$RP^{[d]}$ is an equivalence relation: An enveloping semigroup proof

TL;DR

This paper provides a new proof using enveloping semigroups to show that the regional proximal relation of order d is an equivalence relation for minimal flows with an abelian group action.

Contribution

It offers a purely enveloping semigroup proof of a known theorem regarding regional proximal relations, expanding the theoretical toolkit.

Findings

Regional proximal relation of order d is an equivalence relation.

The proof is purely enveloping semigroup-based.

Applicable to minimal flows with abelian group actions.

Abstract

We present a purely enveloping semigroup proof of a theorem of Shao and Ye which asserts that for an abelian group $T$, a minimal flow $(X,T)$ and any integer $d \ge 1$, the regional proximal relation of order $d$ is an equivalence relation.