Topological dynamics of piecewise {\lambda}-affine maps
Arnaldo Nogueira, Benito Pires, Rafael A. Rosales

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 and be a piecewise -affine map, that is, there exist points and real numbers such that for every . We prove that, for Lebesgue almost every , the map is asymptotically periodic. More precisely, has at most periodic orbits and the -limit set of every is a periodic orbit.
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