  Period doubling H\'enon renormalization of strongly dissipative maps is
generalized in arbitrary finite dimension. In particular, a small perturbation
of toy model maps with dominated splitting has invariant $C^r$ surfaces
embedded in higher dimension and the Cantor attractor has unbounded geometry
with respect to full Lebesgue measure on the parameter space. It is an
extension of dynamical properties of three dimensional infinitely
renormalizable H\'enon-like map in arbitrary finite dimension.
