# Michal Rams: Cocycles of circle diffeomorphisms

**Time: **
Tue 2023-02-21 14.00 - 16.00

**Location: **
Institut Mittag-Leffler, Seminar Hall Kuskvillan and Zoom

**Video link: **
Meeting ID: 921 756 1880

**Participating: **
Michal Rams - IM PAN, Institute of Mathematics of the Polish Academy of Sciences

**Abstract:**

I will present an introduction to the world of cocycles of circle diffeomorphisms. The system is simple: given a finite family \(\{f_i\}_{i=1}^n\) of \(C^1\) diffeomorphisms of a circle, we consider the dynamical system \(F:S^1 \times \Sigma, F(x,\xi) = (f_{\xi_0}(x), \sigma\xi)\) (where \(\Sigma = \{1,\ldots,n\}^{\mathbb Z}\)). In simple words, we apply the maps \(f_i\) in any order we want. Such systems can be of many types, including hyperbolic ones -- but that is the boring case that I'll skip. I will concentrate on the really interesting case of robustly nonuniformly hyperbolic cocycles (that is, the set of possible Lyapunov exponents contains 0 as its interior point, and this property is preserved under small perturbations). I'll describe the invariant measures space of such systems, I'll also present the multifractal descriptio of the Lyapunov exponent. The results presented will be from joint works with Lorenzo Diaz and Katrin Gelfert.