**Title:** Sharp regularity for group actions on the circle

**Speaker:** Professor Andres Navas-Flores

**Speaker Info:** IHES and Universidad de Chile

**Brief Description:**

**Special Note**:

**Abstract:**

The goal of this talk is to give the answer in some particular cases to the following question:Given a finitely generated group of circle homeomorphisms, what is the best regularity that can be achieved by performing topological conjugacies ?

We will be interested mainly in the case of small regularity (between $C^0$ and $C^1$), and in this case the intermediate classes will appear in a natural way. For instance, it is not difficult to prove that every countable group of circle homeomorphisms is topologically conjugated to a group of Lipschitz homeomorphisms. However, several groups cannot be donne $C^1$, as a consequence of the famous Thurston's stability theorem. We will give an example of a group of $C^1$ circle diffeomorphisms that cannot be donne $C^{1+\alpha}$ for any $\alpha > 0$, and also several examples of groups that can be donne $C^{1+\alpha}$ but not $C^{1+\alpha'}$ for some $1 > \alpha' > \alpha > 0$. The techniques to study this quind of phenomena combine methods from classical one dimensional dynamics and probabilistic arguments. Several open problems will be presented.

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