Periodic orbits and the continuity of rotation numbers

Swanson, Richard

doi:10.1090/S0002-9939-1993-1112502-X

Periodic orbits and the continuity of rotation numbers
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by Richard Swanson

Proc. Amer. Math. Soc. 117 (1993), 269-273

DOI: https://doi.org/10.1090/S0002-9939-1993-1112502-X

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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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