Title: Self-similar groups, dynamical systems and applications

Abstract: Iterated monodromy groups provide a strong link between self-similar groups and complex dynamics. In fact, iterated monodromy groups were used by Bartholdi and Nekrashevych to solve Hubbard’s question in complex dynamics on Thurston equivalence of certain topological polynomials.

Our goal is to provide a different link between self-similar groups and dynamics. We consider self-similar profinite groups and regard them as probability spaces via their Haar measure and study when the natural action of the regular rooted tree on these probability spaces is measure-preserving. We further study ergodicity and mixing properties of the arising dynamical systems. This has different consequences and applications to both the structure of self-similar groups and to the study of measure-preserving dynamical systems. We shall discuss some of these applications as time permits.