Topological dynamics of piecewise {\lambda}-affine maps

TL;DR

This paper studies the dynamics of piecewise affine maps with contraction, showing that for almost all shifts, these maps are asymptotically periodic with a bounded number of periodic orbits.

Contribution

It proves that almost all shifted maps of a piecewise fine map are asymptotically periodic with at most 2n periodic orbits, extending understanding of their long-term behavior.

Findings

Almost every shifted map is asymptotically periodic.

Each such map has at most 2n periodic orbits.

The fine maps' finite -limit sets are periodic orbits.

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 $b_1,\ldots,b_n$ 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 $2n$ periodic orbits and the $\omega$-limit set of every $x\in [0,1)$ is a periodic orbit.