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

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.