Tame definable topological dynamics

(Joint work with Pierre Simon) I will present some new results on definably amenable groups in NIP theories (typical examples of which are definably amenable groups in o-minimal theories, algebraically closed valued fields and p-adics). In particular I will demonstrate that in this context various notions of genericity coincide (answering some questions of Newelski and Petrykowski) and a characterization of ergodic measures will be given. Arguments rely on the theory of forking for types and measures in NIP theories and the so-called (p,q)-theorem from combinatorics.If time permits, I will describe how these results generalize to homogeneous spaces, ind-definable groups and action of the group of automorphisms, and how these developments can be viewed as a study of the definable case of abstract tame dynamical systems introduced by Glasner.