Bowen's Problem 32 and the conjugacy problem for systems with specification

TL;DR

The paper investigates Bowen's Problem 32 by constructing a parametric class of symbolic systems with the specification property and proving that the conjugacy relation on this class is highly nontrivial in the descriptive-set-theoretic sense (not smooth, not hyperfinite, not treeable). It uses Borel-reducibility, turbulence, and condensed-point arguments to establish the intrinsic complexity of classifying such systems, and extends the analysis to pointed systems on Cantor and Hilbert cube spaces, connecting to $E_{ ext{csc}}$ and $E_{ ext{tt}}$ relations. Key results include a positive resolution to a question of Ding–Gu for zero-dimensional completions, and turbulence-based non-classifiability for pointed transitive Hilbert cube systems. The work thus shows that any attempt at a concrete invariant-based classification for symbolic systems with specification faces fundamental obstacles, while clarifying the landscape of pointed-system classifications and their connections to broader Borel-reducibility hierarchies.

Abstract

We show that Rufus Bowen's Problem 32 on the classification of symbolic systems with the specification property does not admit a solution that would use concrete invariants. To this end, we construct a class of symbolic systems with the specification property and show that the conjugacy relation on this class is too complicated to admit such a classification. More generally, we gauge the complexity of the classification problem for symbolic systems with the specification property. Along the way, we also provide answers to two questions related to the classification of pointed systems with the specification property: to a question of Ding and Gu related to the complexity of the classification of pointed Cantor systems with the specification property and to a question of Bruin and Vejnar related to the complexity of the classification of pointed Hilbert cube systems with the specification property.

Figures (1)

• Figure 1: An Oxtoby sequence.

Theorems & Definitions (77)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 3.1
• Definition 3.2
• Proposition 3.3
• proof
• Lemma 4.1
• proof
• Proposition 4.2