Hyperbolic Rank, Carnot Metrics and Rigidity of Slow Foliations

Séminaire « Géométrie dynamique »
Salle Visio M3
Orateur: Ralf Spatzier Affiliation: Université de Michigan, USA.   Lieu: Salle Visio M3. L'exposé sera online projeté sur l'écran de la salle. Date: Vendredi, 21 Janvier, 2022 - 16.00-17.00   Titre :   Hyperbolic Rank, Carnot Metrics and Rigidity of Slow Foliations. Abstract: I will discuss joint work with Chris Connell and Thang Nguyen.  We developed new techniques for studying smooth dynamical systems in the presence of a Carnot metric.   These metrics arise from totally non-integrable distributions on manifolds. Principally, one measures distances by the length of curves tangent to the distributions.  Then we employ the theory of Margulis-Mostow, M\'etivier, Mitchell and Pansu on tangent cones and differentiation properties  to establishresonances 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 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.

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