  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.
