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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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