  We consider the iterates of a generic injective piecewise contraction of the
interval defined by a finite family of contractions. Let $\phi_i:[0,1]\to
(0,1)$, $1\le i\le n$, be $C^2$-diffeomorphisms with $\sup_{x\in (0,1)} \vert
D\phi_i(x)\vert<1$ whose images $\phi_1([0,1]), \ldots, \phi_n([0,1])$ are
pairwise disjoint. Let $0<x_1<\cdots<x_{n-1}<1$ and let $I_1,\ldots, I_n$ be a
partition of the interval $[0,1)$ into subintervals $I_i$ having interior
$(x_{i-1},x_i)$, where $x_0=0$ and $x_n=1$. Let $f_{x_1,\ldots,x_{n-1}}$ be the
map given by $x\mapsto \phi_i(x)$ if $x\in I_i$, for $1\le i\le n$. Among other
results we prove that for Lebesgue almost every $(x_1,\ldots,x_{n-1})$, the
piecewise contraction $f_{x_1,\ldots,x_{n-1}}$ is asymptotically periodic.
