| Anosov Magnetic Flows, Critical Values And Topological Entropy (2007) | |||||||||||||||
Abstract | |||||||||||||||
| . We study the magnetic ow determined by a smooth Riemannian metric g and a closed 2-form on a closed manifold M . If the lift of to the universal cover f M is exact, we can dene a critical value c(g; in the sense of Ma~ne [28] for the lift of the ow to f M . We have c(g; < 1 if and only if the lift of has a bounded primitive. This critical value can be expressed in terms of an isoperimetric constant dened by (g; 2 which coincides with Cheeger's isoperimetric constant when M is an oriented surface and is the area form of g. When the magnetic ow of (g; is Anosov on the unit tangent bundle SM , we show that 1=2 > c(g; and any closed bounded form in f M of degree 2 has a bounded primitive. Next we consider the 1-parameter family of magnetic ows on SM associated with the pair (g; for 0, where is such that its lift to f M has a bounded primitive. We introduce a volume entropy h v () dened as the exponential growth rate of the volume of certain minimal b... | |||||||||||||||
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