Revista Matemática Iberoamericana


Full-Text PDF (276 KB) | Metadata | Table of Contents | RMI summary
Volume 20, Issue 1, 2004, pp. 107–130
DOI: 10.4171/RMI/382

Published online: 2004-04-30

Transitive flows on manifolds

Víctor Jiménez López[1] and Gabriel Soler López[2]

(1) Universidad de Murcia, Spain
(2) Universidad Politécnica de Cartagena, Spain

In this paper we characterize manifolds (topological or smooth, compact or not, with or without boundary) which admit flows having a dense orbit (such manifolds and flows are called transitive) thus fully answering some questions by Smith and Thomas. Namely, it is shown that a surface admits a transitive flow (which can be got smooth) if and only if it is connected and it is neither homeomorphic to the sphere nor the projective plane nor embeddable in the Klein bottle (or, alternatively, if it is connected and includes two orientable topological circles intersecting transversally at exactly one point). We also prove that any (connected) manifold with dimension at least 3 admits a transitive flow, which can be got smooth if the manifold admits a smooth structure. In particular, this allows us to characterize $\omega$-limit sets with nonempty interior for flows in a given $n$-manifold (as they can be described by the property of being the closure of its transitive $n$-submanifolds).

Keywords: Manifold, transitive flow

Jiménez López Víctor, Soler López Gabriel: Transitive flows on manifolds. Rev. Mat. Iberoam. 20 (2004), 107-130. doi: 10.4171/RMI/382