Let
$-1<\unicode[STIX]{x1D706}<1$ and let
$f:[0,1)\rightarrow \mathbb{R}$ be a piecewise
$\unicode[STIX]{x1D706}$-affine contraction: that is, let there exist points
$0=c_{0}<c_{1}<\cdots <c_{n-1}<c_{n}=1$ and real numbers
$b_{1},\ldots ,b_{n}$ such that
$f(x)=\unicode[STIX]{x1D706}x+b_{i}$ for every
$x\in [c_{i-1},c_{i})$. We prove that, for Lebesgue almost every
$\unicode[STIX]{x1D6FF}\in \mathbb{R}$, the map
$f_{\unicode[STIX]{x1D6FF}}=f+\unicode[STIX]{x1D6FF}\,(\text{mod}\,1)$ is asymptotically periodic. More precisely,
$f_{\unicode[STIX]{x1D6FF}}$ has at most
$n+1$ periodic orbits and the
$\unicode[STIX]{x1D714}$-limit set of every
$x\in [0,1)$ is a periodic orbit.