Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Keplerian shear with Rajchman property

Arthur Boos, Benoit Saussol

TL;DR

The paper develops a Rajchman-measure framework to characterize Keplerian shear in non-ergodic dynamical systems, notably flows on torus bundles and their extensions to compact Lie group bundles. It establishes precise equivalences between Keplerian shear and Rajchman decay of associated push-forward measures, and derives both qualitative and quantitative decay rates for conditional correlations in discrete and continuous settings. The results cover regular and singular fiber measures, yield invariant-function characterizations, and reveal connections to Diophantine approximation, shrinking-target properties, and dynamical Borel-Cantelli lemmas. The work thereby extends Keplerian shear beyond classical Lebesgue contexts, providing new tools for limit theorems in non-ergodic systems and enriching the interaction between harmonic analysis and dynamical systems on fiber bundles.

Abstract

The Keplerian shear was introduced within the context of measure preserving dynamical systems by Damien Thomine, as a version of mixing for non ergodic systems. In this study we provide a characterization of the Keplerian shear using Rajchman measure, for some flows on tori bundles. Our work applies to dynamical systems with singularities or with non-absolutely continuous measures. We relate the speed of decay of conditional correlations with the Rajchman order of the measures. Some of these results are extended to the case of compact Lie group bundles.

Keplerian shear with Rajchman property

TL;DR

The paper develops a Rajchman-measure framework to characterize Keplerian shear in non-ergodic dynamical systems, notably flows on torus bundles and their extensions to compact Lie group bundles. It establishes precise equivalences between Keplerian shear and Rajchman decay of associated push-forward measures, and derives both qualitative and quantitative decay rates for conditional correlations in discrete and continuous settings. The results cover regular and singular fiber measures, yield invariant-function characterizations, and reveal connections to Diophantine approximation, shrinking-target properties, and dynamical Borel-Cantelli lemmas. The work thereby extends Keplerian shear beyond classical Lebesgue contexts, providing new tools for limit theorems in non-ergodic systems and enriching the interaction between harmonic analysis and dynamical systems on fiber bundles.

Abstract

The Keplerian shear was introduced within the context of measure preserving dynamical systems by Damien Thomine, as a version of mixing for non ergodic systems. In this study we provide a characterization of the Keplerian shear using Rajchman measure, for some flows on tori bundles. Our work applies to dynamical systems with singularities or with non-absolutely continuous measures. We relate the speed of decay of conditional correlations with the Rajchman order of the measures. Some of these results are extended to the case of compact Lie group bundles.
Paper Structure (24 sections, 44 theorems, 310 equations, 1 figure)

This paper contains 24 sections, 44 theorems, 310 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Settings
  3. Mixing property and Keplerian shear
  4. Rajchman measure
  5. Functional equivalences
  6. Fourier-Rajchman decay
  7. Radon-Nikodym-Lebesgue decomposition
  8. Example of singular Rajchman measures
  9. Tori bundle
  10. Main results in discrete dynamical systems
  11. Previous results
  12. Generalizations to singular measures
  13. Invariant functions and $\sigma-$algebra
  14. Main result in continuous dynamical systems
  15. Invariant functions and $\sigma-$algebra
  16. ...and 9 more sections

Key Result

Proposition 2.6

The Keplerian shear property is equivalent to the convergence to $0$ of the expectation of the conditional correlation, in other words, for all $f_1,f_2\in \mathbb{L}^2_\mu(\Omega)$,

Figures (1)

  • Figure 1: Push forward of measure ${((\mu)_{\pi})}_{\vert U}$ on $\mathbb{R}$ by $\langle \xi\vert v_U\rangle$

Theorems & Definitions (126)

  • Definition 2.1: Mixing system
  • Example 2.2: Non ergodic systems
  • Definition 2.3: Invariant $\sigma-$algebra
  • Definition 2.4: Keplerian shear
  • Definition 2.5: Conditional correlation
  • Proposition 2.6: DaTho
  • Definition 2.7: Rajchman measure
  • Proposition 2.8
  • proof
  • Lemma 2.9: Weak$-*$ convergence and Rajchman property
  • ...and 116 more