Arnaldo Nogueira
Institut de Mathematiques de Marseille, France
February 21, 2019
Dynamics of 2-interval piecewise linear contraction maps and Hecke-Mahler series: Let $I=[0,1)$ be the unity interval. Let $0 < a <1$ and $0 < b < 1$ with $a+b>1$. Let $f=f_{a,b} :x\in I \mapsto ax+b \mod 1$. Once the parameter $a$ is fixed, we are interested in the family $f_{a,b}$, where $b$ runs on the interval $I$. We use the fact that, as in the case of circle homeomorphisms, any map $f_{a,b}$ has a rotation number which depends only on the parameters $a$ et $b$. We will discuss the dynamical and diophantine aspects of the subject. In particular, we will show that, if $a$ and $b$ are algebraic numbers, the rotation number is rational using a transcendance theorem about the value of the Hecke-Mahler series at an algebraic point. If we have time, we will discuss other cases of $2$-interval piecewise linear contractions which have the same property.
Institut de Mathematiques de Marseille, France
February 21, 2019
Dynamics of 2-interval piecewise linear contraction maps and Hecke-Mahler series: Let $I=[0,1)$ be the unity interval. Let $0 < a <1$ and $0 < b < 1$ with $a+b>1$. Let $f=f_{a,b} :x\in I \mapsto ax+b \mod 1$. Once the parameter $a$ is fixed, we are interested in the family $f_{a,b}$, where $b$ runs on the interval $I$. We use the fact that, as in the case of circle homeomorphisms, any map $f_{a,b}$ has a rotation number which depends only on the parameters $a$ et $b$. We will discuss the dynamical and diophantine aspects of the subject. In particular, we will show that, if $a$ and $b$ are algebraic numbers, the rotation number is rational using a transcendance theorem about the value of the Hecke-Mahler series at an algebraic point. If we have time, we will discuss other cases of $2$-interval piecewise linear contractions which have the same property.
(The talk is based on a joint work with Michel Laurent.)