## 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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