Operator Theory Seminar - Assistant Professor Olena Karpel; Department of Applied Mathematics, AGH University of Science and Technology

Operator Theory Seminar - Assistant Professor Olena Karpel; Department of Applied Mathematics, AGH University of Science and Technology promotional image

Invariant measures and dynamics for reducible generalized Bratteli diagrams

Assistant Professor Olena Karpel; Department of Applied Mathematics, AGH University of Science and Technology

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 nonstationary reducible generalized Bratteli diagrams with infinitely many simple standard subdiagrams, in particular, with infinitely many odometers, characterize the sets of all probability ergodic invariant measures for such diagrams and study orders under which the diagrams can support a Vershik homeomorphism.

In contrast to the case of standard Bratteli diagrams, there are examples of generalized Bratteli diagrams with a unique minimal and a unique maximal path such that the corresponding Vershik map cannot be prolonged to a homeomorphism. We show that in the class of ordered generalized Bratteli diagrams with a unique infinite minimal path and a unique infinite maximal path, one can find examples of diagrams such that (i) both the Vershik map φB and its inverse φ−1 B are not continuous; (ii) the Vershik map φB is continuous but the inverse φ−1 B is discontinuous; (iii) both the Vershik map φB and its inverse φ−1 B are continuous.

The talk is based on results obtained together with Sergey Bezuglyi, Palle E.T. Jorgensen, Jan Kwiatkowski and Shrey Sanadhya. The work is supported by the NCN (National Science Center, Poland) Grant 2019/35/D/ST1/01375 and the program “Excellence initiative - research university” for the AGH University of Science and Technology.

Tuesday, March 26, 2024 1:30pm to 2:30pm
Van Allen Hall
309
30 North Dubuque Street, Iowa City, IA 52242
View on Event Calendar
Individuals with disabilities are encouraged to attend all University of Iowa–sponsored events. If you are a person with a disability who requires a reasonable accommodation in order to participate in this program, please contact Department of Physics & Astronomy in advance at 319-335-1686 or physics-astronomy@uiowa.edu.