Renormalization of H\'enon map in arbitrary dimension I : Universality and reduction of ambient space

TL;DR

This paper extends the theory of period doubling Hénon map renormalization to arbitrary finite dimensions, revealing universal properties and complex geometric structures of attractors in higher-dimensional dissipative systems.

Contribution

It generalizes Hénon renormalization to higher dimensions, demonstrating invariant surfaces and unbounded attractor geometry in a broad class of dissipative maps.

Findings

Invariant $C^r$ surfaces in higher dimensions

Unbounded geometry of Cantor attractors

Extension of 3D renormalization properties to arbitrary dimensions

Abstract

Period doubling H\'enon renormalization of strongly dissipative maps is generalized in arbitrary finite dimension. In particular, a small perturbation of toy model maps with dominated splitting has invariant $C^r$ surfaces embedded in higher dimension and the Cantor attractor has unbounded geometry with respect to full Lebesgue measure on the parameter space. It is an extension of dynamical properties of three dimensional infinitely renormalizable H\'enon-like map in arbitrary finite dimension.