Rotation number of interval contracted rotations

Abstract

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