Dynamics of the universal area-preserving map associated with period doubling: hyperbolic sets. Nonlinearity 22(10

Abstract. 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 ≈ 0.00013 e−7499