Dynamics of piecewise increasing contractions

Let I1=[a0,a1), horizontal ellipsis ,Ik=[ak-1,ak)$I_1=[a_0,a_1),\ldots ,I_{k}= [a_{k-1},a_k)$ be a partition of the interval I=[0,1)$I=[0,1)$ into k$k$ subintervals. Let f:I -> I$f:I\rightarrow I$ be a map such that each restriction f|Ii$f|_{I_i}$ is an increasing Lipschitz contraction. We prove that any f$f$ admits at most k$k$ periodic orbits, where the upper bound is sharp. We are also interested in the dynamics of piecewise linear lambda$\lambda$-affine maps, where 0 R$F_\lambda : I\rightarrow \mathbb {R}$ be a function such that each restriction F lambda|Ii(x)=lambda x+bi$F_\lambda |_{I_i}(x)=\lambda x +b_i$. Under a generic assumption on the parameters a1, horizontal ellipsis ,ak-1,b1, horizontal ellipsis ,bk$a_1,\ldots ,a_{k-1},b_1,\ldots ,b_k$, we prove that, up to a zero Hausdorff dimension set of slopes lambda$\lambda$, the omega$\omega$-limit set of the piecewise lambda$\lambda$-affine maps f lambda:x is an element of I -> F lambda(x)(mod1)$f_\lambda :x\in I \rightarrow F_\lambda (x)\pmod {1}$ at every point equals a periodic orbit and there exist at most k$k$ periodic orbits. Moreover, let E(k)$\mathfrak {E}<^>{(k)}$ be the exceptional set of parameters lambda,a1, horizontal ellipsis ,ak-1,b1, horizontal ellipsis ,bk$\lambda ,a_1,\ldots ,a_{k-1},b_1,\ldots ,b_k$ which define non-asymptotically periodic f$f$, we prove that E(k)$\mathfrak {E}<^>{(k)}$ is a Lebesgue null measure set whose Hausdorff dimension is large or equal to k$k$.