J. A. G. Miranda | Searcharxiv

arXiv2020

A note on Tonelli Lagrangian systems on $\mathbb{T}^2$ with positive topological entropy on high energy level

In this work we study the dynamical behavior Tonelli Lagrangian systems defined on the tangent bundle of the torus $\mathbb{T}^2=\mathbb{R}^2 / \mathbb{Z}^2$. We prove that the Lagrangian flow restricted to a high energy level $ E_L^{-1}(c)$ (i.e $ c> c_0(L)$) has positive topological entropy if the flow satisfies the Kupka-Smale propriety in $ E_L^{-1}(c)$ (i.e, all closed orbit with energy $c$ are hyperbolic or elliptic and all heteroclinic intersections are transverse on $E_L^{-1}(c)$). The proof requires the use of well-known results in Aubry-Mather's Theory.

math.DS↗

arXiv2012

Jets of closed orbits of Mañé generic Hamiltonian flows

We prove a perturbation theorem for the $k$-jets, $k\geq 2$, of the Poincaré map of a closed orbit of the Hamiltonian flow of a Tonelli Hamiltonian $H: T^*M\to \R$, on a closed manifold $M$. As a consequence we obtain Mañé generic properties of Hamiltonian and Lagrangian flows.

math.DS↗