Ralf Spatzier

Rigidity of the measurable structure for algebraic actions of higher-rank Abelian groups

We investigate rigidity of measurable structure for higher-rank Abelian algebraic actions. In particular, we show that ergodic measures for these actions fiber over a zero entropy measure with Haar measures along the leaves. We deduce various rigidity theorems for isomorphisms and joinings as corollaries.

Spherical rank rigidity and Blaschke manifolds

Let M be a complete Riemannian manifold whose sectional curvature is bounded above by 1. We say that M has positive spherical rank if along every geodesic one hits a conjugate point at t=\pi. The following theorem is then proved: If M is a complete, simply connected Riemannian manifold with upper curvature bound 1 and positive spherical rank, then M is isometric to a compact, rank one symmetric space (CROSS) i.e., isometric to a sphere, complex projective space, quaternionic projective space or to the Cayley plane. The notion of spherical rank is analogous to the notions of Euclidean rank and hyperbolic rank studied by several people (see references). The main theorem is proved in two steps: first we show that M is a so called Blaschke manifold with extremal injectivity radius (equal to diameter). Then we prove that such M is isometric to a CROSS.

GLOBAL RIGIDITY OF HIGHER RANK ANOSOV ACTIONS ON TORI AND NILMANIFOLDS

We show that sufficiently irreducible Anosov actions of higher rank abelian groups on tori and nilmanifolds are smoothly conjugate to affine actions.

Smooth classification of Cartan actions of higher rank semisimple Lie groups and their lattices

Let G be a connected semisimple Lie group without compact factors whose real rank is at least 2, and let ⊂ G be an irreducible lattice. We provide a C ∞ classification for volume-preserving Cartan actions of an d G. Also, if G has real rank at least 3, we provide a C ∞ classification for volume-preserving, multiplicity free, trellised, Anosov actions on compact manifolds.

On Livšic’s theorem, superrigidity, and Anosov actions of semisimple Lie groups

We prove a generalization of Livsic’s Theorem on the vanishing of the cohomology of certain types of dynamical systems. As a consequence, we strengthen a result due to Zimmer concerning algebraic hulls of Anosov actions of semisimple Lie groups. Combining this with Topological Superrigidity, we find a Holder geometric structure for multiplicity free Anosov actions.

Totally nonsymplectic Anosov actions on tori and nilmanifolds

We show that sufficiently irreducible totally non-symplectic Anosov actions of higher rank abelian groups on tori and nilmanifolds are C 1 -conjugate to actions by affine automorphisms.

Positively curved manifolds with large spherical rank

Rigidity results are obtained for Riemannian d-manifolds with sec > 1 and spherical rank at least d − 2 > 0. Conjecturally, all such manifolds are locally isometric to a round sphere or complex projective space with the (symmetric) Fubini– Study metric. This conjecture is verified in all odd dimensions, for metrics on dspheres when d 6= 6, for Riemannian manifolds satisfying the Rakic duality principle, and for Kahlerian manifolds.

Nonstationary normal forms and rigidity of group actions

We develop a proper “nonstationary” generalization of the classical theory of normal forms for local contractions. In particular, it is shown under some assumptions that the centralizer of a contraction in an extension is a particular Lie group, determined by the spectrum of the linear part of the contractions. We show that most homogeneous Anosov actions of higher rank abelian groups are locally C∞ rigid (up to an automorphism). This result is the main part in the proof of local C∞ rigidity for two very different types of algebraic actions of irreducible lattices in higher rank semisimple Lie groups: (i) the actions of cocompact lattices on Furstenberg boundaries, in particular projective spaces, and (ii) the actions by automorphisms of tori and nilmanifolds. The main new technical ingredient in the proofs is the centralizer result

Equilibrium measures for certain isometric extensions of Anosov systems

We prove that for the frame flow on a negatively curved, closed manifold of odd dimension other than 7, and a Holder continuous potential that is constant on fibers, there is a unique equilibrium measure. We prove a similar result for automorphisms of the Heisenberg manifold fibering over the torus. Our methods also give an alternate proof of Brin and Gromovs result on the ergodicity of these frame flows.

On the work of Rodriguez Hertz on rigidity in dynamics

This paper is a survey about recent progress in measure rigidity and global rigidity of Anosov actions, and celebrates the profound contributions by Federico Rodriguez Hertz to rigidity in dynamical systems.