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Dynamics of the Universal Area-Preserving Map Associated with Period Doubling: Hyperbolic Sets
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2009 (English)In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 22, no 10, 2487-2520 p.Article in journal (Refereed) Published
Abstract [en]

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).

Place, publisher, year, edition, pages
2009. Vol. 22, no 10, 2487-2520 p.
Keyword [en]
2000 Mathematics Subject Classification. 37E20, 37F25, 37D05, 37D20, 37C29, 37A05, 37G15, 37M99
National Category
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-107533DOI: 10.1088/0951-7715/22/10/010ISI: 000269717000010OAI: oai:DiVA.org:uu-107533DiVA: diva2:231605
Available from: 2009-08-14 Created: 2009-08-14 Last updated: 2017-12-13Bibliographically approved

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Gaidashev, DenisJohnson, Tomas

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