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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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