Characterization of uniform hyperbolicity for fiber-bunched cocycles

Abstract

We prove a new characterization of uniform hyperbolicity for fiber-bunched cocycles. Specifically, we show that the existence of a uniform gap between the Lyapunov exponents of a fiber-bunched $SL(2,\mathbb{R})$-cocycle defined over a subshift of finite type or an Anosov diffeomorphism implies uniform hyperbolicity. In addition, we construct an $\alpha$-H\"{o}lder cocycle which has uniform gap between the Lyapunov exponents, however it is not uniformly hyperbolic.