Asymptotically periodic piecewise contractions of the interval

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$C^2$-diffeomorphismswith$\sup_{x\in (0,1)} \vert D\phi_i(x)\vert<1$whoseimages$\phi_1([0,1]), \ldots, \phi_n([0,1])$ are pairwise disjoint. Let $0<x_1<\cdots <x_{n-1}<1$ and let $I_1,\dots ,I_n$ 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 $f_{x_1,\dots ,x_{n-1}}$ be the map given by $x\mapsto {\varphi }_i(x)$ if $x\in I_i$, for $1\le i\le n$. Among other results we prove that for Lebesgue almost every $(x_1,\dots ,x_{n-1})$, the piecewise contraction $f_{x_1,\dots ,x_{n-1}}$ is asymptotically periodic.