For this last talk in the series, I will discuss the details of how spectrum of the Schrödinger operator can be defined, and the proof that the spectrum can consist of an infinite number of intervals. 

We begin by discussing a conjectured version of a non-stationary stable manifold theorem for a hyperbolic horseshoe and small perturbations of it, and discuss the techniques needed to achieve such a result. With these techniques in mind, we will conjecture a similar result for the trace maps on a particular family of cubic surfaces, and explain what deductions this result would make about the spectrum of related discrete Schrodinger operators with Sturmian potential.

A hyperbolic locus $\mathcal{H} \subset SL(2,R)^n$ is a connected open set such that for all $x\in\mathcal{H}$, $\{x_i\}_1^n$ is a uniformly hyperbolic set of matrices.  In $SL(2,R)^2$, the geometry of the loci was studied in Avila, Bochi, and Yoccoz's 2008 work. In this talk, some of the details of the geometry in higher dimensions will be discussed, as well as the relevance with Schrodinger operators. 

The Lorentz gas was originally introduced as a model for the movement of electrons in metals. It consists of a massless point particle (electron) moving through Euclidean space bouncing off a given set of scatterers $\mathcal{S}$ (atoms of the metal) with elastic collisions at the boundaries $\partial \mathcal{S}$. If the set of scatterers is periodic in space, then the quotient system, which is compact, is known as the Sinai billiard. There is a great body of work devoted to Sinai billiards and in many ways their dynamics is well understood.
In contrast, very little is known about the behavior of the Lorentz gases with aperiodic configurations of scatterers which model quasicrystals and other low-complexity aperiodic sets. This case is the focus of our joint work with Rodrigo Trevino.
We establish some dynamical properties which are common for the periodic and quasiperiodic billiards. We also point out some significant differences between the two. The novelty of our approach is the use of tiling spaces to obtain a compact model of the aperiodic Lorentz gas on the plane.

In this talk, I will introduce some recent joint work with A. Avila and D. Damanik in showing positivity and large deviations of the Lyapunov exponent for Schrodinger operators with potentials generated by hyperbolic transformations. Specifically, we consider the base dynamics which is a subshift of finite type with an ergodic measure admitting a bounded distortion property and which has a fixed point. We show that if the potentials are locally constant or globally fiber bunched, then the set of zero Lyapunov exponent is finite. Moreover, we have a uniform large deviation estimate away from this finite set.