Dynamics of piecewise increasing contractions

Let $I_1=[a_0,a_1),\ldots,I_{k}= [a_{k-1},a_k)$ be a partition of the interval $I=[0,1)$ into $k$ subintervals. Let $f:I\to I$ be a map such that each restriction $f|_{I_i}$ is an increasing Lipschitz contraction. We prove that any $f$ admits at most $k$ periodic orbits, where the upper bound is sharp. We are also interested in the dynamics of piecewise linear $\lambda$-affine maps, where $0<\lambda<1$. Let $b_1,\ldots,b_k$ be real numbers and let $F_\lambda: I\to \mathbb{R}$ be a function such that each restriction $F_\lambda|_{I_i}(x)=\lambda x +b_i$. Under a generic assumption on the parameters $a_1,\ldots,a_{k-1},b_1,\ldots,b_k$, we prove that, up to a zero Hausdorff dimension set of slopes $\lambda$, the $\omega$-limit set of the piecewise $\lambda$-affine maps $f_\lambda:x\in I \mapsto F_\lambda(x)\pmod{1}$ at every point equals a periodic orbit and there exist at most $k$ periodic orbits. Moreover, let $\mathfrak{E}^{(k)}$ be the exceptional set of parameters $\lambda,a_1,\ldots,a_{k-1},b_1,\ldots,b_k$ which define non-asymptotically periodic $f$, we prove that $\mathfrak{E}^{(k)}$ is a Lebesgue null measure set whose Hausdorff dimension is large or equal to $k$.