Minimal PI-systems with all points are multiply minimal

TL;DR

This paper constructs a specific minimal subshift that is an open proximal extension of its maximal equicontinuous factor, demonstrating that all points are multiply recurrent minimal, thus solving an open problem in the field.

Contribution

It introduces a new example of a minimal PI-system where every point is multiply minimal, addressing a previously unresolved question.

Findings

Constructed a minimal subshift with all points multiply minimal

Proved the subshift is an open proximal extension of its maximal equicontinuous factor

Solved an open problem regarding the existence of such systems

Abstract

We construct a minimal subshift \((X^{*},\sigma)\) that serves as an open proximal extension of its maximal equicontinuous factor. We establish that every point in this subshift is multiply recurrent minimal. This work solves an open problem raised by Huang, Shao and Ye regarding the existence of minimal PI-systems such that each point is multiply minimal.