Dynamical pair assignments

TL;DR

This paper introduces dynamical pair assignments to unify various relations in topological dynamical systems, establishing conditions for their Borel measurability and realizability, thus generalizing concepts like CPE, weak mixing, and UPE.

Contribution

It defines the concept of dynamical pair assignments and explores their properties, providing a unified framework that generalizes multiple existing dynamical system concepts.

Findings

The space of ull systems is always a Borel set.

The space of realizable systems is Borel if and only if a natural rank is bounded.

The framework generalizes concepts like CPE, weak mixing, and UPE.

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.