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Three dimensional H\'non-like map $$ F(x,y,z) = (f(x) - \epsilon (x,y,z),\x,\ \delta (x,y,z)) $$ is defined on the cubic box $ B $. An invariant spaceunder renormalization would appear only in higher dimension. Considerrenormalizable maps each of which satisfies the condition $$ \partial_y \delta\circ F(x,y,z) + \partial_z \delta \circ F(x,y,z) \cdot \partial_x \delta(x,y,z) \equiv 0 $$ for $ (x,y,z) \in B $. Denote the set of maps satisfyingthe above condition be $ \mathcal N $. Then the set $ \mathcal N \cap \mathcalI(\bar \epsilon) $ is invariant under the renormalization operator where $\mathcal I(\bar \epsilon) $ is the set of infinitely renormalizable maps.H\'enon like diffeomorphism in $ \mathcal N \cap \mathcal I(\bar \epsilon) $has universal numbers, $ b_2 \asymp | \partial_z \delta | $ and $ b_1 = b_F/b_2 $ where $ b_F $ is the average Jacobian of $ F $. The Cantor attractor of$ F \in \mathcal N \cap \mathcal I(\bar \epsilon) $, $ \mathcal O_F $ has {\emunbounded geometry} almost everywhere in the parameter space of $ b_1 $. If twomaps in $ \mathcal N $ has different universal numbers $ b_1 $ and $ \widetildeb_1 $, then the homeomorphism between two Cantor attractor is at most H\"oldercontinuous, which is called {\em non rigidity}.

Invariant space under H\'enon renormalization : Intrinsic geometry of Cantor attractor