## Institute of Mathematical Statistics Lecture Notes - Monograph Series

- Lecture Notes--Monograph Series
- Volume 48, 2006, 237-247

### Differentiable equivalence of fractional linear maps

#### 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***Dynamics & Stochastics: Festschrift in honor of M. S. Keane* (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2006)

**Dates**

First available in Project Euclid: 28 November 2007

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1214/074921706000000257

**Mathematical Reviews number (MathSciNet)**

MR2306204

**Zentralblatt MATH identifier**

1132.11043

**Subjects**

Primary: 37A05: Measure-preserving transformations

Secondary: 11K55: Metric theory of other algorithms and expansions; measure and Hausdorff dimension [See also 11N99, 28Dxx] 37E05: Maps of the interval (piecewise continuous, continuous, smooth)

**Keywords**

measure preserving maps interval maps

**Rights**

Copyright © 2006, Institute of Mathematical Statistics

#### Citation

Schweiger, Fritz. Differentiable equivalence of fractional linear maps. Dynamics & Stochastics, 237--247, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2006. doi:10.1214/074921706000000257. https://projecteuclid.org/euclid.lnms/1196285824