Local rigidity of Lyapunov spectrum for toral automorphisms

Abstract

We study the regularity of the conjugacy between an Anosov automorphism \(L\) of a torus and its small perturbation. We assume that \(L\) has no more than two eigenvalues of the same modulus and that \(L^4\) is irreducible over \(\mathbb Q\). We consider a volume-preserving \(C^1\)-small perturbation \(f\) of \(L\). We show that if Lyapunov exponents of \(f\) with respect to the volume are the same as Lyapunov exponents of \(L\), then \(f\) is \(C^{1+\text{H\"older}}\) conjugate to \(L\). Further, we establish a similar result for irreducible partially hyperbolic automorphisms with two-dimensional center bundle.