Horizontally stationary generalized Bratteli diagrams
Sergey Bezuglyi, Palle E. T. Jorgensen, Olena Karpel, Jan Kwiatkowski
TL;DR
This paper studies tail-invariant probability measures on the path space $X_B$ of horizontally stationary generalized Bratteli diagrams and identifies when ergodic tail measures on subdiagrams extend to finite measures on $X_B$. A Fourier-transform framework is used to reformulate the tail-invariance criterion (Proposition inv meas). A finite-extension condition is established: the extension $\widehat{\mu}_{\overline i}$ is finite iff $\sum_{n=0}^{\infty} \frac{\sigma^{(n)}}{f^{(n)}_{i_{n+1} i_n}} < \infty$, where $\sigma^{(n)}$ denotes the sum of all entries in the $i_{n+1}$-row except $f^{(n)}_{i_{n+1} i_n}$ (Theorem ext_from_odom), and the paper also shows the existence of diagrams with no finite invariant measures (Prop. prop:no_inv_meas). For a certain class of horizontally stationary diagrams, all ergodic tail-invariant measures arise from extensions from odometers (Theorem thm all erg meas), and there is a necessary-and-sufficient condition for the corresponding Vershik map to be a homeomorphism (Theorem Thm:continVmap).
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.