Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Partial section II: classification for general flows

Théo Marty

TL;DR

This work extends Fried’s cross-section theory to partial cross-sections of general flows by marrying dynamical and topological data. It develops a cohomological framework, including a germ-based relative homology $H_1(M,\mathcal{g}(\mathcal{R}_\alpha),\mathbb{Z})$, to classify partial cross-sections via $\alpha$-recurrent sets and quasi-Lyapunov conditions; a relative asymptotic direction set $D_{\varphi,\alpha}$ guides existence and uniqueness. The paper proves existence criteria, establishes bijections with pre-Lyapunov maps and Conley’s order, and provides a detailed cardinality analysis of $\mathcal{PS}_\varphi(\alpha)$, along with a robust homological criterion that links dynamics to germ-relative topology. Together, these results generalize Fried’s desingularization and Fried’s asymptotic-direction criterion to partial cross-sections, enabling computation in practical settings and connecting dynamical recurrence with topological invariants.

Abstract

This is the second article in a series that aims at classifying partial sections of flows, that is a general family of transverse surfaces. In this part, we classify partial cross-sections for all continuous flows, in the spirit of Schwartzman-Fried-Sullivan theory. We give a dynamical criterion for the existence of partial cross-sections, which is a direct consequence of part I of the series. Then we describe all partial cross-sections using a cohomological criterion, resembling Fried's criterion. We also characterize the cardinality of the set of partial cross-sections in a given cohomology class.

Partial section II: classification for general flows

TL;DR

This work extends Fried’s cross-section theory to partial cross-sections of general flows by marrying dynamical and topological data. It develops a cohomological framework, including a germ-based relative homology , to classify partial cross-sections via -recurrent sets and quasi-Lyapunov conditions; a relative asymptotic direction set guides existence and uniqueness. The paper proves existence criteria, establishes bijections with pre-Lyapunov maps and Conley’s order, and provides a detailed cardinality analysis of , along with a robust homological criterion that links dynamics to germ-relative topology. Together, these results generalize Fried’s desingularization and Fried’s asymptotic-direction criterion to partial cross-sections, enabling computation in practical settings and connecting dynamical recurrence with topological invariants.

Abstract

This is the second article in a series that aims at classifying partial sections of flows, that is a general family of transverse surfaces. In this part, we classify partial cross-sections for all continuous flows, in the spirit of Schwartzman-Fried-Sullivan theory. We give a dynamical criterion for the existence of partial cross-sections, which is a direct consequence of part I of the series. Then we describe all partial cross-sections using a cohomological criterion, resembling Fried's criterion. We also characterize the cardinality of the set of partial cross-sections in a given cohomology class.

Paper Structure

This paper contains 30 sections, 56 theorems, 29 equations, 2 figures.

Table of Contents

  1. Acknowledgments.
  2. Preliminary: the topology part
  3. $\mathbb{Z}$-covering
  4. Descripiton of the rank 1 cohomology.
  5. $\mathbb{Z}$-covering
  6. Equivariance.
  7. Partial cross-section
  8. Preliminary: the dynamic part
  9. Conley's theory
  10. Asymptotic pseudo-directions
  11. Quasi-Lyapunov class
  12. The $\alpha$-recurrent set
  13. Characterization of partial cross-sections
  14. Existence of partial cross-section
  15. Equivalence with pre-Lyapunov maps
  16. ...and 15 more sections

Key Result

Theorem 1

Let $M$ be a compact connected manifold, $\varphi$ be a continuous flow on $M$ and $\alpha$ be in $H^1(M,\mathbb{Z})$. There exists a partial cross-section cohomologous to $\alpha$ if and only if either we have $\alpha\neq 0$ and $-\alpha$ is quasi-Lyapunov, or if we have $\alpha=0$ and $\varphi$ is

Figures (2)

  • Figure 1: Existence of partial cross-sections, in green and blue, cohomologous to $dx$ and $-dx$.
  • Figure 2: Classification of partial cross-sections.

Theorems & Definitions (76)

  • Theorem 1
  • Theorem 2
  • Theorem 3: Concatenation of Theorems \ref{['thm-ps-countable']}, \ref{['thm-finite-pa']} and \ref{['thm-unique-ps']}
  • Remark 1.1
  • Lemma 1.2
  • Lemma 1.3
  • Theorem 1.4: Fried Fried1982
  • Definition 2.1
  • Lemma 2.2: See martyPS1 Corollary 1.6 for instance
  • Theorem 2.3: Conley1978
  • ...and 66 more