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

Abstract

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}.