Journal of Statistical Physics, Vol. 121, Nos. 5/6, December 2005. p. 611-669.
DOI: 10.1007/s10955-005-8668-4
In this paper geometric properties oi infinitely renormahzable real Henon-like maps F in R^2 are studied. It is shown that the appropriately defined renormal-izations 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.