Bratteli diagrams in Borel dynamics

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 are as follows. (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 and (b) are not uniquely ergodic with respect to the tail equivalence relation. (iv) We describe classes of generalized Bratteli diagrams which can or cannot admit a continuous Vershik map and construct a Vershik map which is a minimal homeomorphism of a (non-locally compact) Polish space. (v) We provide an application of the theory of stochastic matrices to analyze diagrams with positive recurrent incidence matrices.