Prime ends rotation numbers and periodic points

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 Carathéodory’s prime ends rotation number, similar to Poincaré’s theory for circle h...

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Bibliographic Details
Published in:Duke mathematical journal 2015-02, Vol.164 (3), p.403-472
Main Authors: Koropecki, Andres, Le Calvez, Patrice, Nassiri, Meysam
Format: Article
Language:English
Subjects:
30D40
37B45
37E30
37E45
boundary dynamics
Dynamical Systems
dynamics of surface homeomorphisms
generic area-preserving diffeomorphisms
Mathematics
periodic points
Primes ends
rotation numbers
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Summary: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 Carathéodory’s prime ends rotation number, similar to Poincaré’s theory for circle homeomorphisms. In particular, we prove the converse of a classic result of Cartwright and Littlewood. The results are proved in a general context for homeomorphisms of arbitrary surfaces with a weak nonwandering-type hypothesis, which allows for applications in several different settings. The most important consequences are in the C^{r} -generic area-preserving context, building on previous work of Mather.
ISSN:0012-7094
1547-7398
DOI:10.1215/00127094-2861386