Dynamical pair assignments
Udayan B. Darji, Felipe García-Ramos
TL;DR
The paper introduces dynamical pair assignments $\mathcal{P}$ to unify various dynamical-pair notions (e.g., entropy pairs, regionally proximal pairs) under a descriptive-set-theoretic framework. It defines $\mathcal{P}$-full and $\mathcal{P}$-realizable systems and develops the $\mathcal{P}$-rank via the $\Gamma$-closure to characterize when the realizability class is Borel; specifically, $R(\mathcal{P}_X)$ is Borel iff the $\mathcal{P}$-rank is bounded on $X$. The paper proves that the $\mathcal{P}$-rank is a coanalytic rank and shows that the $\Gamma$-rank is a coanalytic rank, providing a unifying approach to how dynamical-pair structures behave across factors and extensions. Through concrete examples with entropy pairs and regionally proximal pairs, the results connect classical dynamical properties (CPE, UPE, weak mixing) to the descriptive-set-theoretic complexity of corresponding realizability sets, yielding a rank-based criterion for Borelness. This framework offers a potent lens for studying entropy, equicontinuity, and related dynamical phenomena in a unified, rigorous manner.
Abstract
Relations between points in the phase space are central to the study of topological dynamical systems. Since many of these relations share common properties, it is natural to study them within a unified framework. To this end, we introduce the concept of \textit{dynamical pair assignments} $\mathcal{P}$. We then introduce the notions of a dynamical system being $\mathcal{P}$-full and $\mathcal{P}$-realizable, which generalize several existing concepts in the field like CPE, weak mixing and UPE. Our results establish that the space of $\mathcal{P}$-full systems is always a Borel set, while the space of $\mathcal{P}$-realizable systems is Borel if and only if an associated natural rank is bounded.