Dynamique topologique de groupoïdes pour l’étude des aléatorisations
Jorge Muñoz Carvajal · HAL (Le Centre pour la Communication Scientifique Directe) · 2022
The primary objective of this thesis is the study of the preservation of model-theoretic properties by the randomization using dynamical methods, building on the work done by T. Ibarlucía. Randomizations are continuous structures consisting of random functions from an atomless probability space into a family of structures. Given a theory T, its randomization T^R, is the common theory of the randomizations of models of T. The new theory preserve many of the properties of T, among them stability and NIP. Recently, Ben Yaacov showed that to any complete, countable theory admitting a universal Skolem sort, we can associate a topological groupoid that determines the theory up to bi-interpretation. We show that if T admits such a sort then so it does its randomization. Subsequently, we study fields of Banach spaces and groupoid actions. We show that weakly almost periodic functions (wap for short) are those coming from representations on fields of reflexive spaces. Finally, we study the action of the groupoid associated to a theory on the space of partial types. We prove that stable formulas correspond to wap functions. Furthermore, we show how to build a representation of the randomization of the action starting from a representation of the original action.