On the Markus-Neumann theorem

TL;DR

This paper examines the Markus-Neumann theorem on flow equivalence on surfaces, identifies gaps in its original proof, and proposes a corrected, more general formulation with illustrative examples.

Contribution

It highlights issues in the original theorem, provides counterexamples, and offers a revised, accurate statement of the result.

Findings

Original theorem has gaps and may not hold as stated.

Counterexamples demonstrate limitations of the original formulation.

Proposed a corrected, more general version of the theorem.

Abstract

A well-known result by L. Markus, later extended by D. A. Neumann, states that two continuous flows on a surface are equivalent if and only if there is a surface homeomorphism preserving orbits and time directions of their separatrix configurations. In this paper we present several examples showing that, as originally formulated, the Markus-Neumann theorem needs not work. Besides, we point out the gap in its proof and show how to restate it in a correct (and slightly more general) way.