Invariant space under Hénon renormalization : Intrinsic geometry of Cantor attractor

Abstract

Three dimensional Hńon-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. Hénon 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 ${\tilde{b}}_1$, then the homeomorphism between two Cantor attractor is at most Hölder continuous, which is called {\em non rigidity}.

43 pages

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