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Minimal PI-systems with all points are multiply minimal

Zijie Lin, Kangbo Ouyang

TL;DR

The paper constructs a minimal subshift $(X^*,\sigma)$ that is an open proximal extension of its maximal equicontinuous factor (an odometer) and proves that every $x\in X^*$ is multiply recurrent minimal, i.e., $x^{(d)}$ is recurrent for $\tau_d$ for all $d\ge2$. The approach realizes multiple recurrence within finite words by an inductive word-construction guided by a key combinatorial Lemma $l:\mathrm{combin}$, inspired by Oprocha's doubly minimal example. This yields a minimal PI-system with all points multiply minimal, answering HSY22 in the affirmative. The paper also clarifies the structure around PI-systems via the structure theorem and demonstrates openness of the maximal equicontinuous-factor map to the odometer. Overall, the construction provides a new canonical example of a minimal PI-system with strong recurrence properties and a proximal-open extension to an equicontinuous factor.

Abstract

We construct a minimal subshift \((X^{*},σ)\) 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.

Minimal PI-systems with all points are multiply minimal

TL;DR

The paper constructs a minimal subshift that is an open proximal extension of its maximal equicontinuous factor (an odometer) and proves that every is multiply recurrent minimal, i.e., is recurrent for for all . The approach realizes multiple recurrence within finite words by an inductive word-construction guided by a key combinatorial Lemma , inspired by Oprocha's doubly minimal example. This yields a minimal PI-system with all points multiply minimal, answering HSY22 in the affirmative. The paper also clarifies the structure around PI-systems via the structure theorem and demonstrates openness of the maximal equicontinuous-factor map to the odometer. Overall, the construction provides a new canonical example of a minimal PI-system with strong recurrence properties and a proximal-open extension to an equicontinuous factor.

Abstract

We construct a minimal subshift \((X^{*},σ)\) 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.

Paper Structure

This paper contains 12 sections, 19 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. Main ideas
  3. Preliminaries
  4. Structure theorem of minimal systems
  5. Odometer
  6. Symbolic dynamics
  7. Permutations
  8. Combinatorial lemmas
  9. Proof of Theorem 1.2
  10. Inductive construction of words
  11. Multiple minimality
  12. Open and proximal extensions

Key Result

Theorem 1.2

There is a minimal subshift $(X^*,\sigma)$ such that it is an open proximal extension of its maximal equicontinuous factor, and for $d\ge 2$ and $x\in X^*$, $x^{(d)}$ is $\sigma_d$-minimal.

Theorems & Definitions (36)

  • Theorem 1.2
  • Theorem 2.1: Structure theorem for minimal systems
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 4.1
  • ...and 26 more