Entropia Topológica positiva de fluxos Lagrangianos do tipo Tonelli

Abstract

Let be M a smooth manifold of dimension n + 1 and consider a TonelliLagrangian(...) be the set of smooth potentials (...), fixed with C^2-topology. Given a potential (...), consider the flow (...) of the perturbed Lagrangian (...) be the set of all periodic orbits (...) on the energy level (...) and define (...). We prove that if (...) and under certain conditions for the potencial u, then the set (...) is a hyperbolic set. In particular, if (...) has an infinite number of periodic orbits then it has positive topological entropy. The proof of this result is based on an analogue of Franks' Lemma for Euler-Lagrange ow on closed manifolds, that is proven in this work, and R. Mañé's techniques on dominated splitting. We also show that if M is a closed surface and (...), the Euler-Lagrange flow admits a perturbation by potencial u, with C^2-norm arbitrarily small, such that the perturbed flow (...) has positive topological entropy.