Positive Topological Entropy for Magnetic Flows on Surfaces

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^cM= E^{-1}(c), then there exists an exact $ C^\infty$-perturbation of the 2-form $ Ω$ such that the new magnetic flow has positive topological entropy in T^cM. We also prove that if the magnetic flow has an infinite number of closed orbits in T^cM, then there exists an exact C^1-perturbation of $ Ω$ with positive topological entropy in T^cM. 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^\infty $-perturbation with positive topological entropy.