My talk will be devoted to a (quick and very brief) introduction to the domino tilings (intensively studied during the last fifty years), the subject that is very simple in the origin, while giving almost immediately very beautiful images. My goal will be to explain (roughly), where does the "arctic circle" effect in tilings come from, meanwhile mentioning asymptotic shape of Young diagrams, entropy, height function and variational problems. If the time permits, I will speak about computation of determinants and permanents.

Take a finitely-generated group of (analytic) circle diffeomorphisms. Since the times of Poincaré we know that any such action admits either a finite orbit, or a Cantor minimal set, or the action is minimal on all the circle. But what else can be said on such a group?

In this direction, there are well-known questions due to Sullivan, Ghys and Hector: assuming that such an action is minimal, is it necessarily Lebesgue-ergodic? If there is a Cantor minimal set, is it necessarily of a zero Lebesgue measure?

Our results provide a positive answer to the latter question, in some cases allow to resolve the former one and, more generally speaking, give some kind of understanding how a general characterization of an action can look like. This is a joint project with B. Deroin, D. Filimonov, and A. Navas.

As is well known, a class of one-dimensional lattice models, such as Ising models, Jacobi and CMV operators and others, are susceptible to renormalization analysis that can be carried out via the transfer matrix formalism. As a result, models on the one-dimensional lattice of a certain quasi-periodic type (namely, those generated by primitive substitutions) can be studied via dynamics of so-called trace maps, which are polynomial maps acting on the real (or complex) Euclidean space of appropriate dimension. A prominent example is the widely studied Fibonacci model. Much work has been done in this direction. At some point it became apparent that a model-independent framework, based on the dynamics of trace maps, can be built, that would cover essentially all models the relevant information of which is encapsulated in the traces of the associated transfer matrices (and, as experience has shown, this information is very difficult if not impossible to obtain via techniques other than the trace map). The purpose of this talk is to give a broad overview of past and very recent results on the dynamics of the Fibonacci trace map in a model-independent fashion, motivated by a class of models from physics, and with a view towards applications to those models. We shall cover hyperbolicity and partial hyperbolicity of the trace map and implications in spectral theory of Jacobi operators; some applications to Ising models; recent advances in understanding invariant measures on the invariant hyperbolic sets and implications for the density of states measures for the Jacobi operators; the Newhouse phenomenon and mixed behavior with large (in the sense of Hausdorff dimension) chaotic sea, and some connections with kicked two-level systems. Time permitting, we shall also state some open problems.

We prove results about the Holder continuity of the spectral measures of the extended CMV matrix, given power law bounds of the solution of the eigenvalue equation. We thus arrive at a unitary analogue of the results of Damanik, Killip and Lenz about the spectral measure of the discrete Schrodinger operator. This is joint work with Darren Ong. 

Interval translation maps (ITMs) are non-invertible generalizations of interval exchanges (IETs). The dynamics of finite type ITMs is similar to IETs, while infinite type ITMs are known to exhibit new interesting effects. The finiteness conjecture says that the subset of ITMs of finite type is open, dense, and has full Lebesgue measure. In my talk, I will prove the conjecture for the ITMs of three intervals and discuss some open problems.