Statistical properties of unimodal maps: smooth families with negative Schwarzian derivative

Abstract

We prove that there is a residual set of families of smooth or analytic unimodal maps with quadratic critical point and negative Schwarzian derivative such that almost every non-regular parameter is Collet-Eckmann with subexponential recurrence of the critical orbit. Those conditions lead to a detailed and robust statistical description of the dynamics. This proves the Palis conjecture in this setting.

33 pages, no figures, third version, to appear in Astérisque

Cited by in corpus (8)

• Fractional susceptibility functions for the quadratic family: Misiurewicz-Thurston parameters
• Normal forms and Misiurewicz renormalization for dissipative surface diffeomorphisms
• Shy shadows of infinite-dimensional partially hyperbolic invariant sets
• Bifurcations of unimodal maps
• Quasisymmetric robustness of the Collet-Eckmann condition in the quadratic family
• A parameter ASIP for the quadratic family
• Non-uniform hyperbolicity in polynomial skew products
• Physical measures for infinite-modal maps