H\'enon renormalization in arbitrary dimension : Invariant space under renormalization operator

TL;DR

This paper extends the theory of Hénon renormalization to arbitrary finite dimensions, showing that a specific invariant set of maps exhibits unbounded geometric complexity almost everywhere in parameter space.

Contribution

It introduces an invariant space of infinitely renormalizable Hénon-like maps in any finite dimension and demonstrates unbounded geometry of their attractors, generalizing previous 3D results.

Findings

Invariant set of maps under renormalization in arbitrary dimensions.

Unbounded geometry of attractors for these maps in parameter space.

Extension of 3D results to higher dimensions.

Abstract

Infinitely renormalizable H\'enon-like map in arbitrary finite dimension is considered. The set, $\mathcal N$ of infinitely renormalizable H\'enon-like maps satisfying the certain condition is invariant under renormalization operator. The Cantor attractor of infinitely renormalizable H\'enon-like map, $F$ in $\mathcal N$ has {\em unbounded geometry} almost everywhere in the parameter space of the universal number which corresponds to the average Jacobian of two dimensional map. This is an extension of the same result in $\mathcal N$ for three dimensional infinitely renormalizable H\'enon-like maps.