Henon renormalization I
Strongly dissipative Henon maps can be considered as
perturbations of unimodal maps on the interval. Families of such
maps exhibit period doubling cascades which have some universal
metric properties also observed in the unimodal period doubling cascades.
Although strongly dissipative Henon maps are perturbations, the global
geometry of infinitely renormalizable Henon maps is surprisingly
different from to the geometry of infinitely renormalizable unimodal
maps. For example, the Cantor attractors are non-rigid, maps with
different average Jacobian are at most Holder conjugated. The Cantor
attractors are not contained in a smooth curve, the maps can not be
understood as one-dimensional unimodal maps.
However, the geometry of Henon period doubling Cantor attractors can be
understood in a probabilistic sense with respect to the unique invariant
measure on the attractor. The asymptotic scaling structure in almost
every point is the same as scaling structure of the corresponding point
in the unimodal Cantor attractor. This phenomenon is called
probabilistic universality. Similarly there is the phenomenon of
probabilistic rigidity. In particular, the Hausdorff dimension of the
invariant measure is universal.
The theory of universality and rigidity in one-dimensional dynamics
became a probabilistic geometric theory for Henon dynamics.
The talk consist of two parts: the first describes the phenomena and the
second will address the tools and techniques.