Lyapunov exponents for families of rotated linear cocycles

Abstract

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 $$A_{\theta }(x):=A(x)R_{\theta },x\in X,$$ 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.