Partial section II: classification for general flows

TL;DR

This paper classifies partial cross-sections for all continuous flows using cohomological criteria, extending Schwartzman-Fried-Sullivan theory, and provides conditions for their existence and characterization.

Contribution

It introduces a cohomological criterion for classifying partial cross-sections of continuous flows, building upon previous dynamical criteria and extending existing theory.

Findings

Dynamical criterion for existence of partial cross-sections

Cohomological classification of all partial cross-sections

Characterization of the number of cross-sections in a cohomology class

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.