Invariant measures and dynamics for reducible generalized Bratteli diagrams

13.06.2024 15:15 - 17:00

Olena Karpel (Krakow)

Abstract: Bratteli diagrams are a powerful tool for the study of dynamical systems in
measurable, Cantor and Borel dynamics. The set of invariant measures, minimal
components, structure of the orbits of the transformation become more transpar-
ent when one deals with the corresponding Bratteli-Vershik dynamical systems.
In 2010, S. Bezuglyi, J. Kwiatkowski, K. Medynets and B. Solomyak found a
complete description of the set of probability ergodic tail invariant measures
on the path space of a standard stationary reducible Bratteli diagram. It was
shown that every distinguished eigenvalue for the incidence matrix determines
a probability ergodic invariant measure. We will show that this result does
not hold for stationary reducible generalized Bratteli diagrams. We consider
classes of stationary and non-stationary reducible generalized Bratteli diagrams
with infinitely many simple standard subdiagrams, in particular, with infinitely
many odometers, and characterize the sets of all probability ergodic invariant
measures for such diagrams. We also study orders under which the diagrams
can support a Vershik homeomorphism.
The talk is based on results obtained together with S. Bezuglyi, P. Jorgensen,
J. Kwiatkowski and S. Sanadhya. The work is supported by the NCN (National
Science Center, Poland) Grant 2019/35/D/ST1/01375 and the program “Ex-
cellence initiative - research university” for the AGH University of Krakow


H. Bruin, R. Zweimüller


BZ 9, 9. OG, OMP1