Mathematics – Dynamical Systems
Scientific paper
2011-06-24
Mathematics
Dynamical Systems
69 pages, incl. references; 15 figures; continues arXiv:math/0508477 and arXiv:0804.0780
Scientific paper
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.
Lyubich Mikhail
Martens Marco
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