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Spectrum of invariant measures via generic points

Sejal Babel, Melih Emin Can, Dominik Kwietniak, Piotr Oprocha

TL;DR

The paper develops a framework linking the spectral data of a $T$-invariant measure to the ergodic behavior of its generic points by introducing regular Wiener–Wintner generic points and connecting orbit averages to the Kronecker factor. It introduces the Besicovitch pseudometric $D_B$ and a new metric $\bar{\rho}$ on invariant measures to study limits and closedness of spectral properties under convergence, proving that discrete spectrum, total ergodicity, weak mixing, and related classes are preserved under $D_B$-limits and $\bar{\rho}$-limits. A key outcome is that limit measures inherit spectral characteristics from almost all approximating measures, enabling uniform, non-symbolic proofs of spectral phenomena and enabling applications to Mirsky measures for $\mathscr{B}$-free shifts. The results unify and extend prior discrete-spectrum characterizations, provide new proofs of known spectral properties, and yield applications to shifts and specification-type dynamical systems with broad potential impact in ergodic theory and topological dynamics.

Abstract

We describe the spectrum of an ergodic invariant measure by examining the behaviour of its generic points. We define regular Wiener--Wintner generic points for a measure to generalise the characterisation of generic points for discrete spectrum measure from Lenz et al. [Ergodic Theory and Dynamical Systems vol. \textbf{44} (2024), no. 2, 524--568]. We also study limits of sequences of generic points with respect to the Besicovitch pseudometric. This translates to results about limits of measures with respect to the metric rho-bar $\barρ$ generalising Ornstein's d-bar metric. We study how the spectrum behaves when passing to the limit and we prove that points generic for discrete spectrum, totally ergodic, or (weakly) mixing measures, property K, zero entropy measures form a closed set with respect to the Besicovitch pseudometric. Hence, the same holds for corresponding measures with respect to the rho-bar metric. Our methods have already been used to prove existence of ergodic measures with desired properties, in particular with discrete spectrum. They also lead to a new proof of rational discrete spectrum of the Mirsky measure associated with a given set of $\mathscr B$-free numbers.

Spectrum of invariant measures via generic points

TL;DR

The paper develops a framework linking the spectral data of a -invariant measure to the ergodic behavior of its generic points by introducing regular Wiener–Wintner generic points and connecting orbit averages to the Kronecker factor. It introduces the Besicovitch pseudometric and a new metric on invariant measures to study limits and closedness of spectral properties under convergence, proving that discrete spectrum, total ergodicity, weak mixing, and related classes are preserved under -limits and -limits. A key outcome is that limit measures inherit spectral characteristics from almost all approximating measures, enabling uniform, non-symbolic proofs of spectral phenomena and enabling applications to Mirsky measures for -free shifts. The results unify and extend prior discrete-spectrum characterizations, provide new proofs of known spectral properties, and yield applications to shifts and specification-type dynamical systems with broad potential impact in ergodic theory and topological dynamics.

Abstract

We describe the spectrum of an ergodic invariant measure by examining the behaviour of its generic points. We define regular Wiener--Wintner generic points for a measure to generalise the characterisation of generic points for discrete spectrum measure from Lenz et al. [Ergodic Theory and Dynamical Systems vol. \textbf{44} (2024), no. 2, 524--568]. We also study limits of sequences of generic points with respect to the Besicovitch pseudometric. This translates to results about limits of measures with respect to the metric rho-bar generalising Ornstein's d-bar metric. We study how the spectrum behaves when passing to the limit and we prove that points generic for discrete spectrum, totally ergodic, or (weakly) mixing measures, property K, zero entropy measures form a closed set with respect to the Besicovitch pseudometric. Hence, the same holds for corresponding measures with respect to the rho-bar metric. Our methods have already been used to prove existence of ergodic measures with desired properties, in particular with discrete spectrum. They also lead to a new proof of rational discrete spectrum of the Mirsky measure associated with a given set of -free numbers.
Paper Structure (13 sections, 35 theorems, 126 equations)

This paper contains 13 sections, 35 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Wiener--Wintner generic points
  4. Frequencies of Wiener--Wintner Generic Points and Spectra of Measures
  5. A characterisation of discrete spectrum and Besicovitch almost periodic points
  6. Besicovitch pseudometric and its relatives
  7. A new metric on the space of invariant measures
  8. On sequences of generic points converging with respect to the Besicovitch pseudometric
  9. Disjointness of measure preserving systems
  10. Applications
  11. Application for $\mathscr{B}$-free shifts
  12. Density of ergodic discrete rational spectrum measures for topological dynamical systems with the specification property
  13. Density of ergodic measures among all invariant measures for topological dynamical systems with the vague specification property

Key Result

Lemma 3.3

Let $x\in X$ and $\xi\in\mathbb{S}^1$. If $(n(k))_{k=1}^\infty\subseteq\mathbb{N}$ is strictly increasing and $\mathcal{D}\subseteq C(X)$ are such that for every $f\in\mathcal{D}$ the sequence $(\mathop{\mathrm{A}}\nolimits_{n(k)}[f,\bar{\xi}](x))_{k=1}^\infty$ has a limit $\Phi_\xi(f)$, then the ma

Theorems & Definitions (65)

  • Definition 3.1
  • Remark 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • proof
  • Remark 3.6
  • Lemma 3.7
  • proof
  • Remark 3.8
  • ...and 55 more