## Institute for Mathematical Sciences

## Preprint ims05-07

** A. de Carvalho, M. Lyubich, M. Martens**
* Renormalization in the Henon family, I: universality but non-rigidity*

Abstract: In this paper geometric properties of infinitely renormalizable real H\'enon-like maps $F$ in $\R^2$ 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-exponential 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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