Carnot metrics, dynamics and local rigidity

Chris Connell,Thang Nguyen,Ralf Spatzier

doi:10.1017/etds.2021.116

Abstract

AbstractThis paper develops new techniques for studying smooth dynamical systems in the presence of a Carnot–Carathéodory metric. Principally, we employ the theory of Margulis and 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.

Highlights

In the present paper we explore these tangent cones with the goal of establishing new techniques for the study of Lyapunov exponents of dynamical systems respecting sub-Riemannian metrics
Mitchell [Mit85] showed that the tangent cones at generic points of N are graded nilpotent Lie groups equipped with left-invariant Carnot–Carathéodory metrics which are unique by Margulis and Mostow [MM00]
We show that automorphism groups vary semicontinuously in this topology for nilpotent G

Summary

Introduction

Mitchell [Mit85] showed that the tangent cones at generic points of N are graded nilpotent Lie groups equipped with left-invariant Carnot–Carathéodory metrics which are unique by Margulis and Mostow [MM00]. (Tangent cone structure theorem) Let f : N → N be a local C∞ diffeomorphism of a sub-Riemannian manifold N preserving the horizontal distribution E.

Results

Conclusion