Michal Rams

Cocycles of circle diffeomorphisms

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 try to explain where that kind of systems comes from, what are they useful for, and what we know about them. The results presented will be from joint works with Lorenzo Diaz and Katrin Gelfert.