Local Lyapunov Spectrum Rigidity of Nilmanifold Automorphisms

We study the regularity of a conjugacy between an Anosov automorphism $L$ of a nilmanifold $N\mathrm{/}\mathrm{\Gamma }$ and a volume-preserving, $C^1$-small perturbation $f$. We say that $L$ is locally Lyapunov spectrum rigid if this conjugacy is $C^{1+}$ whenever $f$ is $C^{1+}$ 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.