Rotation number of contracted rotations

Abstract

Let \begin{document} $0 . We consider the one-parameter family of circle \begin{document} $\lambda$ \end{document} -affine contractions \begin{document} $f_\delta:x \in [0,1) \mapsto \lambda x + \delta \; {\rm mod}\,1 $ \end{document} , where \begin{document} $0 \le \delta . Let \begin{document} $\rho$ \end{document} be the rotation number of the map \begin{document} $f_\delta$ \end{document} . We will give some numerical relations between the values of \begin{document} $\lambda,\delta$ \end{document} and \begin{document} $\rho$ \end{document} , essentially using Hecke-Mahler series and a tree structure. When both parameters \begin{document} $\lambda$ \end{document} and \begin{document} $\delta$ \end{document} are algebraic numbers, we show that \begin{document} $\rho$ \end{document} is a rational number. Moreover, in the case \begin{document} $\lambda$ \end{document} and \begin{document} $\delta$ \end{document} are rational, we give an explicit upper bound for the height of \begin{document} $\rho$ \end{document} under some assumptions on \begin{document} $\lambda$ \end{document} .