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Flexibility of measurable and topological nilfactors in dynamical systems

Seljon Akhmedli

TL;DR

The paper investigates the interplay between measurable and topological nilfactors in dynamical systems by constructing strictly ergodic examples that realize all proposed configurations of $Z_k$ and $X_k$. Building on Furstenberg’s classical minimal yet non-uniquely ergodic example, the authors develop coboundary techniques to create controlled separations between measurable and topological nilfactors at various orders, producing systems with nontrivial measurable nilfactors but trivial topological ones, as well as cases where the two notions align up to a point and diverge afterward. A key methodological contribution is the systematic use of intermediate extensions and regionally proximal relations to analyze the structure of nilfactors, together with a robust appeal to Furstenberg coboundaries to realize prescribed isomorphism patterns. The results significantly illuminate how nilfactors can be flexibly orchestrated within minimal and uniquely ergodic frameworks, and they extend to $\mathbb{Z}^{d}$ actions, highlighting broad applicability to higher-rank dynamics. Overall, the work advances understanding of when measurable and topological nilfactors coincide, and provides explicit constructions exhibiting a full range of behaviors for $Z_k$ and $X_k$ across sections.

Abstract

We construct examples of minimal and uniquely ergodic systems realizing all possible behaviors in the interplay of measurable and topological nilfactors. To build such examples, we adapt an idea that stems from Furstenberg's construction of a minimal but not uniquely ergodic system on $\mathbb{T}^2$.

Flexibility of measurable and topological nilfactors in dynamical systems

TL;DR

The paper investigates the interplay between measurable and topological nilfactors in dynamical systems by constructing strictly ergodic examples that realize all proposed configurations of and . Building on Furstenberg’s classical minimal yet non-uniquely ergodic example, the authors develop coboundary techniques to create controlled separations between measurable and topological nilfactors at various orders, producing systems with nontrivial measurable nilfactors but trivial topological ones, as well as cases where the two notions align up to a point and diverge afterward. A key methodological contribution is the systematic use of intermediate extensions and regionally proximal relations to analyze the structure of nilfactors, together with a robust appeal to Furstenberg coboundaries to realize prescribed isomorphism patterns. The results significantly illuminate how nilfactors can be flexibly orchestrated within minimal and uniquely ergodic frameworks, and they extend to actions, highlighting broad applicability to higher-rank dynamics. Overall, the work advances understanding of when measurable and topological nilfactors coincide, and provides explicit constructions exhibiting a full range of behaviors for and across sections.

Abstract

We construct examples of minimal and uniquely ergodic systems realizing all possible behaviors in the interplay of measurable and topological nilfactors. To build such examples, we adapt an idea that stems from Furstenberg's construction of a minimal but not uniquely ergodic system on .
Paper Structure (13 sections, 13 theorems, 68 equations)

This paper contains 13 sections, 13 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions and notations
  4. Necessary constraints
  5. Intermediate factors
  6. Coboundaries
  7. Eigenfunctions
  8. Furstenberg coboundaries
  9. Flexibility
  10. Strong orbit equivalence
  11. A system with nontrivial measurable nilfactors but trivial topological nilfactors
  12. Proof of Theorem \ref{['the theorem in introduction']}
  13. Flexibility for $\mathbb{Z}^{d}$ actions

Key Result

Theorem 1.1

For all $0 \leq j \leq \ell \leq k,\; k\geq 1$, there exists a strictly ergodic system which admits the following: Furthermore, for every minimal, ergodic, and non-weak mixing $(X,\mu,T)$ there exist $j,\ell$, and $k$, such that $X$ satisfies properties (1)-(4).

Theorems & Definitions (26)

  • Theorem 1.1
  • Lemma 2.1: Host and Kra, Chapter 5, Lemma 19 HK2
  • Definition 2.2
  • Remark 2.3
  • Lemma 3.1
  • proof
  • Corollary 3.2: Furstenberg Fur61
  • Remark 3.3
  • Lemma 4.1
  • proof
  • ...and 16 more