Prime ends rotation numbers and periodic points

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 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.