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

Three dimensional H'non-like map $$F(x,y,z)=(f(x)-ϵ(x,y,z),x,\delta (x,y,z))$$ is defined on the cubic box $B$. An invariant space under renormalization would appear only in higher dimension. Consider renormalizable maps each of which satisfies the condition $${\mathrm{\partial }}_y\delta \circ F(x,y,z)+{\mathrm{\partial }}_z\delta \circ F(x,y,z)\cdot {\mathrm{\partial }}_x\delta (x,y,z)\equiv 0$$ for $(x,y,z)\in B$. Denote the set of maps satisfying the above condition be $\mathcal{N}$. Then the set $ \mathcal N \cap \mathcal I(\bar \epsilon) $isinvariantundertherenormalizationoperatorwhere$ \mathcal I(\bar \epsilon) $ is the set of infinitely renormalizable maps. H'enon like diffeomorphism in $\mathcal{N}\cap \mathcal{I}(\overline{ϵ})$ has universal numbers, $b_2\asymp \mathrm{\mid }{\mathrm{\partial }}_z\delta \mathrm{\mid }$ and $ b_1 = b_F /b_2 $where$ b_F $istheaverageJacobianof$ F $\mathrm{.}TheCantorattractorof$ F \in \mathcal N \cap \mathcal I(\bar \epsilon) $,$ \mathcal O_F $ has {\em unbounded geometry} almost everywhere in the parameter space of $b_1$. If two maps in $\mathcal{N}$ has different universal numbers $b_1$ and $ \widetilde b_1 $, then the homeomorphism between two Cantor attractor is at most H"older continuous, which is called {\em non rigidity}.