The talk will be devoted to the study of finitely generated groups acting on the circle. We will start with joint results with A. Navas and B. Deroin: if such an action by analytic diffeomorphisms admits a Cantor minimal set, then this set is of Lebesgue measure zero, and if such an action by free group of analytic diffeomorphisms is minimal, then it is also Lebesgue-ergodic.

If the time permits, we will discuss the new results and state of art of an ongoing joint project with B. Deroin, A. Navas, D. Filimonov, M. Triestino, D. Malicet, S. Alvarez, P. G. Barrientos and C. Meniño, devoted to the further study of such actions, and the different kingdoms of locally discrete and locally non-discrete actions.