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.Comment: Published at http://dx.doi.org/10.1214/074921706000000257 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org