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

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 R-2. A renormalization approach has been used in Eckmann et al (1982 Phys. Rev. A 26 720-2) and Eckmann et al (1984 Mem. Am. Math. Soc. 47 1-121) 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 is an element of 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 {z(k)}, k is an element of Z, vertical bar z(k vertical bar) ->(k ->infinity) 0, vertical bar z(k vertical bar) k ->(k ->infinity) infinity, (1) whose stable and unstable manifolds intersect transversally; and, for any N is an element of 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 >= dim(H) (C-F) >= epsilon approximate to 0.00013 e(-7499).

Johnson, Tomas,1979-Uppsala universitet,Matematiska institutionen(Swepub:uu)tomjo390 (author)
Uppsala universitetMatematiska institutionen (creator_code:org_t)

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