American Institute of Mathematical Sciences

2018, 12: 175-191. doi: 10.3934/jmd.2018007

Rotation number of contracted rotations

 Aix-Marseille Université, CNRS, Centrale Marseille, Institut de Mathématiques de Marseille, 163 avenue de Luminy, Case 907, 13288, Marseille Cédex 9, France

Received  April 18, 2017 Revised  February 28, 2018 Published  June 2018

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 some assumptions on $\lambda$.

Citation: Michel Laurent, Arnaldo Nogueira. Rotation number of contracted rotations. Journal of Modern Dynamics, 2018, 12: 175-191. doi: 10.3934/jmd.2018007
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A plot of $f_{\lambda, \delta}: I \to I$ , where $\lambda + \delta > 1$
Plot of $F_{1/2, 3/4}(x)$ in the interval $-1\le x < 1$
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