11:00 Nicolas Vichery (Lyon)
Non convex Aubry Mather theory, invariant measures and rotation vectors..
Abstract: After a few reminders about symplectic homogenization, we will present an extension of the fundational result of Mather about the existence of invariant mesures with rotation vector prescribed by the subdifferential of the effective Hamiltonian.
This can be done by replacing the effective hamiltonian by the symplectic homegenization. We will finish by some applications in the context of classical KAM theory.
Invariant global surfaces of section and symmetric periodic orbits.
Abstract: A global surface of section is a crucial tool to understand low dimensional dynamical systems. I will explain joint work with Urs Frauenfelder on the construction of global disk-like surfaces of section which are invariant under symmetries in reversible dynamical systems. Then the Poincaré return map on this obeys a reversibility condition and sees symmetric features of a reversible dynamical system. If time permits, I will explain how to find symmetric periodic orbits using the reversibility of the Poincaré return map.
16:00 Marco Mazzucchelli (Lyon)
Periodic orbits of exact magnetic flows on surfaces.
Abstract: This talk is about periodic obits of exact magnetic flows on the cotangent bundle of closed surfaces. The dynamics of these Hamiltonian systems on high energy levels is well known: it is conjugated to a Reeb flow, and actually to a Finsler geodesic flow. In this talk, I will focus on low energies, more precisely on energies below the so-called Mañé critical value of the universal covering. After introducing the setting, I will present a recent result asserting the existence of infinitely many periodic orbits on almost all energy levels in this range. This is a joint work with A. Abbondandolo, L. Macarini, and G. P. Paternain.
Prochaines séances: 9/01 (J. Fine, D. Gayet, A. Mandini).