## 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 h_{ν}(λ) ≤ h _{top}(λ), where h_{top} (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 λ_{1} ≤ λ_{2} and the magnetic flows for (g, λ_{1}Ω) and (g, λΩ) are both Anosov on SM, then h_{ν}(λ_{1}) ≥ h_{ν}(λ _{2}). 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} < λ_{2} with the property that φ ^{λ1} has conjugate points and φ^{λ2} 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 φ^{λ1} is greater than the topological entropy of the geodesic flow, which in turn is greater than the topological entropy of φ^{λ2}.

Original language | English (US) |
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Pages (from-to) | 281-314 |

Number of pages | 34 |

Journal | Nonlinearity |

Volume | 15 |

Issue number | 2 |

DOIs | |

State | Published - Mar 2002 |

## ASJC Scopus subject areas

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