Carnot metrics, Dynamics and Local Rigidity

TL;DR

This paper introduces new techniques using Carnot metrics and tangent cone theory to analyze rigidity and stability properties in smooth dynamical systems across various geometric settings.

Contribution

It develops novel methods employing tangent cone theory to establish rigidity results for dynamical systems with Carnot metrics, covering Anosov systems, hyperbolic rank metrics, and lattice actions.

Findings

Rigidity of smooth dominated splittings for Anosov systems.

Local rigidity of higher hyperbolic rank metrics.

Structural stability of Brin-Pesin holonomy groups.

Abstract

This paper develops new techniques for studying smooth dynamical systems in the presence of a \CC metric. Principally, we employ the theory of Margulis-Mostow, M\'etivier, Mitchell and Pansu on tangent cones to establish resonances between Lyapunov exponents. We apply these results in three different settings. First, we explore rigidity properties of smooth dominated splittings for Anosov diffeomorphisms and flows via associated smooth \CC metrics. Second, we obtain local rigidity properties of higher hyperbolic rank metrics in a neighborhood of a locally symmetric one. For the latter application we also prove structural stability of the Brin-Pesin asymptotic holonomy group for frame flows. Finally, we obtain local rigidity properties for uniform lattice actions on the ideal boundary of quaternionic and octonionic symmetric spaces.