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arXiv · 1206.4707

Prime ends rotation numbers and periodic points

Andres Koropecki·Patrice Le Calvez·Meysam Nassiri
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Abstract

We study the problem of existence of a periodic point in the boundary of an invariant domain for a surface homeomorphism. In the area-preserving setting, a complete classification is given in terms of rationality of Carath\'eordory's prime ends rotation number, similar to Poincar\'e's theory for circle homeomorphisms. In particular, we prove the converse of a classic result of Cartwright and Littlewood. This has a number of consequences for generic area preserving surface diffeomorphisms. For instance, we extend previous results of J. Mather on the boundary of invariant open sets for $C^r$-generic area preserving diffeomorphisms. Most results are proved in a general context, for homeomorphisms of arbitrary surfaces with a weak nonwandering-type hypothesis. This allows us to prove a conjecture of R. Walker about co-basin boundaries, and it also has applications in holomorphic dynamics.

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Andres Koropecki, Patrice Le Calvez, Meysam Nassiri. 2012-06-20. Prime ends rotation numbers and periodic points. https://doi.org/10.1215/00127094-2861386

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