Renormalization in the Henon family, I: universality but non-rigidity

Details Renormalization in the Henon family, I: universality but non-rigidity Renormalization in the Henon family, I: universality but non-rigidity

2005-08-24

arxiv.org/abs/math/0508477v1

Mathematics

Dynamical Systems

42 pages, 5 pictures

Scientific paper

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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