  Let $-1<\lambda<1$ and $f:[0,1)\to\mathbb{R}$ be a piecewise $\lambda$-affine
map, that is, 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 $f(x)=\lambda x+b_i$ for every $x\in
[c_{i-1},c_i)$. We prove that, for Lebesgue almost every $\delta\in\mathbb{R}$,
the map $f_{\delta}=f+\delta\,({\rm mod}\,1)$ is asymptotically periodic. More
precisely, $f_{\delta}$ has at most $2n$ periodic orbits and the $\omega$-limit
set of every $x\in [0,1)$ is a periodic orbit.
