We will discuss a dynamical approach by Qinghui Liu and Yanhui Qu to study the spectrum of the Thue-Morse Hamiltonian, a Schrodinger operator on l^2. They use the trace map formalism to describe the spectrum, and study the corresponding polynomials to characterize subsets of their zeros that converge to a Cantor set. In doing this they establish bounds of the Hausdorff dimension. We will also discuss a possiblility to obtain similar bounds on the thickness of the spectrum. 

Cantor sets and their fractional dimensions are related to many unsolved conjectures. One of these conjectures, known as Palis' Conjecture, involves the dimension of the Minkowski sums of Cantor sets, and in this talk we will describe the details of the Palis Conjecture. In addition, we will use some of the tools involving Cantor sets to describe of the some properties of the Lagrange and Markov spectra, in particular, related to existence of the Hall's ray. The talk will be mostly based on the work done by Moreira in recent years.

We consider discrete Schr\"odinger operators on $\ell^2(\mathbb{Z})$ with bounded random but not necessarily identically distributed values of the potential. The distribution at a given site is not assumed to be absolutely continuous (or to contain an absolutely continuous component). We prove spectral localization (with exponentially decaying eigenfunctions) as well as dynamical localization for this model.

An important ingredient of the proof is a non-stationary analog of the Furstenberg Theorem on random matrix products,
which is also of independent interest. 

This is a joint project with A.Gorodetski.

About 40 years ago Arnold suggested  a  construction of so called  complex rotation numbers. Consider a circle diffeomorphism, lifted to a map of the real line that commutes with the shift by 1. For each complex number in the  open upper half-plane consider the lifted map combined with the shift by this number. Glue the  real line and the shifted real line via this map  and factorize the cylinder obtained by the unit shift. Consider the modulus of the elliptic curve thus obtained. It is sometimes called the complex rotation number of the original map combined with the shift. We get the moduli map of the upper half plane into itself.  The questions due to Arnold and Ghys can be formulated as follows:

How the boundary values of the moduli map are related to the rotation number of the original shifted map considered as a function on the absolute?

It appeared that for generic circle diffeomorphisms the boundary values of the moduli map form a fractal set ``Bubbles'' located inside the upper half plane: any rational point on the real axis belongs to a closed curve from this fractal set; this curve is  called a bubble. The boundary values of the moduli map coincide with  the rotation number when these values are irrational. The way of approximate calculation of bubbles is suggested, their possible bizarre shapes are drawn, and it is proved that the bubbles can intersect and self-intersect.

Self-similarity of the fractal set ``bubbles'' is established.

Recently it was  found out that the fractal set ``bubbles'' does not depend continuously on the original circle diffeomorphism. This drastically distinguishes it from Arnold tongues that depend on the original diffeomorphism continuously.

Most part of these results belongs to N. Goncharuk; other part is obtained by Buff, Moldavskis, Rissler and others.