Rotation number of interval contracted rotations

Abstract

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