Renormalization of $C^r$ Hénon map : Two dimensional embedded map in three dimension

Abstract

We study renormalization of highly dissipative analytic three dimensional Hénon maps $$F(x,y,z)=(f(x)-\epsilon (x,y,z),x,\delta (x,y,z))$$ where $\epsilon (x,y,z)$ is a sufficiently small perturbation of ${\epsilon }_{2d}(x,y)$. Under certain conditions, $C^r$ single invariant surfaces each of which is tangent to the invariant plane field over the critical Cantor set exist for $2\le r<\infty$. The $C^r$ conjugation from an invariant surface to the $xy-$plane defines renormalization two dimensional $C^r$ Hénon-like map. It also defines two dimensional embedded $C^r$ Hénon-like maps in three dimension. In this class, universality theorem is re-constructed by conjugation. Geometric properties on the critical Cantor set in invariant surfaces are the same as those of two dimensional maps --- non existence of the continuous line field and unbounded geometry. The set of embedded two dimensional Hénon-like maps is open and dense subset of the parameter space of average Jacobian, $b_{F_{2d}}$ for any given smoothness, $2\le r<\infty$.

24 pages