Existence of minimal flows on nonorientable surfaces

TL;DR

This paper characterizes minimal nonorientable surfaces of finite genus and constructs an example with infinite genus, advancing understanding of flow dynamics on nonorientable surfaces.

Contribution

It provides the first complete characterization of minimal nonorientable surfaces and introduces a conjecture about minimality in nonorientable surfaces without boundary.

Findings

Characterization of minimal nonorientable surfaces of finite genus

Construction of a minimal nonorientable surface with infinite genus

Conjecture on minimality of nonorientable surfaces without boundary

Abstract

Surfaces admitting flows all whose orbits are dense are called minimal. Minimal orientable surfaces were characterized by J.C. Beni\`{e}re in 1998, leaving open the nonorientable case. This paper fills this gap providing a characterization of minimal nonorientable surfaces of finite genus. We also construct an example of a minimal nonorientable surface with infinite genus and conjecture that any nonorientable surface without combinatorial boundary is minimal.