Dynamics of piecewise contractions of the interval

TL;DR

This paper investigates the long-term behavior of piecewise contractive maps on the interval, showing they have at most as many periodic orbits as the number of pieces and are topologically conjugate to linear contractions.

Contribution

It establishes an upper bound on the number of periodic orbits for piecewise contractions and proves their topological conjugacy to linear contractions, advancing understanding of their dynamics.

Findings

At most n periodic orbits for n-interval piecewise contractions

Every piecewise contraction is topologically conjugate to a linear contraction

Characterization of the asymptotic behavior of these maps

Abstract

We study the asymptotical behaviour of iterates of piecewise contractive maps of the interval. It is known that Poincar\'e first return maps induced by some Cherry flows on transverse intervals are, up to topological conjugacy, piecewise contractions. These maps also appear in discretely controlled dynamical systems, describing the time evolution of manufacturing process adopting some decision-making policies. An injective map $f:[0,1)\to [0,1)$ is a {\it piecewise contraction of $n$ intervals}, if there exists a partition of the interval $[0,1)$ into $n$ intervals $I_1$,..., $I_n$ such that for every $i\in{1,...,n}$, the restriction $f|_{I_i}$ is $\kappa$-Lipschitz for some $\kappa\in (0,1)$. We prove that every piecewise contraction $f$ of $n$ intervals has at most $n$ periodic orbits. Moreover, we show that every piecewise contraction is topologically conjugate to a piecewise linear contraction.