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

Let −1 < λ < 1 and f : [0, 1) → R be a piecewise λ-affine map, that is, there exist points 0 = c 0 < c 1

Li-Yorke chaos is a popular and well-studied notion of chaos. Several simple and useful characterizations of this notion of chaos in the setting of linear dynamics were obtained recently. In this note we show that even simpler and more useful characterizations of Li-Yorke chaos can be given in the special setting of composition operators on L p spaces. As a consequence we obtain a simple characterization of weighted shifts which are Li-Yorke chaotic. We give numerous examples to show that our results are sharp.2010 Mathematics Subject Classification. Primary 47A16, 47B33; Secondary 37D45.

We consider the iterates of a generic injective piecewise contraction of the interval defined by a finite family of contractions.

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