Lyapunov exponents for families of rotated linear cocycles

In this work, we are interested in the study of the upper Lyapunov exponent $\lambda^+(\theta)$ associated to the periodic family of cocycles defined by $$A_\theta(x):=A(x)R_\theta,\qquad x\in X,$$ where $A\::\: X\to \mathbb{GL}^+(2,\mathbb{R})$ is a linear cocycle orientation--preser\-ving and $R_\theta$ is a rotation of angle $\theta\in\mathbb{R}$. We show that if the cocycle $A$ has dominated splitting, then there exists a non empty open set $\mathcal{U}$ of parameters $\theta$ such that the cocycle $A_\theta$ has dominated splitting and the function $\mathcal{U}\ni\theta\mapsto\lambda^+(\theta)$ is real analytic and strictly concave. As a consequence, we obtain that the set of parameters $\theta$ where the cocycle $A_\theta$ has not dominated splitting is non empty.