Stably positive Lyapunov exponents for $SL(2,\mathbb{R})$ linear cocycles over partially hyperbolic diffeomorphisms

Mauricio Poletti

Published 2016-04-29Version 1

A result of Avila asserts that $SL(2,\mathbb{R})$ cocycles with non-zero Lyapunov exponents are dense in a very general setting. In this paper, we are concerned with \emph{stably non-zero} exponents. We consider $SL(2,\mathbb{R})$ cocycles over partially hyperbolic diffeomorphisms. Under a hypothesis on the behavior of the cocycles over a compact center leaf of the diffeomorphism, we prove that the cocycle is accumulated by open sets where the Lyapunov exponents are non-zero. Ussing this criteria we give a class of examples where the Lyapunov exponents are non-zero in an open and dense set.