Adriana Sánchez (University of São Paulo, São Carlos)

*“Lyapunov exponents of probability distributions with non-compact support”*

A recent result of Bocker-Viana asserts that the Lyapunov exponents of compactly supported probability distributions in \({\rm GL}(2,\mathbb{R})\) depend continuously on the distribution.

In a joint work with Marcelo Viana we investigate the general, possibly non-compact case. In this talk we are going to see that the Lyapunov exponents are semi-continuous with respect to the Wasserstein topology, but not with respect to the weak-star topology. Moreover, they are not continuous with respect to the Wasserstein topology.

Ao Cai (University of Lisbon)

*“Almost reducibility and reducibility of finitely differentiable quasi-periodic \({\rm SL}(2,\mathbb{R})\) linear cocycles”*

In this talk, we will first go over the development of reducibility theory. Then we will present the process of obtaining quantitative (almost) reducibility in the \(C^k\) perturbative regime via Kolmogorov-Arnold-Moser theory and analytic approximation. Finally, we will talk about applications to Lyapunov exponents.

Catalina Freijo (Federal University of Minas Gerais)

*“Holonomies and invariant sections”*

We consider a fixed hyperbolic dynamic in the base and study different types of holonomies for a linear cocycle in \({\rm SL}(2,\mathbb{R})\). The purpose is to give a simple construction for an invariant section for the cocycle and expose examples where this is applied.

Lorenzo Díaz (Pontifical Catholic University of Rio de Janeiro)

*“Cycles and co.”*

Rewording the results of Avila-Bochi-Yoccoz, heterodimensional cycles appear in a natural way in the boundary of the set of hyperbolic cocycles of matrices. We discuss this fact and study families of heterodimensional cycles originated by those cocycles, with special emphasis in the thermodinamical analysis.

This talk is based on joint results with Gelfert and Rams.

Pedro Duarte (University of Lisbon)

*“Lyapunov Exponents of quasi-periodic cocycles with random noise”*

The quantitative regularity (Hölder, weak Hölder, etc) of the Lyapunov exponents of analytic quasi-periodic linear cocycles over Diophantine torus translations is known, even for fiber non-invertible cocyles.

Fixing the Diophatine base dynamics, what can be said about the stability and regularity of the Lyapunov exponents when one considers a class of cocyles which allows random perturbations of the fiber action?

In this talk we will try to answer this question. Joint work with Ao Cai and Silvius Klein.