Bratteli--Vershik models and graph covering models

TL;DR

This paper introduces weighted and flexible graph covering models that are nearly equivalent to Bratteli--Vershik models, providing new tools for analyzing zero-dimensional dynamical systems and constructing transitive substitution subshifts.

Contribution

It develops new graph covering models that are almost equivalent to Bratteli--Vershik models, enabling analysis and construction of zero-dimensional dynamical systems.

Findings

Every invertible zero-dimensional system admits a Bratteli--Vershik model.

An analogue of Krieger's lemma is established for these systems.

A method for constructing transitive substitution subshifts is demonstrated.

Abstract

Based on our previous graph covering method, we introduce weighted graph covering models and flexible graph covering models that are almost equivalent to the well-known Bratteli--Vershik models. These models play important roles in showing that every invertible dynamical system on compact metrizable zero-dimensional space admits a non-trivial Bratteli--Vershik model and a basic set. We can also obtain an analogue of Krieger's lemma for compact metrizable zero-dimensional systems. The flexible graph covering models enable us to consider "stationary" graph covering models, by which some portion of the substitution subshifts can be expressed. As an application, we show a way of constructing some class of transitive substitution subshifts.