Let $0 < a < 1$, $0 \le c <1$ and $I = [0,1)$. We call contracted rotation the interval map $\phi_{a,c} : x \in I \mapsto ax + c \mod 1$. Once $a$ is fixed, we are interested in the dynamics of the one-parameter family $\phi_{a,c}$, where $c$ runs on the interval $[0,1)$. Any contracted rotation has a rotation number $\rho_{a,c}$ which describes the asymptotic behavior of $\phi_{a,c}$. In the first part of the talk, we analyze the numerical relation between the parameters $a,c$ and $\rho_{a,c}$ and discuss some applications of this map. Then, we introduce a generalization of the contracted rotations. Let $-1 < \lambda < 1$ and $f : [0,1) \to \R$ be a piecewise $\lambda$-affine contraction, that is, there exist points $0 = c_0 <c_1 < ... < c_{n-1} < c_n = 1$ and real numbers $b_1, ..., b_n$ such that $f(x) = \lambda x + b_i$ for every $x \in [c_{i-1}, c_i)$. We prove that, for Lebesgue-almost every $\delta \in \R$, the map $f_{\delta} = f + \delta ({\rm mod} 1)$ is asymptotically periodic. More precisely, $f_{\delta}$ has at most $n + 1$ periodic orbits and the $\omega$-limit set of every $x \in [0,1)$ is a periodic orbit.