Renormalization of $C^r$ H\'enon map : Two dimensional embedded map in three dimension

Abstract

We study renormalization of highly dissipative analytic three dimensional H\'enon maps $$ F(x,y,z) = (f(x) - \varepsilon(x,y,z),\ x,\ \delta(x,y,z)) $$ where $ \varepsilon(x,y,z) $ is a sufficiently small perturbation of $ \varepsilon_{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 \leq r < \infty $. The $ C^r $ conjugation from an invariant surface to the $ xy- $plane defines renormalization two dimensional $ C^r $ H\'enon-like map. It also defines two dimensional embedded $ C^r $ H\'enon-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 H\'enon-like maps is open and dense subset of the parameter space of average Jacobian, $ b_{F_{2d}} $ for any given smoothness, $ 2 \leq r < \infty $.