Piecewise contractions defined by iterated function systems

Abstract

Let ${\varphi }_1,\dots ,{\varphi }_n:[0,1]\to (0,1)$ be Lipschitz contractions. Let $I=[0,1)$, $x_0=0$ and $x_n=1$. We prove that for Lebesgue almost every $(x_1,...,x_{n-1})$ satisfying $0<x_1<\cdots <x_{n-1}<1$, the piecewise contraction $f:I\to I$ defined by $x\in [x_{i-1},x_i)\mapsto {\varphi }_i(x)$ is asymptotically periodic. More precisely, $f$ has at least one and at most $n$ periodic orbits and the $\omega$-limit set ${\omega }_f(x)$ is a periodic orbit of $f$ for every $x\in I$.

16 pages, two figures

References in corpus (1)

• Periodicity in the transient regime of exhaustive polling systems

Cited by in corpus (1)

• Extreme Value Theory for Piecewise Contracting Maps with Randomly Applied Stochastic Perturbations