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Renormalization in the Hénon Family, I: Universality But Non-Rigidity

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Abstract

In this paper geometric properties of infinitely renormalizable real Hénon-like maps F in \(\mathbb{R} ^2\) are studied. It is shown that the appropriately defined renormalizations R n F converge exponentially to the one-dimensional renormalization fixed point. The convergence to one-dimensional systems is at a super-exponen- tial rate controlled by the average Jacobian and a universal function a(x). It is also shown that the attracting Cantor set of such a map has Hausdorff dimension less than 1, but contrary to the one-dimensional intuition, it is not rigid, does not lie on a smooth curve, and generically has unbounded geometry

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Authors and Affiliations

  1. University of São Paolo, Brazil

    A. De Carvalho

  2. Stony Brook University and University of Toronto, Canada

    M. Lyubich

  3. University of Groningen, The Netherland

    M. Martens

Authors
  1. A. De Carvalho
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  2. M. Lyubich
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  3. M. Martens
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Correspondence to M. Lyubich.

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Dedicated to Mitchell Feigenbaum on the occasion of his 60th birthday

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Carvalho, A.D., Lyubich, M. & Martens, M. Renormalization in the Hénon Family, I: Universality But Non-Rigidity. J Stat Phys 121, 611–669 (2005). https://doi.org/10.1007/s10955-005-8668-4

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  • DOI: https://doi.org/10.1007/s10955-005-8668-4

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