This will be an introductory talk devoted to the study of random dynamics on the circle. I will discuss the (exponential) contraction (and Baxendale’s theorem), stationary measures, and generic alternative between minimality and existence of no-exit domains (in particular, following our work with Yu. Kudryashov and A. Okunev), and Yu. Ilyashenko and A. Negut’s «invisible parts of attractors».

Abstract: Furstenberg’s theorem for random matrix products has been a key tool in many contexts, including mathematical physics. Of particular interest is the 1-dimensional Anderson model of electron diffusion in random media. In this talk, we will discuss how to apply a version of Furstenberg’s theorem where matrices which are independent but not necessarily identically distributed (non-stationary). In particular, we will discuss how to prove spectral and dynamical localization in the non-stationary Anderson model with unbounded potentials using this version of Furstenberg’s theorem.

Abstract: We provide an example of a one-parameter family of Cookie-Cutter-Like sets that are generated by a one-parameter family of sequences of analytic maps (varying analytically in the parameter), but for which, the Hausdorff dimension is not even differentiable as a function of the parameter. This motivates an interesting conjecture concerning the regularity of the dimension of the spectrum of a Sturmian Hamiltonian operator as a function of the coupling constant. This is a joint work with Victor Kleptsyn. 

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.