Nonsingular transformations and dimension spaces

TL;DR

This paper constructs explicit dimension spaces for adic transformations on Bratteli diagrams with Markov measures, linking them to Poisson boundaries and illustrating their use in analyzing nonsingular transformations.

Contribution

It introduces a method to associate dimension spaces with adic transformations, providing new tools for studying nonsingular dynamical systems.

Findings

Dimension spaces correspond to matrix-valued random walks on 

Poisson boundaries identified with the original dynamical system

Examples demonstrate applications in nonsingular transformation analysis

Abstract

For any adic transformation $T$ defined on the path space $X$ of an ordered Bratteli diagram, endowed with a Markov measure $\mu$, we construct an explicit dimension space (which corresponds to a matrix values random walk on $\mathbb{Z}$) whose Poisson boundary can be identified as a $\mathbb{Z}$-space with the dynamical system $(X,\mu,T)$. We give a couple of examples to show how dimension spaces can be used in the study of nonsingular transformations.