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 | References | Similar Articles | Additional Information

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