Lyapunov exponents for families of rotated linear cocycles

Consider a compact metric space X, a homeomorphism T: X → X and an ergodic T-invariant measure μ. In this work, we are interested in the study of the upper Lyapunov exponent λ⁺(θ) associated to the periodic family of cocycles defined by where is a linear cocycle orientation-preserving and R_θ is a rotation of angle θεℝ. We show that if the cocycle A has dominated splitting, then there exists a non empty open set of parameters θ such that the cocycle A_θ has dominated splitting and the function is real analytic and strictly concave. As a consequence, we obtain that the set of parameters θ where the cocycle A_θ does not have dominated splitting is non empty.