Differentiable equivalence of fractional linear maps

Fritz Schweiger

doi:10.1214/074921706000000257

paper

Differentiable equivalence of fractional linear maps

arXiv:math/0608250 · doi:10.1214/074921706000000257

Abstract

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 $T x = \frac{c _{k} + d _{k} x}{a _{k} + b _{k} x}$ , $x \in J_{k}$ 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 $h (x) = \int_{B^{*}} \frac{d y}{( 1 + x y ) ^{2}}$ 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.

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)