Topological dynamics of piecewise λ-affine maps

Abstract

Let $-1<\lambda <1$ and $f:[0,1)\to \mathbb{R}$ be a piecewise $\lambda$-affine map, that is, there exist points $0=c_0<c_1<\cdots <c_{n-1}<c_n=1$ and real numbers $b_1,\dots ,b_n$ 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 $f_{\delta }=f+\delta (\mathrm{mod}1)$ is asymptotically periodic. More precisely, $f_{\delta }$ has at most $2n$ periodic orbits and the $\omega$-limit set of every $x\in [0,1)$ is a periodic orbit.

The present article is an extended version of our previous work posted in arXiv:1408.1663v1, entitled "Piecewise Contractions Defined by Iterated Function Systems"

References in corpus (1)

• Asymptotically periodic piecewise contractions of the interval