On some properties of basic sets

TL;DR

This paper systematically studies the fundamental properties of basic sets and quasi-sections in zero-dimensional topological dynamical systems, clarifying their roles in Bratteli--Vershik models and $C^*$-algebras.

Contribution

It provides a systematic analysis of the topological properties of basic sets and quasi-sections in zero-dimensional systems, extending the understanding of their role in dynamical and algebraic models.

Findings

Every basic set is continuously decisive under certain conditions.

Existence of a minimal continuously decisive basic set in systems with dense aperiodic orbits.

Characterization of basic sets in relation to minimal sets and dense orbits.

Abstract

In the theory of zero-dimensional systems and their relation to $C^*$-algebras, Poon (1990) introduced a class of closed sets. We call the closed sets quasi-sections. Medynets (2006) introduced basic sets that are part of quasi-sections in his study of aperiodic zero-dimensional systems and their relation to Bratteli--Vershik models and $C^*$-algebras. Downarowicz and Karpel (2019) introduced the notion of decisiveness in the theory of Bratteli--Vershik models. We previously clarified that particular quasi-sections can be the "bases" of the decisive Bratteli--Vershik models for zero-dimensional systems with dense aperiodic orbits. We call them continuously decisive quasi-sections. However, even the basic topological properties of quasi-sections and the basic sets have not been studied systematically. This paper presents such a systematic study. Some properties are defined, stated, and proved in the general settings of compact Hausdorff topological dynamics. For example, if a topological dynamical system has dense aperiodic orbits and no wandering points, then every basic set is continuously decisive. If a zero-dimensional system has dense aperiodic orbits, then there exists a minimal continuously decisive basic set such that for every minimal set, there exists a unique common point.