Renormalization of three dimensional H\'enon map I : Reduction of ambient space

TL;DR

This paper studies the renormalization of three-dimensional Hénon-like maps, revealing universal Jacobian behavior and conditions under which the system decomposes into lower-dimensional dynamics with dominated splitting.

Contribution

It introduces a renormalization framework for 3D Hénon-like maps, identifies universal Jacobian asymptotics, and establishes conditions for invariant plane fields and dimensional reduction.

Findings

Jacobian of renormalized maps exhibits universal asymptotic form.

Invariant plane fields exist under certain derivative conditions.

3D maps can be decomposed into 2D maps with dominated splitting.

Abstract

Three dimensional analytic H\'enon-like map $$ F(x,y,z) = (f(x) - \epsilon(x,y,z),\, x,\, \delta(x,y,z)) $$ and its {\em period doubling} renormalization is defined. If $ F $ is infinitely renormalizable map, Jacobian determinant of $ n^{th} $ renormalized map, $ R^nF $ has asymptotically universal expression $$ Jac R^nF = b_F^{2^n}a(x)(1 + O(\rho^n)) $$ where $ b_F $ is the average Jacobian of $ F $. The toy model map, $ F_{mod} $ is defined as the map satisfying $ \partial_z \epsilon \equiv 0 $. The set of toy model map is invariant under renormalizaton. Moreover, if $ \| \partial_z \delta \| \ll \| \partial_y \epsilon \| $, then there exists the continuous invariant plane field over $ \mathcal O_F $ with dominated splitting. Under this condition, three dimensional H\'enon-like map %with the dominated splitting is dynamically decomposed into two dimensional map with contraction along the strong stable direction. Any invariant line field on this plane filed over $ \mathcal O_{F_{mod}} $ cannot be continuous.