Probabilistic Universality in Two-Dimensional Dynamics

In this paper we continue to explore infinitely renormalizable Hénon maps with small Jacobian. It was shown in a previous paper by the authors, joint with A. de Carvalho, (J Stat Phys 121(5/6):611–669, 2005) that contrary to the one-dimensional intuition, the Cantor attractor of such a map is non-rigid and the conjugacy with the one-dimensional Cantor attractor is at most 1/2-Hölder. Another formulation of this phenomenon is that the scaling structure of the Hénon Cantor attractor differs from its one-dimensional counterpart. However, in this paper we prove that the unique invariant measure on the attractor assigns a weight to these bad spots which tends to zero on microscopic scales. This phenomenon is called Probabilistic Universality. It implies, in particular, that the Hausdorff dimension of the invariant measure on the attractor is universal. In this way, universality and rigidity phenomena of one-dimensional dynamics assume a probabilistic nature in the two-dimensional world.