The structure of periodic point free distal homeomorphisms on the annulus

TL;DR

This paper characterizes the structure of boundary-preserving, periodic point free, distal homeomorphisms on an annulus, showing they decompose into invariant circles with shared irrational rotation numbers and a linear order.

Contribution

It provides a detailed topological and dynamical description of such homeomorphisms, revealing their decomposition into invariant circles with common rotation properties.

Findings

Decomposition into invariant circles with shared irrational rotation number

Invariant circles are linearly ordered by inclusion

Homeomorphisms have no periodic points and are distal

Abstract

Let $A$ be an annulus in the plane $\mathbb R^2$ and $g:A\rightarrow A$ be a boundary components preserving homeomorphism which is distal and has no periodic points. Then there is a continuous decomposition of $A$ into $g$-invariant circles such that all the restrictions of $g$ on them share a common irrational rotation number and all these circles are linearly ordered by the inclusion relation on the sets of bounded components of their complements in $\mathbb R^2$.