Rotation number of interval contracted rotations

TL;DR

This paper investigates the rotation number of a family of circle contractions, revealing algebraic conditions for rationality and providing bounds on the rotation number's complexity.

Contribution

It establishes a link between parameters and rotation number, proving rationality when parameters are algebraic and providing explicit bounds for rational parameters.

Findings

Rotation number is rational for algebraic parameters.

Numerical relations between parameters and rotation number are derived.

Explicit bounds on the height of the rotation number are given for rational parameters.

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 $\lambda$.