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Dynamics on the path space of generalized Bratteli diagrams
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Dynamics on the path space of generalized Bratteli diagrams

Sergey Bezuglyi, Palle E. T Jorgensen, Olena Karpel and Shrey Sanadhya
ArXiv.org
12/28/2022
DOI: 10.48550/arxiv.2212.13803
url
https://doi.org/10.48550/arxiv.2212.13803View
Preprint (Author's original)This preprint has not been evaluated by subject experts through peer review. Preprints may undergo extensive changes and/or become peer-reviewed journal articles. Open Access

Abstract

Bratteli-Vershik models have been very successfully applied to the study of dynamics on compact metric spaces and, 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: (i) 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 $\sigma$-finite). (ii) We provide criteria for the topological transitivity of the tail equivalence relation. (iii) We describe classes of stationary generalized Bratteli diagrams (hence Borel dynamical systems) that: (a) do not support a probability tail invariant measure, (b) do not admit a continuous Vershik map, (c) are not uniquely ergodic with respect to the tail equivalence relation. (iv) We provide an application of the theory of stochastic matrices to analyze diagrams with positive recurrent incidence matrices.
 Mathematics - Dynamical Systems

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