The set of fiber-bunched cocycles with nonvanishing Lyapunov exponents over a partially hyperbolic map is open

Abstract

We prove that the set of fiber-bunched $SL(2,\mathbb{R})$-valued Hölder cocycles with nonvanishing Lyapunov exponents over a volume preserving, accessible and center-bunched partially hyperbolic diffeomorphism is open. Moreover, we present an example showing that this is no longer true if we do not assume acessibility in the base dynamics.

Final version. To appear in Mathematical Research Letters