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

TL;DR

This paper investigates the renormalization of highly dissipative three-dimensional Hénon maps, showing the existence of invariant surfaces, defining a class of embedded two-dimensional Hénon-like maps, and establishing universality and geometric properties similar to two-dimensional cases.

Contribution

It introduces a framework for renormalization of 3D Hénon maps via invariant surfaces, extending 2D universality results to three dimensions, and characterizes the geometric properties of the critical Cantor set.

Findings

Existence of $C^r$ invariant surfaces tangent to the invariant plane field.

Construction of two-dimensional $C^r$ Hénon-like maps embedded in three dimensions.

Universality and geometric properties of the critical Cantor set are preserved in the 3D setting.

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 $.