Differentiable equivalence of fractional linear maps. (English) Zbl 1132.11043

Schweiger, Fritz

Denteneer, Dee (ed.) et al., Dynamics and stochastics. Festschrift in honor of M. S. Keane. Selected papers based on the presentations at the conference ‘Dynamical systems, probability theory, and statistical mechanics’, Eindhoven, The Netherlands, January 3–7, 2005, on the occasion of the 65th birthday of Mike S. Keane. Beachwood, OH: IMS, Institute of Mathematical Statistics (ISBN 0-940600-64-1/pbk). Institute of Mathematical Statistics Lecture Notes - Monograph Series 48, 237-247 (2006).

The author considers Möbius systems \((B,T)\) on intervals \(B= [a,b]\) with partition \((J_{k}),\) where \(T: J_{k}\rightarrow B\) is a bijective linear fractional transformation. It is well known that the invariant density can be written in the form
\[ h(x)= \int_{B^{*}}\,\frac{dy}{(1+xy)^{2}} \]
(\(B^*\) a suitable interval) provided that the partition consists of two intervals. This is not true in general. However, in the present paper some special cases (partitions with 3 intervals) allow an extension of this result.
For the entire collection see [Zbl 1113.60008].

Reviewer: Robert F. Tichy (Graz)

MSC:

11K55	Metric theory of other algorithms and expansions; measure and Hausdorff dimension
37A05	Dynamical aspects of measure-preserving transformations
37A45	Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
37E05	Dynamical systems involving maps of the interval

Keywords:

measure preserving maps; interval maps