Program
GDD - JLU Gießen
Viktor Ginzburg (Santa Cruz)
Title: Periodic orbits of Hamiltonian systems: from the Conley conjecture to pseudo-rotations
Abstract: One distinguishing feature of Hamiltonian dynamical systems is that such systems, with few notable exceptions, tend to have numerous periodic orbits and these orbits carry a lot of information about the dynamics of the system. In 1984 Conley conjectured that a Hamiltonian diffeomorphism (i.e., the time-one map of a Hamiltonian flow) of a torus has infinitely many periodic points. This conjecture was proved by Hingston some twenty years later, in 2004. Similar results for Hamiltonian diffeomorphisms of surfaces of positive genus were also established by Franks and Handel. Then the Conley conjecture was proved for a fairly broad class of manifolds in a series of papers by Gurel, Hein and the speaker.
However, the Conley conjecture obviously fails for some, even very simple, manifolds such as the sphere. These spaces admit Hamiltonian diffeomorphisms with finitely many periodic orbits -- the so-called pseudo-rotations -- which are of particular interest and occupy a very special place in dynamics. Recently, symplectic topological methods were used by Bramham to study pseudo-rotations in dimension two. As has been noticed by the speaker and Gurel, Floer theory turns out, quite surprisingly, to be the right tool to study the dynamics of pseudo-rotations well beyond periodic orbits in all dimensions.
In these lectures, we will start with the background results on the Conley conjecture and then focus on the dynamics of Hamiltonian pseudo-rotations.
Michael Hutchings (Berkeley)
Title: Two or infinitely many Reeb orbits
Abstract: In a joint work with Dan Cristofaro-Gardiner and Dan Pomerleano, we showed that under two hypotheses, a contact form on a closed connected three-manifold has either two or infinitely many simple Reeb orbits. The two hypotheses are that the contact form is nondegenerate, and the associated contact structure has torsion first Chern class. The idea of the proof is to assume that there are only finitely many simple Reeb orbits, and use embedded contact homology (ECH) to find a genus zero global surface of section for the Reeb flow. One can then invoke a theorem of Franks to obtain the result. A key role in the proof is played by the ``volume property’’ of ECH, which relates the contact volume of the three-manifold to the symplectic actions of certain finite sets of Reeb orbits homologically selected by ECH. In this series of talks, we will first give an introduction to the theorem on two or infinitely many Reeb orbits, and then we will explain as much as possible about the technical tools from ECH that are used to prove it.
