Renormalization in the Hénon family, I: Universality but non-rigidity

In this paper geometric properties of infinitely renormalizable real Hénon-like maps F in ℝ² 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.