Dynamics of piecewise contractions of the interval

Abstract

We study the asymptotical behaviour of iterates of piecewise contractive maps of the interval. It is known that Poincaré 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.

References in corpus (3)

• Asymptotically periodic piecewise contractions of the interval
• Periodicity in the transient regime of exhaustive polling systems
• Orbit structure of interval exchange transformations with flip

Cited by in corpus (8)

• Asymptotically periodic piecewise contractions of the interval
• Symbolic dynamics of piecewise contractions
• On the Asymptotic Properties of Piecewise Contracting Maps
• Piecewise Contractions
• Rotation number of interval contracted rotations
• Outer Billiards with Contraction: Attracting Cantor Sets
• Periodic attractor in the discrete time best-response dynamics of the Rock-Paper-Scissors game
• A spectral decomposition of the attractor of piecewise contracting maps of the interval