Statistical properties of unimodal maps: physical measures, periodic orbits and pathological laminations

In this work, we relate the geometry of chaotic attractors of typical analytic unimodal maps to the behavior of the critical orbit. Our main result is an explicit formula relating the combinatorics of the critical orbit with the exponents of periodic orbits. This connection between topological and smooth invariants is obtained through an analysis of the physical measure. Since the exponents of periodic orbits form a complete set of smooth invariants in this setting, we have ``typical geometric rigidity'' of the dynamics of such chaotic attractors. This unexpected result implies that the lamination structure of spaces of analytic maps (obtained by the partition into topological conjugacy classes, see \cite {ALM}) has an absolutely singular nature.