Poincar\'e theory for decomposable cofrontiers

TL;DR

This paper extends Poincaré's theory to complex topological structures called decomposable cofrontiers, establishing conditions under which their dynamics resemble irrational rotations and analyzing the uniqueness and structure of associated semiconjugacies.

Contribution

It generalizes classical circle homeomorphism results to decomposable circloids, providing new insights into their dynamics and semiconjugacy structures.

Findings

Dynamics are semiconjugate to irrational rotations when boundary is decomposable.

Semiconjugacy is unique up to rotation, with almost one-to-one correspondence under certain conditions.

Constructs a smooth example with non-trivial fibers of the semiconjugacy.

Abstract

We extend Poincar\'e's theory of orientation-preserving homeomorphisms from the circle to circloids with decomposable boundary. As special cases, this includes both decomposable cofrontiers and decomposable cobasin boundaries. More precisely, we show that if the rotation number on an invariant circloid $A$ of a surface homeomorphism is irrational and the boundary of $A$ is decomposable, then the dynamics are monotonically semiconjugate to the respective irrational rotation. This complements classical results by Barge and Gillette on the equivalence between rational rotation numbers and the existence of periodic orbits and yields a direct analogue to the Poincar\'e Classification Theorem for circle homeomorphisms. Moreover, we show that the semiconjugacy can be obtained as the composition of a monotone circle map with a `universal factor map', only depending on the topological structure of the circloid. This implies, in particular, that the monotone semiconjugacy is unique up to post-composition with a rotation. If, in addition, $A$ is a minimal set, then the semiconjugacy is almost one-to-one if and only if there exists a biaccessible point. In this case, the dynamics on $A$ are almost automorphic. Conversely, we use the Anosov-Katok method to build a $C^\infty$-example where all fibres of the semiconjugacy are non-trivial.