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 Rθ is a rotation of angle . 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 Aθ 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 Aθ does not have dominated splitting is non empty.