A variety of questions and results on Cantor sets revolved around the Minkowski sums of Cantor sets and the topological structure or Hausdorff dimension of these sumsets.  For example, Shmeling and Schmerkin showed that given an increasing sequence {x_n} bounded by 0 and 1, there exists a Cantor set C such that x_n is the Hausdorff dimension of C added to itself n times. 

Given any integer n, we will provide a construction for a Cantor set with zero logarithmic capacity such that the Cantor set added to itself n times is a single interval, while a sum of any smaller number of copies of that set is still a Cantor set.

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.