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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Periodic orbits and the continuity of rotation numbers


Author: Richard Swanson
Journal: Proc. Amer. Math. Soc. 117 (1993), 269-273
MSC: Primary 58F20; Secondary 54H20
DOI: https://doi.org/10.1090/S0002-9939-1993-1112502-X
MathSciNet review: 1112502
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Abstract: The main result is that an annulus homeomorphism homotopic to the identity either has a well-defined and continuous assignment of rotation numbers on its chain recurrent set or there exists an interval of rotation numbers and periodic points corresponding to each reduced rational number in the interval. As a corollary, rotational discontinuities force the mapping to admit periodic points of all sufficiently large periods $ n$. In a related result, we provide a criterion for the rotation set of an annulus homeomorphism to be nowhere dense.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1993-1112502-X
Keywords: Annulus maps, rotation numbers, chain recurrence
Article copyright: © Copyright 1993 American Mathematical Society