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Rotation intervals for chaotic sets

Authors: Kathleen T. Alligood and James A. Yorke
Journal: Proc. Amer. Math. Soc. 126 (1998), 2805-2810
MSC (1991): Primary 58Fxx
MathSciNet review: 1443368
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Abstract: Chaotic invariant sets for planar maps typically contain periodic orbits whose stable and unstable manifolds cross in grid-like fashion. Consider the rotation of orbits around a central fixed point. The intersections of the invariant manifolds of two periodic points with distinct rotation numbers can imply complicated rotational behavior. We show, in particular, that when the unstable manifold of one of these periodic points crosses the stable manifold of the other, and, similarly, the unstable manifold of the second crosses the stable manifold of the first, so that the segments of these invariant manifolds form a topological rectangle, then all rotation numbers between those of the two given orbits are represented. The result follows from a horseshoe-like construction.

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  • 1. D. Aronson, M. Chory, G. Hall, and R. McGehee, Bifurcations from an invariant circle for two-parameter families of maps of the plane: A computer assisted study, Comm. Math. Phys. 83 (1982), 303-354. MR 83j:58078
  • 2. M. Barge, Periodic points for an orientation-preserving homeomorphism of the plane near an invariant immersed line, Topology Proceedings 20 (1995), 1-13. MR 98b:54051
  • 3. M. Casdagli, Periodic orbits for dissipative twist maps, Ergodic Theory Dynamical Systems 7 (1987), 165-173. MR 89b:58177
  • 4. R. Devaney, An introduction to chaotic dynamical systems, Benjamin/Cummings, Menlo Park, CA, 1986. MR 87e:58142
  • 5. K. Hockett and P. Holmes, Josephson's junction, annulus maps, Birkhoff attractors, horseshoes and rotation sets, Ergodic Theory Dynamical Systems 6 (1986), 205-239. MR 88m:58103
  • 6. A. Katok, Some remarks on Birkhoff and Mather twist map theorems, Ergodic Theory Dynamical Systems 2 (1982), 185-194. MR 84m:58041
  • 7. P. LeCalvez, Properties des attracteurs de Birkhoff, Ergodic Theory Dynamical Systems 8 (1988), 241-310. MR 90a:58103
  • 8. J. Mather, Existence of quasi-periodic orbits for twist homeomorphisms of the annulus, Topology 21 (1982), 457-467. MR 84g:58084
  • 9. K. Meyer and G. Hall, Introduction to Hamiltonian dynamical systems and the $N$-body problem, Springer-Verlag, New York, 1992. MR 93b:70002
  • 10. J. Palis and W. deMelo, Geometric theory of dynamical systems: An introduction, Springer-Verlag, New York, 1982. MR 84a:58004

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Additional Information

Kathleen T. Alligood
Affiliation: Department of Mathematical Sciences, George Mason University, Fairfax, Virginia 22030

James A. Yorke
Affiliation: Institute for Physical Sciences and Technology, University of Maryland, College Park, Maryland 20742

Received by editor(s): January 24, 1997
Additional Notes: The authors’ research was partially supported by the National Science Foundation. The second author’s research was also supported by the Department of Energy (Office of Scientific Computing)
Communicated by: Linda Keen
Article copyright: © Copyright 1998 American Mathematical Society

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