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

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 space under renormalization would appear only in higher dimension. Consider renormalizable 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 satisfying the above condition be $\mathcal N$. Then the set $\mathcal N \cap \mathcal I(\bar \epsilon)$ 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 {\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}.