  Let $\phi_1,\ldots,\phi_n:[0,1]\to (0,1)$ be Lipschitz contractions. Let
$I=[0,1)$, $x_0=0$ and $x_n=1$. We prove that for Lebesgue almost every
$(x_1,...,x_{n-1})$ satisfying $0<x_1<\cdots <x_{n-1}<1$, the piecewise
contraction $f:I\to I$ defined by $x\in [x_{i-1},x_i)\mapsto \phi_i(x)$ is
asymptotically periodic. More precisely, $f$ has at least one and at most $n$
periodic orbits and the $\omega$-limit set $\omega_f(x)$ is a periodic orbit of
$f$ for every $x\in I$.
