  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.
