Analiticity of the Lyapunov exponents of perturbed toral automorphisms

Abstract

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