Topological dynamics of piecewise lambda-affine maps of the interval

Tuesday, March 28, 2017
Ora Inizio: 
Ora Fine: 
Arnaldo Nogueira
Institut de Mathematiques de Marseille
Let $ 0 < a < 1 $, $ 0 \leq c<1 $ and $I=[0,1)$. We call {\em contracted rotation} the interval map $\phi_{a,c} :x\in I \mapsto ax+c \mod 1$. Once $a$ is fixed, we are interested in the dynamics of the one-parameter family $\phi_{a,c}$, where $c$ runs on the interval interval $[0,1)$. Any contracted rotation has a rotation number $\rho_{a,c}$ which describes the asymptotic behavior of $\phi_{a,c}$. In the first part of the talk, we analyze the numerical relation between the parameters $a,c$ and $\rho_{a,c}$ and discuss some applications of this map. Next, we introduce a generalization of contracted rotations. Let $-1<\lambda<1$ and $f:[0,1)\to\mathbb{R}$ be a piecewise $\lambda$-affine contraction, that is, there exist points $ 0=c_0 < c_1 < \cdots < c_{n-1} < c_n = 1 $ and real numbers $ b_1, \ldots,b_n$ such that $f(x)=\lambda x+b_i$ for every $ x \in [c_{i-1},c_i)$. We prove that, for Lebesgue almost every $\delta\in\mathbb{R}$, the map $f_{\delta}=f+\delta\,({\rm mod}\,1)$ is asymptotically periodic. More precisely, $f_{\delta}$ has at most $n+1$ periodic orbits and the $\omega$-limit set of every $x\in [0,1)$ is a periodic orbit.