Olena Karpel (AGH University of Krakow)
Bratteli diagrams and the dynamical systems arising on their associated path spaces provide a flexible and powerful framework for modeling a broad class of equivalence relations and dynamical phenomena. Originally introduced in the study of AF C∗-algebras, Bratteli diagrams have since become an important tool in the construction of models in measurable, Cantor, and Borel dynamics, where they encode orbit structures, invariant measures, and asymptotic properties of dynamical systems in a combinatorial manner. The talk contributes to the study of invariant measures of Borel dynamical systems that can be modeled using generalized Bratteli diagrams. In this context, we study tail invariant measures on the path spaces of generalized Bratteli diagrams, allowing countably infinite vertex sets at each level. Our main focus is on subdiagrams of generalized Bratteli diagrams
and the problem of extending tail invariant probability measures from vertex and edge subdiagrams to the ambient diagram. We establish necessary and sufficient conditions for the finiteness of such extensions, formulated in terms of incidence matrices and associated stochastic matrices. Several classes of generalized Bratteli diagrams and their subdiagrams are analyzed in detail, including simple, stationary, and bounded size diagrams. We develop constructive, step-by-step procedures for measure extension and for approximating invariant measures by measures supported on suitable subdiagrams. In addition, we provide explicit examples of generalized Bratteli diagrams that admit no probability tail invariant measures, a phenomenon absent for standard Bratteli diagrams with finite vertex sets. Finally, we address convergence questions for sequences of invariant measures arising from approximations by subdiagrams, clarifying the relationship between combinatorial structure and measure-theoretic behavior. The talk is based on results obtained together with S. Bezuglyi, P. Jorgensen, J. Kwiatkowski, T. Raszeja, S. Sanadhya, and M. Wata