Anosov magnetic flows, critical values and topological entropy

Abstract:
We study the magnetic flow determined by a smooth Riemannian metric $g$ and a closed 2-form $\Omega$ on a closed manifold $M$. If the lift of $\Omega$ to the universal cover is exact, we can define a critical value in the sense of Mañé for the lifted flow. This critical value is finite if and only if the lift of $\Omega$ has a bounded primitive. The critical value can be expressed in terms of an isoperimetric constant defined by $g$ and $\Omega$, which coincides with Cheeger's isoperimetric constant when the manifold is an oriented surface and $\Omega$ is the area form for $g$. When the magnetic flow (restricted to the unit tangent bundle) is Anosov, we show that the critical value is less than 1/2 and any closed bounded form on $M$ of degree at least 2 has a bounded primitive.

Next we consider the 1-parameter family of magnetic flows associated with $g$ and $\lambda\Omega$ for $\lambda \geq 0$, under the assumption that the lift of $\Omega$ to the universal cover has a bounded primitive. We introduce an entropy defined by the growth rate of the average volume of certain minimal balls and we show that this volume entropy is a nonincreasing function of $\lambda$. We also show that the volume entropy is a lower bound for the topological entropy of the magnetic flow (restricted to the unit tangent bundle) and that equality holds if the magnetic flow of $(g,\la\Omega)$ is Anosov on the unit tangent bundle.

We construct an example of a Riemannian metric of negative curvature on a closed oriented surface of higher genus such that there are values of the parameter $0 < \lambda_1 < \lambda_2$ with the property that if $\phi^i$ is the magnetic flow on the unit tangent bundle defined by $\lambda_i$ times the area form, then $\phi^1$ has conjugate points and $\phi^2$ 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 positive $\lambda$. Finally we have a version of the example (in which $\Omega$ is no longer the area form) such that the topological entropy of $\phi^1$ is greater than the topological entropy of the geodesic flow, which in turn is greater than the topological entropy of $\phi^2$.

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Click here to see a picture of the curved path of a charged particle moving in a magnetic field (and then click on the image to see it enlarged).

Authors' addresses:

	


  
	
	Keith  Burns 
	Department of Mathematics
	Northwestern University
	Evanston, IL 60208-2730
        U.S.A.
        burns followed by math.northwestern.edu


	
	
	
        Gabriel Paternain
        Department of Pure Mathematics and Mathematical Statistics
        University of Cambridge
        Cambridge CB3 0WB
	England
	gpp24 followed by dpmms.cam.ac.uk