Rotation number of interval contracted rotations

Abstract

Let $0<\lambda <1$. We consider the one-parameter family of circle $\lambda$-affine contractions $f_{\delta }:x\in [0,1)\mapsto \lambda x+\delta \mathrm{mod}1$, 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 $\lambda$.

References in corpus (1)

• Dynamics of piecewise contractions of the interval

Cited by in corpus (2)

• A piecewise contractive dynamical system and election methods
• Piecewise contractions and b-adic expansions