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On orthogonality to uniquely ergodic systems

Martyna Górska, Mariusz Lemańczyk, Thierry de la Rue

TL;DR

This work resolves Boshernitzan's problem by characterizing sequences orthogonal to all uniquely ergodic systems through Furstenberg systems and disjointness principles, and by connecting this orthogonality to averaged Chowla properties via characteristic-class frameworks. The authors develop a robust toolkit—lifting lemmas, orbital uniquely ergodic models, relative ergodic decompositions, and confinement criteria—to reduce global orthogonality questions to fiberwise and factor-level tests, including Markov-image orthogonality. They establish general equivalences for orthogonality to classes of zero-entropy (ZE) and more general characteristic classes, provide concrete instances (e.g., DISP, ID, NIL$_1$), and even exhibit a counterexample showing Veech and Sarnak conditions need not coincide in full generality. Consequently, the paper unifies ergodic-theoretic disjointness, spectral analysis, and arithmetic dynamics to describe when bounded sequences, especially multiplicative ones, remain orthogonal to UE systems, yielding results relevant to averaged Chowla-type phenomena and the structure of Furstenberg systems.

Abstract

We solve Boshernitzan's problem of characterization (in terms of so called Furstenberg systems) of bounded sequences that are orthogonal to all uniquely ergodic systems. Some variations of Boshernitzan's problem involving characteristic classes are considered. As an application, we characterize sequences orthogonal to all uniquely ergodic systems whose (unique) invariant measure yields a discrete spectrum automorphism as those satisfying an averaged Chowla property.

On orthogonality to uniquely ergodic systems

TL;DR

This work resolves Boshernitzan's problem by characterizing sequences orthogonal to all uniquely ergodic systems through Furstenberg systems and disjointness principles, and by connecting this orthogonality to averaged Chowla properties via characteristic-class frameworks. The authors develop a robust toolkit—lifting lemmas, orbital uniquely ergodic models, relative ergodic decompositions, and confinement criteria—to reduce global orthogonality questions to fiberwise and factor-level tests, including Markov-image orthogonality. They establish general equivalences for orthogonality to classes of zero-entropy (ZE) and more general characteristic classes, provide concrete instances (e.g., DISP, ID, NIL), and even exhibit a counterexample showing Veech and Sarnak conditions need not coincide in full generality. Consequently, the paper unifies ergodic-theoretic disjointness, spectral analysis, and arithmetic dynamics to describe when bounded sequences, especially multiplicative ones, remain orthogonal to UE systems, yielding results relevant to averaged Chowla-type phenomena and the structure of Furstenberg systems.

Abstract

We solve Boshernitzan's problem of characterization (in terms of so called Furstenberg systems) of bounded sequences that are orthogonal to all uniquely ergodic systems. Some variations of Boshernitzan's problem involving characteristic classes are considered. As an application, we characterize sequences orthogonal to all uniquely ergodic systems whose (unique) invariant measure yields a discrete spectrum automorphism as those satisfying an averaged Chowla property.
Paper Structure (63 sections, 71 theorems, 381 equations, 1 figure)

This paper contains 63 sections, 71 theorems, 381 equations, 1 figure.

Table of Contents

  1. Introduction
  2. General background and main notations
  3. Presentation of the main problem and motivations
  4. Ergodic theory results
  5. Solution of Boshernitzan's problem
  6. Applications
  7. Auxiliary results
  8. Preparatory material
  9. Lifting lemma
  10. Orbital uniquely ergodic models
  11. Joinings and Markov operators
  12. Weakly ergodic part of a measure-preserving system
  13. Koopman operator and spectral measures
  14. Furstenberg systems of arithmetic functions
  15. Confined extensions
  16. ...and 48 more sections

Key Result

Theorem 2.1

Assume that $(Y,S)$ and $(X,T)$ are topological systems. Let $\nu\in M(X,T)$, $u\in Y$ be quasi-generic along an increasing sequence $(N_m)$ for $\kappa\in M(Y,S)$ and $\rho\in J(\kappa,\nu)$. Then there exist a sequence $(x_n)\subset X$ and a subsequence $(N_{m_\ell})$ such that $(S^nu,x_n)$ is qua

Figures (1)

  • Figure 1: The partition of $X\times X$ into Rokhlin towers. We represented the atoms of $\mathscr{P}\times \mathscr{P}$ inside only one of these Rokhlin towers.

Theorems & Definitions (157)

  • Theorem 2.1: Ka-Ku-Le-Ru
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4: Ab-Le-Ru
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Lemma 2.7: del Junco-Rudolph, Ju-Ru, proof of Prop. 5.3
  • Lemma 2.8
  • proof
  • ...and 147 more