Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the spectrum of non-ergodic measures

Michael Francis, Christopher Ramsey, Nicolae Strungaru

TL;DR

The paper develops a general transfer principle for spectral data in topological dynamical systems with abelian, locally compact groups. By combining Choquet theory with operator-algebraic spectral tools, it shows that if an invariant measure $m$ has its spectrum contained in a Borel set $B$, then almost every ergodic component $\omega$ (in the Choquet representation) also has spectrum contained in $B$, yielding that pure point spectrum persists almost surely. This yields concrete corollaries for mean and Besicovitch almost periodic measures: hulls of mean almost periodic measures contain Besicovitch almost periodic measures, after suitable translations. The framework is then specialized to translation bounded measures, linking dynamical and diffraction spectra via autocorrelation and the Dworkin argument, and establishing that pure point dynamical spectrum implies pure point diffraction spectrum, thereby clarifying the correspondence between spectral and structural order in aperiodic systems.

Abstract

Consider a topological dynamical system where the group is abelian and the topologies are locally compact and second-countable. Given an invariant measure for this system, we show that if its dynamical spectrum is contained in some Borel subset of the dual group then the same holds almost surely for all ergodic measures arising via the Choquet theorem. In particular, if the invariant measure has pure point dynamical spectrum, so do almost all the ergodic measures. As an application, we show that given any mean almost periodic measure, in its hull there exists a Besicovitch almost periodic measure.

On the spectrum of non-ergodic measures

TL;DR

The paper develops a general transfer principle for spectral data in topological dynamical systems with abelian, locally compact groups. By combining Choquet theory with operator-algebraic spectral tools, it shows that if an invariant measure has its spectrum contained in a Borel set , then almost every ergodic component (in the Choquet representation) also has spectrum contained in , yielding that pure point spectrum persists almost surely. This yields concrete corollaries for mean and Besicovitch almost periodic measures: hulls of mean almost periodic measures contain Besicovitch almost periodic measures, after suitable translations. The framework is then specialized to translation bounded measures, linking dynamical and diffraction spectra via autocorrelation and the Dworkin argument, and establishing that pure point dynamical spectrum implies pure point diffraction spectrum, thereby clarifying the correspondence between spectral and structural order in aperiodic systems.

Abstract

Consider a topological dynamical system where the group is abelian and the topologies are locally compact and second-countable. Given an invariant measure for this system, we show that if its dynamical spectrum is contained in some Borel subset of the dual group then the same holds almost surely for all ergodic measures arising via the Choquet theorem. In particular, if the invariant measure has pure point dynamical spectrum, so do almost all the ergodic measures. As an application, we show that given any mean almost periodic measure, in its hull there exists a Besicovitch almost periodic measure.
Paper Structure (15 sections, 29 theorems, 93 equations)

This paper contains 15 sections, 29 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Spaces of measures
  4. Translation bounded measures
  5. Almost periodic functions
  6. Invariant measures on the hull
  7. The Borel structure of probability measures
  8. Review of Choquet theory
  9. Spectral decomposition
  10. Spectral measures and the spectral theorem
  11. Dynamical spectrum
  12. Main results
  13. Dynamical and diffraction spectrum of translation bounded measures
  14. Dynamical spectrum
  15. Diffraction spectrum

Key Result

Lemma 2.16

Let $X$ be a second-countable locally compact Hausdorff space. Then, there exists some countable set $D \subseteq C_{\mathsf{c}}(X)$ such that, for each $\varphi \in C_{\mathsf{c}}(X)$, there exists a sequence $\psi_n \in D$ that converges to $\varphi$ uniformly and whose supports are contained in a

Theorems & Definitions (82)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Remark 2.7
  • Remark 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 72 more