# Dynamics of 2-interval piecewise affine maps and Hecke-Mahler series

## TL;DR

This paper explicitly characterizes the dynamics of a specific class of 2-interval piecewise affine maps using functions related to Hecke-Mahler series, revealing rational rotation numbers for algebraic parameters.

## Contribution

It provides an explicit description of the dynamics of 2-interval piecewise affine maps via functions involving Hecke-Mahler series, linking algebraic parameters to rational rotation numbers.

## Key findings

- Explicit formulas for the dynamics using functions δ and φ
- Rotation number is rational for algebraic parameters
- Connections between dynamics and Hecke-Mahler series

## Abstract

Let $f : [0,1)\rightarrow [0,1)$ be a $2$-interval piecewise affine increasing map which is injective but not surjective. Such a map $f$ has a rotation number and can be parametrized by three real numbers. We make fully explicit the dynamics of $f$ thanks to two specific functions $\delta$ and $\phi$ depending on these parameters whose definitions involve Hecke-Mahler series. As an application, we show that the rotation number of $f$ is rational, when the three parameters are algebraic numbers.