Positive Topological Entropy for Magnetic Flows on Surfaces

We study the topological entropy of the magnetic flow on 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 analogue of Frank's lemma for magnetic flows on surfaces, that is proven in this work, and the Mañe's techniques on dominated splitting. As a consequence of those results, an exact magnetic flows 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 flows on the torus in high energy levels admits a C ∞-perturbation with positive topological entropy.