Prime ends rotation numbers and periodic points

TL;DR

This paper investigates the existence of periodic points on the boundary of invariant domains for surface homeomorphisms, linking prime ends rotation numbers to periodicity, with implications for area-preserving dynamics and holomorphic systems.

Contribution

It provides a complete classification of boundary periodic points based on prime ends rotation numbers and extends classical results to broader surface homeomorphisms.

Findings

Complete classification in area-preserving case based on rotation number rationality

Proved the converse of Cartwright and Littlewood's classic result

Extended results on boundary behavior for generic area-preserving diffeomorphisms

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.