Abstract: Linear dynamical systems in finite dimension are well understood, and their stability relies on the eigenvalues of a representation matrix. This allows to describe some properties (e.g. hyperbolicity or ellipticity) of matrices $A$ in the space $SL(2, R)$ in terms of their eigenvalues. The situation becomes immediately non-trivial when one considers random products of two matrices $A,B$ in $SL(2, R)$, which determine a linear cocycle over the shift. In the paper “Some questions and remarks about $SL(2,R)$-cocycles”, J.-C. Yoccoz initiated the study of the hyperbolicity and ellipticity locus for such cocycles. In this talk I will recall some results by Yoccoz and subsequent work, and show how the a good understanding of the space of linear cocycle impacts in other problems, including a characterization of the space of cocycles whose Lyapunov exponents are everywhere well defined. Join via https://videoconf-colibri.zoom.us/j/96003557028