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

Three dimensional analytic H'enon-like map $$F(x,y,z)=(f(x)-ϵ(x,y,z),x,\delta (x,y,z))$$ and its {\em period doubling} renormalization is defined. If $F$ is infinitely renormalizable map, Jacobian determinant of $n^{th}$ renormalized map, $R^{n}F$ has asymptotically universal expression $$Jac{R}^{n}F=b_F^{2^n}a(x)(1+O({\rho }^n))$$ where $b_F$ is the average Jacobian of $F$. The toy model map, $F_{mod}$ is defined as the map satisfying ${\mathrm{\partial }}_zϵ\equiv 0$. 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.