Hyperbolicity and integral expression of the Lyapunov exponents for linear cocycles

Consider in this paper a linear skew-product system(θ,Θ):T×W×Rn→W×Rn;(t,w,x)↦(t⋅w,Θ(t,w)⋅x) where T=RT=R or ZZ, and θ:(t,w)↦t⋅w is a topological dynamical system on a compact metrizable space W  , and where Θ(t,w)∈GL(n,R)Θ(t,w)∈GL(n,R) satisfies the cocycle condition based on θ and is continuously differentiable in t   if T=RT=R. We show that ‘semi λ  -exponential dichotomy’ of (θ,Θ)(θ,Θ) implies ‘λ-exponential dichotomy.’ Precisely, if Θ has no Lyapunov exponent λ and is almost uniformly λ-contracting along the λ  -stable direction Es(w;λ)Es(w;λ) and if dimEs(w;λ)dimEs(w;λ) is constant a.e., then Θ is almost λ  -exponentially dichotomous. To prove this, we first use Liao's spectrum theorem, which gives integral expression of the Lyapunov exponents, and then use the semi-uniform ergodic theorem by Sturman and Stark, which allows one to derive uniform estimates from nonuniform ones. As a consequence, we obtain the open-and-dense hyperbolicity of eventual GL+(2,R)GL+(2,R)-cocycles based on a uniquely ergodic endomorphism, and of GL(2,R)GL(2,R)-cocycles based on a uniquely ergodic equi-continuous endomorphism, respectively.On the other hand, in the sense of C0C0-topology we obtain the density of SL(2,R)SL(2,R)-cocycles having positive Lyapunov exponent based on a minimal subshift satisfying the Boshernitzan condition.