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Partial section I: $α$-recurrence and equivariant Lyapunov maps

Théo Marty

TL;DR

The paper develops a framework to classify partial cross-sections of flows via α-equivariant Lyapunov maps on Abelian covers. It introduces the notions of quasi-Lyapunov classes and the α-recurrent set, then proves a spectral-decomposition principle in the integer case and extends to real coefficients through rational approximation and cone analysis. Key technical tools include reduction of pseudo-orbits, Conley-order based pre-Lyapunov maps, and a detailed study of how the α-recurrent set R_ depends on α. Together, these results lay groundwork for a cohomological classification of partial cross-sections and provide structural insight into how recurrence and Lyapunov properties interact over coverings and varying cohomology data.

Abstract

This is the first article in a series that aims at classifying partial sections of flows, that is a general family of transverse surfaces. In this part, we deal with the dynamical aspect of the question. Given a flow on a compact manifold $M$ and a cohomology class $α$ of rank 1, we give a criterion for the existence of an $α$-equivariant Lyapunov map on an Abelian covering of $M$ associated to $α$. One important aspect of the existence of such Lyapunov maps, and of the classification of partial sections, is a type of recurrence set relative to $α$. We describe how that set depends on $α$.

Partial section I: $α$-recurrence and equivariant Lyapunov maps

TL;DR

The paper develops a framework to classify partial cross-sections of flows via α-equivariant Lyapunov maps on Abelian covers. It introduces the notions of quasi-Lyapunov classes and the α-recurrent set, then proves a spectral-decomposition principle in the integer case and extends to real coefficients through rational approximation and cone analysis. Key technical tools include reduction of pseudo-orbits, Conley-order based pre-Lyapunov maps, and a detailed study of how the α-recurrent set R_ depends on α. Together, these results lay groundwork for a cohomological classification of partial cross-sections and provide structural insight into how recurrence and Lyapunov properties interact over coverings and varying cohomology data.

Abstract

This is the first article in a series that aims at classifying partial sections of flows, that is a general family of transverse surfaces. In this part, we deal with the dynamical aspect of the question. Given a flow on a compact manifold and a cohomology class of rank 1, we give a criterion for the existence of an -equivariant Lyapunov map on an Abelian covering of associated to . One important aspect of the existence of such Lyapunov maps, and of the classification of partial sections, is a type of recurrence set relative to . We describe how that set depends on .

Paper Structure

This paper contains 22 sections, 60 theorems, 14 equations, 1 figure.

Table of Contents

  1. Acknowledgments.
  2. Preliminary
  3. Abelian covering
  4. Descripiton of the rank 1 cohomology.
  5. Equivariance.
  6. Compactification.
  7. Pseudo-orbit and recurrence chains
  8. Lyapunov functions
  9. Asymptotic pseudo-directions
  10. Reducing pseudo-orbits
  11. Quasi-Lyapunov classes
  12. Characterizations of quasi-Lyapunov classes
  13. The $\alpha$-recurrent set
  14. Quasi-Lyapunov with integer coefficient
  15. Spectral decomposition: the integer case
  16. ...and 7 more sections

Key Result

Theorem 1

Let $\alpha$ be in $H^1(M,\mathbb{Z})$. Then there exists a $\mathbb{Z}$-equivariant Lyapunov map $h\colon\widehat{M}_\alpha\to\mathbb{R}$ if and only if $\alpha$ is quasi-Lyapunov.

Figures (1)

  • Figure 1: Flow on the torus which satisfies $\mathcal{R}_{dx+dy}\varsubsetneq\mathcal{R}_{dx}\cap\mathcal{R}_{dy}$.

Theorems & Definitions (73)

  • Theorem 1: Equivariant spectral decomposition
  • Theorem 2
  • Theorem 3: Equivariant spectral decomposition, real case
  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3: Conley Conley1978
  • Lemma 1.4
  • Lemma 1.5
  • Corollary 1.6
  • Lemma 1.7
  • ...and 63 more