Mauricio Poletti
Université Paris 13

Abstract: Continuity and positiveness of Lyapunov exponents where widely studied in the recent years. A classical result of Bochi-Mañe proves that in the  topology cocyles that don't exhibit a uniform hyperbolic behavior can be approximated by zero exponents, also Avila proves that there exists a dense set of cocycles with non-vanishing Lyapunov exponents, these 2 results implie that the Lyapunov exponent is not a continuous function of the cocycle in  topology. For cocycles with more regularity, results of continuity where proved with different hypothesis, for  cocycles with hyperbolic base by Bocker-Viana, Backes-Butler-Brown, for analytic cocycles over rotations by Avila-Krikorian, Duarte-Klein. Here we will prove that if the map is volume preserving partially hyperbolic accessible then the Bochi-Mañe phenomena can not happen (non-vanishing exponents are open), we will also give examples showing that without accessibility this phenomena can happen.