Nonautonomous Dynamical Systems

Friday, June 24, 2016 - 2:00pm - 2:50pm
Keller 3-180
Russell Johnson (Università di Firenze)
George Sell was one of the first mathematicians to realize that a broad class of nonautonomous differential and difference equations can be studied in an effective way using dynamical systems methods, via a compactification of the time variable. In the last 50 years or so, a good deal of effort has gone into giving substance to this insight through the development of basic techniques, and into applying those techniques to problems in which time-dependent differential/discrete systems play a role.

In this talk we will discuss some fundamental tools for studying time-dependent linear differential systems, namely the Oseledets theorem, the Sacker-Sell spectral theory and the theory of rotation numbers. Then we will show how they can be applied to study the class of generalized reflectionless Schroedinger potentials.

This class GR of potentials was introduced by Lundina in 1985. It contains the classical reflectionless potentials and (certain translates of) the algebro-geometric potentials of Dubrovin-Novikov-Matveev. It was subsequently studied by Marchenko (1991) and by Kotani (2008), who used the Sato-Segal-Wilson theory to show that each q(x) in GR gives rise to a solution u(t,x) of the Korteweg-de Vries equation, with initial value q(x), which is meromorphic in the complex (t,x)-plane.

We will consider stationary ergodic subsets of GR which are reflectionless in the sense of Craig. It turns out that one can determine subsets of this type which are counterexamples to the so-called Kotani-Last conjecture; they are almost automorphic in the sense of Bochner-Veech but not almost periodic. Other subsets of this type have a Lyapunov exponent with singular behavior. We will discuss as many examples as time permits. Our discussion is based on joint work with L. Zampogni.

Thanks for collaboration concerning the material of this talk are due to R. Fabbri, M. Franca, S. Novo, C. Nunez, L. Zampogni and many other colleagues.