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 R2. A renormalization approach has been used in (Eckmann et al 1982) and (Eckmann et al 1984) in a computer-assisted proof of existence of a “universal” area-preserving map F∗ — a map with orbits of all binary periods 2k, k ∈ N. 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 {zk}, k ∈ Z, |zk| k→∞ −→ 0, |zk| k→−∞ −→ ∞, (1) whose stable and unstable manifolds intersect transversally; and, for any N ∈ N, 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 unbounded and 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 ≥ dimH(CF ) ≥ ε ≈ 0.00044 e−1797. AMS classification scheme numbers: 37E20, 37F25, 37D05, 37D20, 37C29, 37A05, 37G15, 37M99 ar X iv :0 90 5. 13 90 v3 [ m at h. D S] 1 2 A ug 2 00 9 Dynamics of the Universal Area-Preserving Map: Hyperbolic Sets 1