The complexity of topological conjugacy of pointed Cantor minimal systems

TL;DR

This paper investigates the complexity of classifying pointed Cantor minimal systems up to topological conjugacy, revealing its equivalence to a known Borel relation and exploring implications for ordered Bratteli diagrams.

Contribution

It establishes the Borel complexity of topological conjugacy for pointed Cantor minimal systems and connects it to ordered Bratteli diagrams, providing new insights into their classification.

Findings

Topological conjugacy is Borel bireducible with a specific equivalence relation on real sequences.

This relation serves as a lower bound for the complexity of classifying Cantor minimal systems.

Results have applications in understanding the structure of Bratteli diagrams.

Abstract

In this paper, we analyze the complexity of topological conjugacy of pointed Cantor minimal systems from the point of view of descriptive set theory. We prove that the topological conjugacy relation on pointed Cantor minimal systems is Borel bireducible with the Borel equivalence relation $\Delta_{\mathbb{R}}^+$ on $\mathbb{R}^{\mathbb{N}}$ defined by $x \Delta_{\mathbb{R}}^+ y \Leftrightarrow \{x_i:i \in \mathbb{N}\}=\{y_i:i \in \mathbb{N}\}$. Moreover, we show that $\Delta_{\mathbb{R}}^+$ is a lower bound for the Borel complexity of topological conjugacy of Cantor minimal systems. Finally, we interpret our results in terms of properly ordered Bratteli diagrams and discuss some applications.