Analiticity of the Lyapunov exponents of perturbed toral automorphisms

TL;DR

This paper proves that for certain perturbed dynamical systems on tori, the invariant splitting and Lyapunov exponents depend analytically on the perturbation parameter, extending classical results to more general settings.

Contribution

It introduces a partial conjugation that preserves the invariant splitting under perturbation and shows Lyapunov exponents are analytic functions of the perturbation.

Findings

Invariant splitting extends analytically under perturbation.

Lyapunov exponents are analytic functions of the perturbation.

Partial conjugation preserves the invariant splitting.

Abstract

We consider a dynamical system generated by an analytic perturbation $A_\varepsilon$ 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 $dA_0$ 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.