American Institute of Mathematical Sciences

2010, 17: 68-79. doi: 10.3934/era.2010.17.68

Local rigidity of partially hyperbolic actions

 1 Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, United States

Received  February 2010 Revised  July 2010 Published  September 2010

We prove the local differentiable rigidity of partially hyperbolic abelian algebraic high-rank actions on compact homogeneous spaces obtained from simple indefinite orthogonal and unitary groups. The conclusions are based on geometric Katok-Damjanovic way and progress towards computations of the Schur multipliers of these non-split groups.
Citation: Zhenqi Jenny Wang. Local rigidity of partially hyperbolic actions. Electronic Research Announcements, 2010, 17: 68-79. doi: 10.3934/era.2010.17.68
References:
 [1] M. Brin, Y. Pesin, Partially hyperbolic dynamical systems,, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170. [2] D. Damjanovic and A. Katok, Periodic cycle functionals and Cocycle rigidity for certain partially hyperbolic $\RR^k$ actions,, Discr. Cont. Dyn.Syst., 13 (2005), 985. [3] D. Damjanovic and A. Katok, Local rigidity of partially hyperbolic actions. II. The geometric method and restrictions of Weyl Chamber flows on $SL(n,\RR)/\Gamma$,, , (). [4] D. Damjanovic and A. Katok, Local rigidity of partially hyperbolic actions. I. KAM method and $\ZZ^k$ actions on the torus,, Annals of Math, (2010). [5] D. Damjanovic, Central extensions of simple Lie groups and rigidity of some abelian partially hyperbolic algebraic actions,, J. Modern Dyn., 1 (2007), 665. [6] Vinay V. Deodhar, On central extensions of rational points of algebraic groups,, Amer. J. Math., 100 (1978), 303. doi: doi:10.2307/2373853. [7] A. J. Hahn and O. T. O'Meara, The classical groups and K-theory,, Springer Verlag, (1980), 55. [8] M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds. Lecture Notes in Mathematics,", 583, (1977). [9] A. Katok and R. Spatzier, Differential rigidity of Anosov actions of higher rank abelian groups and algebraic lattice actions,, Proc. Steklov Inst. Math., 216 (1997), 287. [10] G. A. Margulis, "Discrete Subgroups of Semisimple Lie Groups,", Springer-Verlag, (1991). [11] G. A. Margulis and N. Qian, Rigidity of weakly hyperbolic actions of higher real rank semisimple Lie groups and their lattices,, Ergodic Theory Dynam. Systems, 21 (2001), 121. doi: doi:10.1017/S0143385701001109. [12] R. Steinberg, Generateurs, relations et revetements de groupes algebriques,, Colloque de Bruxelles, (1962), 113.

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References:
 [1] M. Brin, Y. Pesin, Partially hyperbolic dynamical systems,, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170. [2] D. Damjanovic and A. Katok, Periodic cycle functionals and Cocycle rigidity for certain partially hyperbolic $\RR^k$ actions,, Discr. Cont. Dyn.Syst., 13 (2005), 985. [3] D. Damjanovic and A. Katok, Local rigidity of partially hyperbolic actions. II. The geometric method and restrictions of Weyl Chamber flows on $SL(n,\RR)/\Gamma$,, , (). [4] D. Damjanovic and A. Katok, Local rigidity of partially hyperbolic actions. I. KAM method and $\ZZ^k$ actions on the torus,, Annals of Math, (2010). [5] D. Damjanovic, Central extensions of simple Lie groups and rigidity of some abelian partially hyperbolic algebraic actions,, J. Modern Dyn., 1 (2007), 665. [6] Vinay V. Deodhar, On central extensions of rational points of algebraic groups,, Amer. J. Math., 100 (1978), 303. doi: doi:10.2307/2373853. [7] A. J. Hahn and O. T. O'Meara, The classical groups and K-theory,, Springer Verlag, (1980), 55. [8] M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds. Lecture Notes in Mathematics,", 583, (1977). [9] A. Katok and R. Spatzier, Differential rigidity of Anosov actions of higher rank abelian groups and algebraic lattice actions,, Proc. Steklov Inst. Math., 216 (1997), 287. [10] G. A. Margulis, "Discrete Subgroups of Semisimple Lie Groups,", Springer-Verlag, (1991). [11] G. A. Margulis and N. Qian, Rigidity of weakly hyperbolic actions of higher real rank semisimple Lie groups and their lattices,, Ergodic Theory Dynam. Systems, 21 (2001), 121. doi: doi:10.1017/S0143385701001109. [12] R. Steinberg, Generateurs, relations et revetements de groupes algebriques,, Colloque de Bruxelles, (1962), 113.
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