Relations among Hamiltonian, area-preserving, and non-wandering flows on surfaces

TL;DR

This paper characterizes Hamiltonian flows with finitely many singular points on compact surfaces and explores their relationships with divergence-free and non-wandering flows, providing answers to existing open problems.

Contribution

It offers a topological characterization of Hamiltonian flows on surfaces and clarifies the connections among Hamiltonian, divergence-free, and non-wandering flows.

Findings

Topological characterization of Hamiltonian flows with finitely many singular points.

Relationships among Hamiltonian, divergence-free, and non-wandering flows clarified.

Affirmative answer to Nikolaev and Zhuzhoma's problem under specific conditions.

Abstract

This paper gives a topological characterization of Hamiltonian flows with finitely many singular points on compact surfaces, using the concept of ``demi-caract\'eristique'' in the sense of Poincar\'e. Furthermore, we describe the relationships and distinctions among the Hamiltonian, divergence-free, and non-wandering properties for continuous flows, which gives an affirmative answer to the problem posed by Nikolaev and Zhuzhoma under the assumption of finitely many singular points.