The famous Baxendale Theorem states that for a random dynamical system by diffeomorphisms of a compact manifold M^d, unless the system possesses a measure that is invariant under all the maps of the system (that is quite rare), there exists an ergodic stationary measure with strictly negative «volume» Lyapunov exponent 
\lambda_vol = \lambda_1+…+\lambda_d. 
My talk will be devoted to a recent joint result with V. P. H. Goverse, generalising this theorem to a non-invertible (and only piecewise-continuous) setting. Now, the upper bound for the volume Lyapunov exponent is logarithm of the average number of preimages of a point. In particular, once this number does not exceed 1 (``\mu-injectivity’’ by Brofferio, Oppelmeyer and Szarek), the volume Lyapunov exponent can again be claimed to be negative.

     For an arbitrary irrational angle \alpha and arbitrary coupling constant \lambda, the Lebesgue measure of the spectrum of the Almost Mathieu operator with these parameters is equal to 4 |1-\lambda|. This was first conjectured in works by S. Aubry and G. Andre (1980), and later established in a series of results by J. Avron, P. H. M. v. Mouche & B. Simon (1990), Jitomirskaya and Last (1998), Jitomirskaya and Krasovsky (2001), Avila and Krikorian (2006).
     The main result of this talk, that is a work in progress with Anton Gorodetski, is devoted to the answer to the following question: what can be said about the measure of the part of the spectrum that is comprised between two given gaps? We show that the dependence on the parameters is piecewise-analytic: it is analytic on any domain when the bounding gaps do not bifurcate. We also show that the moments of the Lebesgue measure, restricted on the spectrum, are polynomials in \lambda and trigonometric polynomials in \alpha.
 

We will discuss Schrodinger operators generated by hyperbolic transformations of tori and show that they cannot have any essential spectral gaps.The key ingredient is a topological argument using Johnson's gap-labelling theorem.

Schrödinger operators with Sturmian potential are highly analogous to the Almost Matthieu ones: the potential is also quasi-periodic with some frequency $\alpha$. In particular, the corresponding spectrum (plotted as a function of $\alpha$) forms so-called Kohmoto butterfly, that can be seen as a sibling to the Hofstadter butterfly, associated to the Almost Matthieu operator.

These operators and their spectra have been already extensively studied by many authors; in particular, many results have been obtained previously for the frequencies $\alpha$ that are quadratic irrationalities or of bounded type. A key element of many of these works is the study of corresponding renormalisation operators, acting on the associated Markov surface.

My talk will be devoted to our joint result with Anton Gorodetski and Seung uk Jang: we study the dynamics of the skew product that joins the renormalisation of the (traces of the) transition matrices and of the frequency $\alpha$. For this skew product, we construct explicitly a ``stable cone field’’ (over all irrational $\alpha$ and for an arbitrary coupling constant $\lambda$). This is a first step in our strategy of obtaining a dynamical/renormalization point of view on the self-similarity of the Kohmoto butterfly.

Stationary measures are one of the key tools in the theory of random dynamical systems, and this naturally motivates the study of their properties. It is known that for an invertible RDS with no finite invariant sets, the corresponding stationary measures cannot have atoms. There are also generalizations of this property; in particular, in a recent joint work with A. Gorodetski and G. Monakov, we have established Hölder regularity of stationary measures for a system of bi-Lipschitz homeomorphisms under very mild assumptions: absence of a common invariant measure and some positive moment condition for the Lipschitz constant. It was also further recently generalized by G. Monakov, obtaining log-Hölder regularity under even milder assumptions for the system.

A next natural direction of generalization would be to attempt to remove the invertibility assumption, considering the maps that are only locally invertible. However, it turns out that such a generalization is impossible! Namely, in a recent joint work with V. Goverse, we construct an example of a smooth random dynamical system on the circle, consisting of locally invertible maps, having no common invariant measure, and for which there is an atomic stationary measure. Moreover, this stationary measure is physical: for a Lebesgue-generic initial point, its trajectory is almost surely distributed with respect to this measure. My talk will be devoted to the presentation of this example.