Linearizations of periodic point free distal homeomorphisms on the annulus

Abstract

Let $\mathbb{A}$ be an annulus in the plane ${\mathbb{R}}^2$ and $g:\mathbb{A}\to \mathbb{A}$ 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 $\alpha$.