The summer school will provide courses from experts in the areas of the scope of the program ''Dynamics with Structures''. It is intended to provide surveys of modern directions in the theory of dynamical systems on the level of graduate students and junior researchers. The goal is to propagate modern directions of research to younger scientists and show the variety of methods and results obtained by imposing different structures on a dynamical system. It will be a unique possibility to give an insight on all three main events of the semester. The topics will reach from Conservative Dynamics and Hamiltonian Systems over Algebraic and Number Theoretic Systems and Methods, Analytic Dynamical Systems and Thermodynamic Formalism. Besides the lectures the audience is encouraged to have active discussions with the lecturers and other experts.
Dynamical approaches to the spectral theory of operators.
Lecturer: David Damanik (Rice University)
Abstract: The goal of these lectures is to give an overview of fundamental techniques and results underlying dynamical approaches to the study of spectra of operators associated to physical systems, with the primary examples given by Schrödinger operators on the line. Topics to be covered include linear cocycles, Lyapunov exponents, rotation numbers, uniform hyperbolicity, and reducibility. A major theme will be the study of systems displaying either almost periodicity or randomness.
Dynamics on homogeneous spaces and interactions with number theory.
Lecturer: Anish Ghosh (Tata Institute of Fundamental Research)
Abstract: I will talk about the ergodic theory of Lie group actions on homogeneous spaces. This subject has seen tremendous activity in the last few years, much of it motivated by its connections to number theory. In this lecture series, we will discuss mixing for group actions and its consequences as well as measure rigidity (especially Ratner's theorems) and its consequences for Diophantine approximation.
Periodic orbits of Hamiltonian systems: the Conley conjecture, pseudo-rotations and holomorphic curves.
Lecturer: Viktor Ginzburg (University of California Santa Cruz)
Abstract: One distinguishing feature of Hamiltonian dynamical systems is that such systems, with few notable exceptions, tend to have numerous periodic orbits. For instance, for many symplectic manifolds, every Hamiltonian diffeomorphism has infinitely many periodic orbits unconditionally. This fact, usually referred to as the Conley conjecture, has by now been established for a broad class of manifolds. However, the Conley conjecture obviously fails for some, even very simple, manifolds such as the sphere. These spaces admit Hamiltonian diffeomorphisms with few periodic orbits -- the so-called pseudo-rotations -- which are of particular interest and occupy a very special place in dynamics. Symplectic topological methods and, in particular, Floer theory turn out to be the right tools to study pseudo-rotations in all dimensions and recently a connection between the existence of pseudo-rotations and the Gromov-Witten invariants has been discovered.
We will start these lectures with the background results on the Conley conjecture and then focus on the dynamics of Hamiltonian pseudo-rotations and the connection between pseudo-rotations and quantum homology.
Dynamics of singular Riemann surface foliations.
Lecturer: Nessim Sibony (Université Paris-Sud, Orsay)
Abstract: Consider a polynomial differential equation in two complex variables. The time is complex. In order to study the global behavior of the solutions, it is convenient to consider the extension as a foliation in the projective plane. This is an example of a singular foliation by Riemann surfaces.
I will discuss some recent results around the following questions. What is the ergodic theory of such systems, in a general compact Kahler manifold? How do the leaves distribute in a generic case?
The main tool is the potential theory of positive ddc-closed currents and their geometry, which I will introduce. They provide the analogue of "invariant measures" and permit to prove unique ergodicity theorems when all the singularities are hyperbolic. The averaging used is inspired by Nevanlinna's theory. I will also give examples of the use of pluri-potential theory in discrete dynamics. I will also describe analogies with the theory of polynomial automorphisms of in two complex variables, which exhibit similar rigidity phenomena.