Dynamics of the Universal Area-Preserving Map Associated with Period Doubling: Hyperbolic Sets

It is known that the famous Feigenbaum-Coullet-Tresser period doubling universality has a counterpart for area-preserving maps of ${\text{fR}}^2$. A renormalization approach has been used in \cite{EKW1} and \cite{EKW2} in a computer-assisted proof of existence of a “universal” area-preserving map $F_{\ast }$ – a map with orbits of all binary periods $2^k,k\in \text{fN}$. In this paper, we consider maps in some neighbourhood of $F_{\ast }$ and study their dynamics. We first demonstrate that the map $F_{\ast }$ admits a “bi-infinite heteroclinic tangle”: a sequence of periodic points $\{z_k\}$, $k\in \text{fZ}$, |z_k| \converge{{k \to \infty}} 0, \quad |z_k| \converge{{k \to -\infty}} \infty, whose stable and unstable manifolds intersect transversally; and, for any $N \in \fN$,acompactinvariantsetonwhich$F_*$ is homeomorphic to a topological Markov chain on the space of all two-sided sequences composed of $N$ symbols. A corollary of these results is the existence of {\it unbounded} and {\it oscillating} orbits. We also show that the third iterate for all maps close to $F_{\ast }$ admits a horseshoe. We use distortion tools to provide rigorous bounds on the Hausdorff dimension of the associated locally maximal invariant hyperbolic set: $$0.7673\ge {\mathrm{d}\mathrm{i}\mathrm{m}}_H({\text{cC}}_F)\ge \epsilon \approx 0.00044e^{-1797}\mathrm{.}$$