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

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.