Groups of homeomorphisms of Cantor sets in the line

(Dominique Malicet ) - LAMA, Université Paris Est Marne la Vallée

Let K be a Cantor set included in the real line. We look at group actions on K by homeomorphisms which locally preserve or invert the orientation of the line (that is, the homeomorphisms are locally monotonic). These actions are similar with group actions in dimension 1, but may be more complex. A natural class of examples is given by the groups of diffeomorphisms of a Cantor set: in the case of the triadic Cantor we obtain an action of the Thompson group V, which contains every finite group. Another interesting class of examples are the finitely generated groups of transformations of [0; 1] with a finite number of discontinuities, which give rise to such actions by blowing up the discontinuity points by a semiconjugacy. In a recent work with Emmanuel Militon (University Côte d’Azur), we studied these group actions, and proved they satisfy a Tits alternative in the Margulis sense: either the group has a non abelian free subgroup, or it preserves some probability measure on K. A large part of the proof consists in the study of random dynamical systems/random walks generated by elements of the group..