Carnot metrics, Dynamics and Local Rigidity

Abstract

This paper develops new techniques for studying smooth dynamical systems in the presence of a Carnot-Carathéodory metric. Principally, we employ the theory of Margulis-Mostow, Métivier, 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 Carnot-Carathéodory 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.