Hyperbolicity and integral expression of the Lyapunov exponents for linear cocycles

Abstract Consider in this paper a linear skew-product system ( θ , Θ ) : T × W × R n → W × R n ; ( t , w , x ) ↦ ( t ⋅ w , Θ ( t , w ) ⋅ x ) where T = R or Z , 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 ) satisfies the cocycle condition based on θ and is continuously differentiable in t if T = R . We show that ‘semi λ-exponential dichotomy’ of ( 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 E s ( w ; λ ) and if dim E s ( 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 ) -cocycles based on a uniquely ergodic endomorphism, and of GL ( 2 , R ) -cocycles based on a uniquely ergodic equi-continuous endomorphism, respectively. On the other hand, in the sense of C 0 -topology we obtain the density of SL ( 2 , R ) -cocycles having positive Lyapunov exponent based on a minimal subshift satisfying the Boshernitzan condition.