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Ergodicity and rigidity for certain subgroups of $$\text{Diff}^\omega(S^1)$$. (English) Zbl 0968.37002
Author’s abstract: We consider the non solvable subgroups of the group of real analytic diffeomorphisms of the circle which admit a finite generating set whose elements belong to an appropriate and fixed neighborhood of the identity. If $$G$$ is such a group, we prove that there are non trivial local analytic vector fields which are a sort of “limit” of some local diffeomorphisms in $$G$$. Finally we apply these vector fields to prove, in particular, that either the group $$G$$ is ergodic or it has a finite orbit. These vector fields also enable us to show that the dynamics of $$G$$ is topologically rigid.

##### MSC:
 37A15 General groups of measure-preserving transformations and dynamical systems 37C85 Dynamics induced by group actions other than $$\mathbb{Z}$$ and $$\mathbb{R}$$, and $$\mathbb{C}$$
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