Lyapunov exponents for families of rotated linear cocycles

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 $A_θ(x):=A(x)R_θ,\qquad x\in X,$ where $A\::\: X\to \mathbb{GL}^+(2,\mathbb{R})$ is a linear cocycle orientation--preser\-ving and $R_θ$ is a rotation of angle $θ\in\mathbb{R}$. We show that if the cocycle $A$ has dominated splitting, then there exists a non empty open set $\mathcal{U}$ of parameters $θ$ such that the cocycle $A_θ$ has dominated splitting and the function $\mathcal{U}\niθ\mapstoλ^+(θ)$ is real analytic and strictly concave. As a consequence, we obtain that the set of parameters $θ$ where the cocycle $A_θ$ has not dominated splitting is non empty.

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