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

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.