletEckmann 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. | Annals of Mathematics Statistical properties of unimodal maps the quadratic family By Artur Avila and Carlos Gustavo Moreira Annals of Mathematics 161 2005 831 881 Statistical properties of unimodal maps the quadratic family By Artur Avila and Carlos Gustavo Moreira 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. Contents Introduction 1. General definitions 2. Real quadratic maps 3. Measure and capacities 4. Statistics of the principal nest 5. Sequences of quasisymmetric constants and trees 6. Estimates on time 7. Dealing with hyperbolicity 8. Main theorems Appendix Sketch of the proof of the phase-parameter relation References Introduction Here we consider the quadratic family fa a x2 where -1 4 a 2 is the parameter and we analyze its dynamics in the invariant interval. The quadratic family has been one of the most studied dynamical systems in the last decades. It is one of the most basic examples and exhibits very Partially supported by Faperj and CNPq Brazil. 832 ARTUR AVILA AND CARLOS GUSTAVO MOREIRA rich behavior. It was also studied through many different techniques. Here we are interested in describing the dynamics of a typical quadratic map from the statistical point of view. 0.1. The probabilistic point of view in dynamics. In the last decade Palis Pa described a general program for dissipative dynamical systems in any dimension. In short he shows that typical dynamical systems can be modeled stochastically in a robust way. More precisely one should show that such typical systems can be described by finitely many attractors each of them .
