Measures and dynamics on Pascal-Bratteli diagrams

TL;DR

This paper explores measures and dynamical systems on generalized Pascal-Bratteli diagrams, characterizing invariant measures, path structures, and conditions for Vershik map homeomorphisms.

Contribution

It introduces a framework for analyzing measures and dynamics on complex Bratteli diagrams, including approximation of measures and diverse path configurations.

Findings

All ergodic tail invariant measures are described.

Conditions for Vershik map being a homeomorphism are established.

Various orderings produce different path and minimal/maximal path structures.

Abstract

We introduce and study dynamical systems and measures on stationary generalized Bratteli diagrams $B$ that are represented as the union of countably many classical Pascal-Bratteli diagrams. We describe all ergodic tail invariant measures on $B$. For every probability tail invariant measure $\nu_p$ on the classical Pascal-Bratteli diagram, we approximate the support of $\nu_p$ by the path space of a subdiagram. By considering various orders on the edges of $B$, we define dynamical systems with various properties. We show that there exist orders such that the sets of infinite maximal and infinite minimal paths are empty. This implies that the corresponding Vershik map is a homeomorphism. We also describe orders on both $B$ and the classical Pascal-Bratteli diagram that generate either uncountably many minimal infinite and uncountably many maximal infinite paths, or uncountably many minimal infinite paths alongside countably infinitely many maximal infinite paths.