Analiticity of the Lyapunov exponents of perturbed toral automorphisms

We consider a dynamical system generated by an analytic perturbation $A_{\epsilon }$ of an analytic Anosov diffeomorphism $A_0$ of ${\text{TTT}}^d$. We show that, if $A_0$ admit a splitting of $\mathrm{T}\text{mathds}{T}^d$ in $k$ invariant subspaces, there exists a {\it partial conjugation} ${\mathcal{H}}_{\text{e}}$ of $d{A}_{\text{e}}$ and $d{A}_0$ that preserves the splitting and is analytic in $\text{e}$. This show that the splitting can be extended to $A_{\text{e}}$. As an application of this results, we obtain that the Lyapunov exponents, if non degenerate, are analytic functions of the perturbation.