Logic Seminar

It is often possible to parametrize a given class of dynamical systems by elements of a Polish space and then it becomes natural to ask what properties hold "generically", i.e., on a comeager set of systems. The most extreme situation is when there is a single comeager isomorphism class: that is, the generic properties are captured by a single system. This does not usually happen in ergodic theory but, somewhat surprisingly and to an extent that is still not understood, this phenomenon does occur in zero-dimensional topological dynamics. For example, it is a result of Kechris and Rosendal that there is a generic action of Z on the Cantor space and of Kwiatkowska that there is such a generic action of the free group F_n. These actions are quite degenerate from dynamical point of view: for example, they cannot be topologically transitive. In this work, we are interested in minimal dynamical systems and show that there is a generic minimal action of F_n and also a generic minimal action of F_n that preserves a probability measure, and we identify these two actions. The tools we use come from symbolic dynamics. We also develop a model-theoretic framework to study this and related questions. This is joint work with Michal Doucha and Julien Melleray.