Linearizations of periodic point free distal homeomorphisms on the annulus

TL;DR

This paper investigates conditions under which boundary-preserving, point-free distal homeomorphisms on an annulus can be simplified to rigid rotations, revealing that continuous sections of certain decompositions enable linearization, but some homeomorphisms resist such simplification.

Contribution

It establishes that the existence of a continuous section in the invariant circle decomposition allows linearization of the homeomorphism, and provides examples where linearization is impossible despite having an irrational rotation number.

Findings

Homeomorphisms with continuous sections are topologically conjugate to rigid rotations.

Existence of distal homeomorphisms with irrational rotation numbers that cannot be linearized.

Characterization of when linearization is possible based on the structure of invariant circle decompositions.

Abstract

Let $\mathbb{A}$ be an annulus in the plane $\mathbb R^2$ and $g:\mathbb{A}\rightarrow \mathbb{A}$ be a boundary components preserving homeomorphism which is distal and has no periodic points. In \cite{SXY}, the authors show that there is a continuous decomposition $\mathcal P$ of $\mathbb{A}$ into $g$-invariant circles such that all the restrictions of $g$ on them share a common irrational rotation number (also called the rotation number of $g$) and all these circles are linearly ordered by the inclusion relation on the sets of bounded components of their complements in $\mathbb R^2$. In this note, we show that if the decomposition $\mathcal P$ above has a continuous section, then $g$ can be linearized, that is it is topologically conjugate to a rigid rotation on $\mathbb{A}$. For every irrational number $\alpha\in (0, 1)$, we show the existence of such a distal homeomorphism $g$ on $\mathbb{A}$ that it cannot be linearized and its rotation number is $\alpha$.