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

DOI:
https://doi.org/10.1090/S0002-9939-98-04267-1

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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Additional Information

**Kathleen T. Alligood**

Affiliation:
Department of Mathematical Sciences, George Mason University, Fairfax, Virginia 22030

Email:
alligood@gmu.edu

**James A. Yorke**

Affiliation:
Institute for Physical Sciences and Technology, University of Maryland, College Park, Maryland 20742

Email:
yorke@ipst.umd.edu

DOI:
https://doi.org/10.1090/S0002-9939-98-04267-1

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