Bratteli diagrams in Borel dynamics
Sergey Bezuglyi, Palle E.T. Jorgensen, Olena Karpel, Shrey Sanadhya

TL;DR
This paper extends Bratteli-Vershik models to non-compact Borel dynamical systems using generalized diagrams with countably infinite vertices, providing criteria for measures, transitivity, and Vershik maps.
Contribution
It introduces a framework for analyzing non-compact Borel systems via generalized Bratteli diagrams, including measure existence, transitivity, and Vershik map construction.
Findings
Criteria for tail-invariant measure existence and uniqueness.
Conditions for topological transitivity of tail equivalence.
Construction of minimal Vershik maps on non-locally compact spaces.
Abstract
Bratteli-Vershik models have been very successfully applied to the study of various dynamical systems, in particular, in Cantor dynamics. In this paper, we study dynamics on the path spaces of generalized Bratteli diagrams that form models for non-compact Borel dynamical systems. Generalized Bratteli diagrams have countably infinite many vertices at each level, thus the corresponding incidence matrices are also countably infinite. We emphasize differences (and similarities) between generalized and classical Bratteli diagrams. Our main results: We utilize Perron-Frobenius theory for countably infinite matrices to establish criteria for the existence and uniqueness of tail-invariant path space measures (both probability and -finite). We provide criteria for the topological transitivity of the tail equivalence relation. We describe classes of stationary…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsTopological and Geometric Data Analysis · Mathematical Dynamics and Fractals · Homotopy and Cohomology in Algebraic Topology
