Cocycle rigidity of abelian partially hyperbolic actions
- 14 Downloads
Suppose G is a higher-rank connected semisimple Lie group with finite center and without compact factors. Let G = G or G = G ⋉ V, where V is a finite-dimensional vector space V. For any unitary representation (π,H) of G, we study the twisted cohomological equation π(a)f − λf = g for partially hyperbolic element a ∈ G and λ ∈ U(1), as well as the twisted cocycle equation π(a1)f − λ1f = π(a2)g − λ2g for commuting partially hyperbolic elements a1, a2 ∈ G. We characterize the obstructions to solving these equations, construct smooth solutions and obtain tame Sobolev estimates for the solutions. These results can be extended to partially hyperbolic flows in parallel.
As an application, we prove cocycle rigidity for any abelian higher-rank partially hyperbolic algebraic actions. This is the first paper exploring rigidity properties of partially hyperbolic that the hyperbolic directions don’t generate the whole tangent space. The result can be viewed as a first step toward the application of KAM method in obtaining differential rigidity for these actions in future works.
Unable to display preview. Download preview PDF.
- D. Damjanović and A. Katok, Local rigidity of partially hyperbolic actions. II: The geometric method and restrictions of Weyl chamber flows on SL(n,R)/Γ, International Mathematics Research Notices (2011), 4405–4430.Google Scholar
- A. Katok, Cocycles, cohomology and combinatorial constructions in ergodic theory, in Smooth Ergodic Theory and its Applications (Seattle, WA, 1999), Proceedings of Symposia in Pure Mathematics, Vol. 69, American Mathematical Society, Providence, RI, 2001, pp. 107–173.MathSciNetCrossRefzbMATHGoogle Scholar
- A. Katok, Combinatorial Constructions in Ergodic Theory and Dynamics, University Lecture Series, Vol. 30, American Mathematical Society, Providence, RI, 2003.Google Scholar
- D. Y. Kleinbock and G. A. Margulis, Bounded orbits of nonquasiunipotent flows on homogeneous spaces, in Sinaĭ’s Moscow Seminar on Dynamical Systems, American Mathematical Society Translations. Series 2, Vol. 171, American Mathematical Society, Providence, RI, 1996, pp. 141–172.MathSciNetzbMATHGoogle Scholar
- G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 17, Springer-Verlag, Berlin, 1991.Google Scholar
- K. Vingahe and Z. J. Wang, Local rigidity of twisted symmetric space examples, in progress.Google Scholar
- K. Vingahe and Z. J. Wang, Local rigidity of higher rank homogeneous abelian actions: a complete solution via the geometric method, submitted (2015).Google Scholar
- Z. J. Wang, Various smooth rigidity examples in SL(2, R)×· · ·×SL(2,R)/Γ, in preparation.Google Scholar