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Rotation numbers and instability sets


Author: John Franks
Journal: Bull. Amer. Math. Soc. 40 (2003), 263-279
MSC (2000): Primary 37E45
DOI: https://doi.org/10.1090/S0273-0979-03-00983-2
Published electronically: April 8, 2003
MathSciNet review: 1978565
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Abstract: Translation and rotation numbers have played an interesting and important role in the qualitative description of various dynamical systems. In this exposition we are especially interested in applications which lead to proofs of periodic motions in various kinds of dynamics on the annulus. The applications include billiards and geodesic flows.

Going beyond this simple qualitative invariant in the study of the dynamics of area preserving annulus maps, G.D. Birkhoff was led to the concept of ``regions of instability'' for twist maps. We discuss the closely related notion of instability sets for a generic area preserving surface diffeomorphism and develop their properties.


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Additional Information

John Franks
Affiliation: Department of Mathematics, Northwestern University, Evanston, Illinois 60208-2730
Email: john@math.northwestern.edu

DOI: https://doi.org/10.1090/S0273-0979-03-00983-2
Received by editor(s): December 31, 2002
Published electronically: April 8, 2003
Additional Notes: Supported in part by NSF grant DMS0099640. This article is the written version of an invited address at the January 2002 AMS meeting in San Diego, California
Article copyright: © Copyright 2003 American Mathematical Society

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