Revista Matemática Iberoamericana

Transitive flows on manifolds

Víctor Jiménez López and Gabriel Soler López

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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).

Article information

Rev. Mat. Iberoamericana, Volume 20, Number 1 (2004), 107-130.

First available in Project Euclid: 2 April 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37C70: Attractors and repellers, topological structure
Secondary: 34C05: Location of integral curves, singular points, limit cycles 37C10: Vector fields, flows, ordinary differential equations

manifold transitive flow


Jiménez López, Víctor; Soler López, Gabriel. Transitive flows on manifolds. Rev. Mat. Iberoamericana 20 (2004), no. 1, 107--130.

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