I will give a proof of Furstenberg's Theorem. We restrict ourselves to SL(2,R) cocycles. This is the second of two lectures. 

I am going to talk about Carleman estimate with Carleman weight first. To prove Carleman estimate, I need to introduce some definitions of semiclassical analysis first. Then I am going to talk about Carleman estimate with limiting Carleman weight and some applications. 

I will introduce the model and structure of the proof,  and give the details next week.

This will be a continuation of the presentation from two weeks ago. The abstract is below.

Last time, we briefly discussed the notion of localization and some of its consequences and during this meeting we will finish with the rank one perturbation material and the promised proof.

Abstract:

The goal of this talk will be to discuss various issues related to the Anderson model as presented in Del Rio et. al "Operators with Singular Continuous Spectrum, IV." Firstly, we will explain the type of localization that allows one to make dynamical statements (i.e. given simple spectrum, we have 'SULE' iff 'SUDL'). We then present various facts relating to rank one perturbations of self adjoint operators. Finally, we connect the above two discussions to give the authors' proof that the singular continuous spectral measures produced by rank one perturbations of the Anderson model are supported on a set of Hausdorff dimension zero.

In this talk, I  will explain the notion of a simple Toeplitz sequence (in the sense of Liu-Qu) and of the subshift associated to it. A description of the elements in the subshift will be given and some basic properties of the subshift will be discussed.