Regionally proximal relation of order $d$ along arithmetic progressions and nilsystems

TL;DR

This paper explores the structure of topological dynamical systems using the regionally proximal relation along arithmetic progressions, revealing connections to nilsystems, zero entropy, and providing new examples of systems with specific spectral properties.

Contribution

It introduces the ${f AP}^{[d]}$ relation, characterizes systems with this relation as nilsystems, and establishes conditions for when these relations coincide, including new examples with unique spectral features.

Findings

Systems with ${f AP}^{[d]}=\Delta$ are isomorphic to $d$-step pro-nilsystems.

Strictly ergodic distal systems with isomorphic maximal pro-nilsystems satisfy ${\bf AP}^{[d]}={\bf RP}^{[d]}$.

Constructed example of a strictly ergodic distal system with discrete spectrum but non-isomorphic maximal equicontinuous factor.

Abstract

The regionally proximal relation of order $d$ along arithmetic progressions, namely ${\bf AP}^{[d]}$ for $d\in \N$, is introduced and investigated. It turns out that if $(X,T)$ is a topological dynamical system with ${\bf AP}^{[d]}=\Delta$, then each ergodic measure of $(X,T)$ is isomorphic to a $d$-step pro-nilsystem, and thus $(X,T)$ has zero entropy. Moreover, it is shown that if $(X,T)$ is a strictly ergodic distal system with the property that the maximal topological and measurable $d$-step pro-nilsystems are isomorphic, then ${\bf AP}^{[d]}={\bf RP}^{[d]}$ for each $d\in {\mathbb N}$. It follows that for a minimal $\infty$-pro-nilsystem, ${\bf AP}^{[d]}={\bf RP}^{[d]}$ for each $d\in {\mathbb N}$. An example which is a strictly ergodic distal system with discrete spectrum whose maximal equicontinuous factor is not isomorphic to the Kronecker factor is constructed.