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).

Subject headings

NATURVETENSKAP  -- Matematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics (hsv//eng)

Keyword

2000 Mathematics Subject Classification. 37E20

37F25

37D05

37D20

37C29

37A05

37G15

37M99

MATHEMATICS

MATEMATIK

Publication and Content Type

ref (subject category)

art (subject category)