## DYNAMICAL SYSTEMS Seminar

### Openness of positive Lyapunov exponents for Linear cocycles over partially hyperbolic maps

Mauricio Poletti
(Université París 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 $C^0$ 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, this 2 results implies that the Lyapunov exponents is not a continuous function of the cocycle in $C^0$ topology. For cocycles with more regularity, results of continuity where proved with different hypothesis, for $C^r$ 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.

Tuesday, December 5, 2017
Time: 11h30
Room: C6.2.33, Faculdade de Ciências, UL
http://cemapre.iseg.ulisboa.pt/seminars/dynsys/