Local Lyapunov spectrum rigidity of nilmanifold automorphisms

We study the regularity of a conjugacy between an Anosov automorphism L of a nilmanifold N /Γ and a volume-preserving, C¹-small perturbation f . We say that L is locally Lyapunov spectrum rigid if this conjugacy is C¹⁺ whenever f is C¹⁺ and has the same volume Lyapunov spectrum as L. For L with simple spectrum, we show that local Lyapunov spectrum rigidity is equivalent to L satisfying both an irreducibility condition and an ordering condition on its Lyapunov exponents.