Horizontally stationary generalized Bratteli diagrams
Volume 271 / 2025
Fundamenta Mathematicae 271 (2025), 195-225
MSC: Primary 37A05; Secondary 37B05, 37A40, 54H05, 05C60
DOI: 10.4064/fm240916-6-6
Published online: 21 November 2025
Abstract
Bratteli diagrams with countably infinite levels exhibit a new phenomenon: they can be horizontally stationary. The incidence matrices of these horizontally stationary Bratteli diagrams are infinite banded Toeplitz matrices. In this paper, we study the fundamental properties of horizontally stationary Bratteli diagrams. In these diagrams, we provide an explicit description of ergodic tail invariant probability measures. For a certain class of horizontally stationary Bratteli diagrams, we prove that all ergodic tail invariant probability measures are extensions of measures from odometers. Additionally, we establish conditions for the existence of a continuous Vershik map on the path space of a horizontally stationary Bratteli diagram.
