Graph covers and ergodicity for 0-dimensional systems

TL;DR

This paper explores the relationship between graph covers, circuits, and ergodic measures in 0-dimensional systems, extending previous results to more general cases and analyzing conditions for ergodicity and measure representation.

Contribution

It generalizes the analysis of ergodic measures and circuit independence in 0-dimensional systems using graph covers beyond Bratteli--Vershik frameworks.

Findings

Demonstrates that all ergodic measures are limits of certain graph cover sequences.

Provides conditions for circuit linear independence in general graph covers.

Extends unique ergodicity criteria to unbounded rank cases.

Abstract

Bratteli--Vershik systems have been widely studied. In the context of general 0-dimensional systems, Bratteli--Vershik systems are homeomorphisms that have Kakutani--Rohlin refinements. Bratteli diagram has a strong power to analyze such systems. Besides this approach, general graph covers can be used to represent any 0-dimensional system. Indeed, all 0-dimensional systems can be described as a certain kind of sequences of graph covers that may not be brought about by the Kakutani--Rohlin partitions. In this paper, we follow the context of general graph covers to analyze the relations between ergodic measures and circuits of graph covers. First, we formalize the condition for a sequence of graph covers to represent minimal Cantor systems. In constructing invariant measures, we deal with general compact metrizable 0-dimensional systems. In the context of Bratteli diagrams with finite rank, it has previously been mentioned that all ergodic measures should be limits of some combinations of towers of Kakutani--Rohlin refinements. We demonstrate this for general 0-dimensional case, and develop a theorem that expresses the coincidence of the time average and the space average for ergodic measures. Additionally, we formulate a theorem that signifies the old relation between uniform convergence and unique ergodicity in the context of graph circuits for general 0-dimensional systems. Unlike previous studies, in our case of general graph covers, there arise a possibility of linear dependence of circuits. We give a condition for a full circuit system to be linearly independent. Previous researches also showed that the bounded combinatorics imply unique ergodicity. We present a lemma that enables us to consider unbounded ranks of winding matrices.