## Institute of Mathematical Statistics Lecture Notes - Monograph Series

### Differentiable equivalence of fractional linear maps

Fritz Schweiger

#### Abstract

A Moebius system is an ergodic fibred system $(B, T)$ (see \cite{r5}) defined on an interval $B=[a, b]$ with partition $(J_k), k \in I, \# I \geq 2$ such that $Tx = \frac{c_k + d_kx}{a_k + b_k x}$, $x \in J_k$ 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 $h(x) =\int_{B^{*}}\frac{dy}{(1+xy)^2}$ 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.

#### Chapter information

Source
Dee Denteneer, Frank den Hollander, Evgeny Verbitskiy, eds., Dynamics & Stochastics: Festschrift in honor of M. S. Keane (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2006), 237-247

Dates
First available in Project Euclid: 28 November 2007

https://projecteuclid.org/euclid.lnms/1196285824

Digital Object Identifier
doi:10.1214/074921706000000257

Mathematical Reviews number (MathSciNet)
MR2306204

Zentralblatt MATH identifier
1132.11043

Rights