Topological dynamics of piecewise {\lambda}-affine maps

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 $b_1,\dots ,b_n$ such that $f(x)=\lambda x+b_i$ for every $x\in [c_{i-1},c_i)$\mathrm{.}Weprovethat,forLebesguealmostevery$\delta\in$\mathbb{R}$$, the map $f_{\delta }=f+\delta (\mathrm{m}\mathrm{o}\mathrm{d}1)$ 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.