Topological dynamics of piecewise -affine maps

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.