|Location:||MSRI: Simons Auditorium|
Ever since the Conley-Zehnder proof of the Arnold conjecture for tori, the study of periodic orbits has arguably been the most important interface between Hamiltonian dynamical systems and symplectic topology. A general feature of Hamiltonian systems is that they tend to have numerous periodic orbits. In fact, for a broad class of closed symplectic manifolds, every Hamiltonian diffeomorphism has infinitely many simple periodic orbits.
There are, however, notable exceptions. Namely, an important class of symplectic manifolds including the two-sphere admits Hamiltonian diffeomorphisms with finitely many periodic orbits — the so-called pseudo-rotations — which are of particular interest in dynamical systems. Furthermore, recent works by Bramham (in dimension two) and by Ginzburg and myself (in dimensions greater than two) show that one can obtain a lot of information about the dynamics of pseudo-rotations, going far beyond periodic orbits, via symplectic techniques.
In this talk I will discuss various aspects of the existence question for periodic orbits of Hamiltonian systems, focusing on recent higher dimensional results about pseudo-rotations.