Hénon renormalization in arbitrary dimension : Invariant space under renormalization operator

Abstract

Infinitely renormalizable Hénon-like map in arbitrary finite dimension is considered. The set, $\mathcal{N}$ of infinitely renormalizable Hénon-like maps satisfying the certain condition is invariant under renormalization operator. The Cantor attractor of infinitely renormalizable Hénon-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 Hénon-like maps.

42 pages. arXiv admin note: substantial text overlap with arXiv:1408.4619

References in corpus (3)

• Renormalization in the Hénon family, II: The heteroclinic web
• Renormalization of three dimensional Hénon map I : Reduction of ambient space
• Invariant space under Hénon renormalization : Intrinsic geometry of Cantor attractor