Horizontally stationary generalized Bratteli diagrams

TL;DR

This paper explores the properties of horizontally stationary Bratteli diagrams with infinite levels, focusing on ergodic measures and conditions for Vershik map existence, revealing new structural phenomena in infinite graph dynamics.

Contribution

It introduces the concept of horizontal stationarity in infinite Bratteli diagrams and characterizes ergodic measures and Vershik map conditions for this class.

Findings

Explicit description of ergodic tail invariant measures.

All ergodic measures are extensions of odometer measures in certain cases.

Conditions established for the existence of a continuous Vershik map.

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.