Differentiable equivalence of fractional linear maps

A Moebius system is an ergodic fibred system $(B,T)$ (see \citer5) defined on an interval $B=[a,b]$ with partition $(J_k),k\in I,#I\geq 2$ such that $Tx=\frac{c_k+d_kx}{a_k+b_kx}$, $x\in J_k$ and $T|_{J_k}$ is a bijective map from $J_k$ onto $B$. It is well known that for $#I=2$ the invariant density can be written in the form $h(x)=\int_{B^*}\frac{dy}{(1+xy)^2}$ where $B^*$ is a suitable interval. This result does not hold for $#I\geq 3$. However, in this paper for $#I=3$ two classes of interval maps are determined which allow the extension of the before mentioned result.