Rotation number of 2-interval piecewise affine maps

TL;DR

This paper investigates the rotation number of a class of two-interval piecewise affine maps, providing explicit formulas and showing rationality of the rotation number for algebraic parameters.

Contribution

It introduces a parametrization of these maps and computes their rotation number using Hecke-Mahler series, linking algebraic parameters to rational rotation numbers.

Findings

Rotation number expressed via Hecke-Mahler series.

Rotation number is rational for algebraic parameters.

Explicit formula for rotation number as a function of parameters.

Abstract

We study maps of the unit interval whose graph is made up of two increasing segments and which are injective in an extended sense. Such maps $f_{\p}$ are parametrized by a quintuple $\p$ of real numbers satisfying inequations. Viewing $f_{\p}$ as a circle map, we show that it has a rotation number $\rho(f_{\p})$ and we compute $\rho(f_{\p})$ as a function of $\p$ in terms of Hecke-Mahler series. As a corollary, we prove that $\rho(f_{\p})$ is a rational number when the components of $\p$ are algebraic numbers.