Local Lyapunov Spectrum Rigidity of Nilmanifold Automorphisms

TL;DR

This paper investigates the conditions under which conjugacies between certain automorphisms of nilmanifolds preserve Lyapunov spectra with high regularity, revealing that simple spectrum automorphisms exhibit rigidity if specific algebraic and spectral conditions are met.

Contribution

It establishes a characterization of local Lyapunov spectrum rigidity for nilmanifold automorphisms with simple spectrum, linking it to irreducibility and exponent ordering conditions.

Findings

Rigidity is equivalent to algebraic and spectral conditions for simple spectrum automorphisms.

Conjugacy regularity matches the smoothness of perturbations when conditions are satisfied.

Provides criteria for when Lyapunov spectrum is preserved under smooth conjugacies.

Abstract

We study the regularity of a conjugacy between an Anosov automorphism $L$ of a nilmanifold $N/\Gamma$ 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.