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On the ergodic properties of Cartan flows in ergodicactions of ${\bf SL}_{\bf 2}{\bf (R)}$ and ${\bf SO}{\bf ({\bi n},1)}$

Published online by Cambridge University Press:  01 December 1997

ALEX FURMAN  and
BENJAMIN WEISS
ALEX FURMAN
Affiliation:
Institute of Mathematics, Hebrew University, Jerusalem, 91904 Israel (e-mail: [email protected]) (e-mail: [email protected]) Current address: Department of Mathematics, Penn State University, 218 McAllister Building, University Park, PA 16802, USA.
BENJAMIN WEISS
Affiliation:
Institute of Mathematics, Hebrew University, Jerusalem, 91904 Israel (e-mail: [email protected]) (e-mail: [email protected])
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Abstract

Let $G={\rm SL_2({\bf R})}$ (or $G={\rm SO}(n,1)$) act ergodically on a probability space $(X,m)$. We consider the ergodic properties of the flow $(X,m,\{g_t\})$, where $\{g_t\}$ is a Cartan subgroup of $G$. The geodesic flow on a compact Riemann surface is an example of such a flow; here $X=G/\Gamma$ is a transitive $G$-space, $G={\rm SL_2({\bf R})}$ and $\Gamma\subset G$ is a lattice. In this case the flow is Bernoullian.

For the general ergodic $G$-action, the flow $(X,m,\{g_t\})$ is always a $K$-flow, however there are examples in which it is not Bernoullian.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 17 , Issue 6 , December 1997 , pp. 1371 - 1382
Copyright
1997 Cambridge University Press

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