We study skew products of circle diffeomorphisms over a shift space. Our primary motivation is the fact that they capture some key mechanisms of nonhyperbolic behavior of robustly transitive dynamical systems. We perform a multifractal analysis of fiber-Lyapunov exponents studying the topological entropy of fibers with equal exponent. This includes the study of restricted variational principles of the entropy of ergodic measures with given fiber-exponent, in particular, with exponent zero. This enables to understand transitive dynamical systems in which hyperbolicities of different type are intermingled. Moreover, it enables to 'quantify of the amount of non-hyperbolicity' in a context where any other tools presently available fail. This is joint work with L.J. Díaz and M. Rams.[-]

We consider a special family of invariant random processes on the infinite d-regular tree, which is closely related to random d-regular graphs, and helps understanding the structure of these finite objects. By using different notions of entropy and finding inequalities between these quantities, we present a sufficient condition for a process to be typical, that is, to be the weak local limit of functions on the vertices of a randomly chosen d-regular graph (with fixed d, and the number of vertices tending to infinity). Our results are based on invariant couplings of the process with another copy of itself. The arguments can also be extended to processes on unimodular Galton-Watson trees. In the talk we present the notion of typicality, the entropy inequalities that we use and the sufficient conditions mentioned above. Joint work with Charles Bordenave and Balázs Szegedy.[-]