It is often possible to parameterize a given class of dynamical systems by the elements of a Polish space; it then becomes natural to ask which properties hold "generically," that is, on a comeager set of systems in the sense of Baire. The most extreme situation is when there is a unique comeager isomorphism class: in other words, generic properties are captured by a single system. This typically does not occur in ergodic theory but can happen in zero-dimensional topological dynamics.

For example, a result by Kechris and Rosendal states that there exists a generic action of ℤ on the Cantor space, and a result by Kwiatkowska shows that there exists such a generic action of the free group $F_n$. In this work, we focus on minimal dynamical systems and show that there exists a generic minimal action of $F_n$, as well as a generic minimal action of $F_n$ preserving a probability measure. We develop a model-theoretic framework to study this question and related problems.

This is joint work with Michal Doucha and Julien Melleray.

Café culturel par Bruno Duchesne