## Institute for Mathematical Sciences

## Preprint ims01-09

** J. C. Rebelo and R. R. Silva**
* The multiple ergodicity of non-discrete subgroups of ${\rm Diff}^{\omega} ({\mathbb S}^1)$*

Abstract: In this work we deal with non-discrete subgroups of $\dif$, the group of orientation-preserving analytic diffeomorphisms
of the circle. If $\Gamma$ is such a group, we consider its
natural diagonal action $\ogama$ on the $n-$dimensional torus
$\tor^n$. It is then obtained a complete characterization of
these groups $\Gamma$ whose corresponding $\ogama-$action on
$\tor^n$ is not piecewise ergodic (cf. Introduction) for all
$n \in \N$ (cf. Theorem~A). Theorem~A can also be interpreted
as an extension of Lie's classification of Lie algebras on
$\s$ to general non-discrete subgroups of $\s$.

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