  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.
