Probabilistic Universality in two-dimensional Dynamics

TL;DR

This paper demonstrates that in two-dimensional Hénon maps with small Jacobian, the measure of irregular regions diminishes at microscopic scales, leading to a universal Hausdorff dimension of the invariant measure, thus revealing probabilistic universality.

Contribution

It introduces the concept of Probabilistic Universality, showing that the measure of irregularities tends to zero, and establishes the universality of the Hausdorff dimension in two-dimensional dynamics.

Findings

The measure of bad spots tends to zero on microscopic scales.

The Hausdorff dimension of the invariant measure is universal.

Two-dimensional dynamics exhibit probabilistic universality phenomena.

Abstract

In this paper we continue to explore infinitely renormalizable H\'enon maps with small Jacobian. It was shown in [CLM] 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-H\"older. Another formulation of this phenomenon is that the scaling structure of the H\'enon Cantor attractor differs from its one-dimensional counterpart. 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 {\it Probabilistic Universality}. 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 probabilistic nature in the two-dimensional world.