PROBABILISTIC UNIVERSALITY IN TWO-DIMENSIONAL DYNAMICS

Abstract. In this paper we continue to explore infinitely renor-malizable Hénon maps with small Jacobian. It was shown in [CLM] that contrary to the one-dimensional intuition, the Cantor attrac-tor of such a map is non-rigid and the conjugacy with the one-dimensional Cantor attractor is at most 1/2-Hölder. Another for-mulation of this phenomenon is that the scaling structure of the Hénon Cantor attractor differs from its one-dimensional counter-part. However, in this paper we prove that the weight assigned by the canonical invariant measure to these bad spots tends to zero on microscopic scales. This phenomenon is called Probabilistic Uni-versality. It implies, in particular, that the Hausdorff dimension of the canonical measure is universal. In this way, universality and rigidity phenomena of one-dimensional dynamics assume a proba-bilistic nature in the two-dimensional world