Positive topological entropy for magnetic flows on surfaces

Abstract

Abstract. We study the topological entropy of the magnetic flow on a closed riemannian surface. We prove that if the magnetic flow has a non-hyperbolic closed orbit in some energy set T c M = E −1 (c), then there exists an exact C ∞ -perturbation of the 2-form Ω such that the new magnetic flow has positive topological entropy in T c M . We also prove that if the magnetic flow has an infinite number of closed orbits in T c M , then there exists an exact C 1 -perturbation of Ω with positive topological entropy in T c M . The proof of the last result is based on an analog of Franks' lemma for magnetic flows on surfaces, that is proven in this work, and Mañé's techniques on dominated splitting. As a consequence of those results, an exact magnetic flow on S 2 in high energy levels admits a C 1 -perturbation with positive topological entropy. In the appendices we show that an exact magnetic flow on the torus in high energy levels admits a C ∞ -perturbation with positive topological entropy.

Generic properties of magnetic flows

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We obtain Kupka-Smale's theorem and Franks' lemma for magnetic flows on manifolds with any dimension. This improves Miranda's result [12,13] on surfaces. However our methods relies on geometric control theory, like in Rifford and Ruggiero articles [17,11].

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“…v,v v. The perturbation analysis was done for both flows, leading to new interest techniques. The Gaussian thermostat was studied by Latosinski [8] in any dimension, but the magnetic flow was studied by Miranda [12,13] but only in dimension two.…”

Chaos and integrability in SL(2,R)-geometry

“…Note that the right-hand side can also be interpreted as the Lorenz force in the magnetic field defined by the 2-form Bdσ = Bdx ∧ dy/y 2 with the density B = κV , so the same equations describe the magnetic geodesics on the hyperbolic plane (see e.g. [4], [33]).…”

Rotation vectors for homeomorphisms of non-positively curved manifolds

“…This differential equation defines a flow on the tangent bundle of the hyperbolic plane and also on the tangent bundle of any oriented hyperbolic surface (this is the so called 'magnetic flow' for the volume 2-form, e.g. see [Mir07]). Fixing a non-constant trajectory and denoting the norm of its velocity (which remains constant) by v we will prove the following (see figure 12.2 ):…”