Local Lyapunov spectrum rigidity of nilmanifold automorphisms

Jonathan DeWitt

doi:10.3934/jmd.2021003

Article Contents

2021, Volume 17: 65-109. Doi: 10.3934/jmd.2021003

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Local Lyapunov spectrum rigidity of nilmanifold automorphisms

Jonathan DeWitt

Department of Mathematics, University of Chicago, 5734 South University Avenue, Chicago, IL 60637, USA

Received: January 22, 2020

Revised: November 04, 2020

Published: February 2021

This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE-1746045

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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.

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Mathematics Subject Classification: Primary:37D20, 37D25;Secondary:51F30.

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Figure 1. A diagrammatic depiction of the weak and strong foliations illustrating the points involved in the definition of the weak and strong distances. The strong distance between $ q $ and $ r $ is the distance from $ z $ to $ q $ along the $ \mathscr{S}_{i+1}^u $ foliation. The weak distance between $ q $ and $ r $ is the distance between $ z $ and $ r $ along the $ \mathscr{W}_i^u $ foliation

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