It is known that the famous Feigenbaum-Coullet-Tresser period doubling universality has a counterpart for area-preserving maps of ${ \fR}^2$. 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 $F_*$ -- a map with orbits of all binary periods $2^k, k \in \fN$. In this paper, we consider maps in some neighbourhood of $F_*$ and study their dynamics. We first demonstrate that the map $F_*$ admits a "bi-infinite heteroclinic tangle": a sequence of periodic points $\{ z_k\}$, $k \in \fZ$, |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 $N \in \fN$, a compact invariant set on which $F_*$ 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 $F_*$ admits a horseshoe. We use distortion tools to provide rigorous bounds on the Hausdorff dimension of the associated locally maximal invariant hyperbolic set: $$ 0.7673 \ge { \rm dim}_H(\cC_F) \ge \varepsilon \approx 0.00044 e^{ -1797}.$$
Categories: math.DS