Renormalization of Hénon Maps

Period doubling cascades are observed at transition to chaos in many models used in the sciences and in physical experiments. These period doubling cascades are very well understood in one-dimensional dynamics. In particular, the microscopic geometrical properties of the attractors do not depend on the actual system, they are universal.Moreover, the attractors of two different maps are smoothly conjugate, they are rigid. Strongly dissipative Hénon maps describe parts of the dynamics of systems close to a homoclinic tangency and are often observed in various models. For these maps the transition to positive entropy also occurs along period doubling cascades. These strongly dissipative Hénon maps can be considered as perturbations of one-dimensional systems. Indeed, some of the universal geometrical properties of the one-dimensional systems are present in the Hénon maps. However, they appear in a much more delicate form: in a probabilistic sense the geometry of the Hénon attractors is the same as their one-dimensional counter part. This phenomenon is revered to as probabilistic universality and rigidity.