Journal of Statistical Physics

, Volume 121, Issue 5–6, pp 611–669 | Cite as

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

  • A. De Carvalho
  • M. Lyubich
  • M. Martens


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


Physical Chemistry Statistical Physic Geometric Property Smooth Curve Hausdorff Dimension 
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© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.University of São PaoloBrazil
  2. 2.Stony Brook University and University of TorontoCanada
  3. 3.University of GroningenThe Netherland

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