Anosov magnetic flows, critical values and topological entropy

Keith Burns*, Gabriel P. Paternain

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

32 Scopus citations

Abstract

We study the magnetic flow determined by a smooth Riemannian metric g and a closed 2-form Ω on a closed manifold M. If the lift of Ω to the universal cover M̃ is exact, we can define a critical value c(g, Ω) in the sense of Mane (1997 Bol. Soc. Bras. Mat. 28 141-53) for the lift of the flow to M̃. We have c(g, Ω) < ∞ if and only the lift of Ω has a bounded primitive. This critical value can be expressed in terms of an isoperimetric constant defined by (g, Ω), which coincides with Cheeger's isoperimetric constant when M is an oriented surface and Ω is the area form of g. When the magnetic flow of (g, Ω) is Anosov on the unit tangent bundle SM, we show that 1 /2 ≥ c(g, Ω) and any closed bounded form in M̃ of degree ≥ 2 has a bounded primitive. Next we consider the one-parameter family of magnetic flows on SM associated with the pair (g, λΩ) for λ, ≥ 0, where Ω is such that its lift to M̃ has a bounded primitive. We introduce a volume entropy h ν(λ) defined as the exponential growth rate of the average volume of certain balls. We show that hν(λ) ≤ h top(λ), where htop (λ) is the topological entropy of the magnetic flow of (g, λΩ) on SM and that equality holds if the magnetic flow of (g, λΩ) is Anosov on SM. If λ1 ≤ λ2 and the magnetic flows for (g, λ1Ω) and (g, λΩ) are both Anosov on SM, then hν1) ≥ hν2). We construct an example of a Riemannian metric of negative curvature on a closed oriented surface of higher genus such that if φ λ is the magnetic flow associated with the area form with intensity λ., then there are values of the parameter 0 < λ 1 < λ2 with the property that φ λ1 has conjugate points and φλ2 is Anosov. Variations of this example show that it is also possible to exit and reenter the set of Anosov magnetic flows arbitrarily many times along the one-parameter family. Moreover, we can start with a Riemannian metric with conjugate points and end up with an Anosov magnetic flow for some λ > 0. Finally we have a version of the example (in which Ω is no longer the area form) such that the topological entropy of φλ1 is greater than the topological entropy of the geodesic flow, which in turn is greater than the topological entropy of φλ2.

Original languageEnglish (US)
Pages (from-to)281-314
Number of pages34
JournalNonlinearity
Volume15
Issue number2
DOIs
StatePublished - Mar 2002

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Physics and Astronomy(all)
  • Applied Mathematics

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