Mathematics Seminar - 10/22/26

Abstract: We say a discrete Schroedinger is ergodic if its potential is given by sampling an ergodic dynamical system along an orbit. Quasi-periodic Schroedinger operators are those wherein the dynamical system is a torus rotation. These operators on the integer line encompass many examples related to the tight-binding model from physics. An important observation to come out of the late 20th century is the idea that these operators can be understood using a simple, fundamental object: the transfer matrix, which is essentially a linear cocycle. A key aspect of the transfer matrix is the Lyapunov exponent, which captures certain asymptotic behavior. In particular, a variety of authors such as Avila, Bourgain, and Jitomirskaya, have shown that positivity and continuity are tied to remarkable spectral information of the original operator. While the behavior of 1-frequency operators (that is, where the dynamics occurs on a 1-dimensional torus) is very well-understood, the behavior of multi-frequency operators (where the dynamics occurs on a high-dimensional torus) is not. In this talk, we will introduce the Lyapunov exponent of multi-frequency quasi-periodic cocycles and discuss our recent and ongoing work understanding their regularity properties in various settings along with applications to the integrated density of states.