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.

 In this talk, we will study random dynamical systems of smooth surface diffeomorphisms. Aaron Brown and Federico Rodriguez Hertz showed that, in this setting, hyperbolic stationary measures have the SRB property, except when certain obstructions occur. Here, the SRB property essentially means that the measure is absolutely continuous along certain “nice” curves (unstable manifolds). In this talk, we want to understand conditions that guarantee that SRB stationary measures are absolutely continuous with respect to the Lebesgue measure of the ambient space. Our approach is inspired by Tsujii's “transversality” method, which he used to show Palis conjecture for partially hyperbolic endomorphisms. This is a joint work with Aaron Brown, Homin Lee and Yuping Ruan.

We will discuss the non-stationary version of the Central Limit Theorem for random products of SL(2, R) matrices. The stationary versions were obtained previously by Tutubalin, Le Page, and Benoist-Quint. In all the previous works the assumption that the random matrices are identically distributed was used in a crucial way. We will explain how the recent results on the rate of growth of non-stationary products of random matrices can be used to overcome this restriction. The talk is based on a project joint with A. Gorodetski and G. Monakov.