# 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.   Google Scholar [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.   Google Scholar [3] D. Damjanovic and A. Katok, Local rigidity of partially hyperbolic actions. II. The geometric method and restrictions of Weyl Chamber flows on $SL(n,\RR)/\Gamma$,, , ().   Google Scholar [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).   Google Scholar [5] D. Damjanovic, Central extensions of simple Lie groups and rigidity of some abelian partially hyperbolic algebraic actions,, J. Modern Dyn., 1 (2007), 665.   Google Scholar [6] Vinay V. Deodhar, On central extensions of rational points of algebraic groups,, Amer. J. Math., 100 (1978), 303.  doi: doi:10.2307/2373853.  Google Scholar [7] A. J. Hahn and O. T. O'Meara, The classical groups and K-theory,, Springer Verlag, (1980), 55.   Google Scholar [8] M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds. Lecture Notes in Mathematics,", 583, (1977).   Google Scholar [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.   Google Scholar [10] G. A. Margulis, "Discrete Subgroups of Semisimple Lie Groups,", Springer-Verlag, (1991).   Google Scholar [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.  Google Scholar [12] R. Steinberg, Generateurs, relations et revetements de groupes algebriques,, Colloque de Bruxelles, (1962), 113.   Google Scholar

show all references

##### References:
 [1] M. Brin, Y. Pesin, Partially hyperbolic dynamical systems,, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170.   Google Scholar [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.   Google Scholar [3] D. Damjanovic and A. Katok, Local rigidity of partially hyperbolic actions. II. The geometric method and restrictions of Weyl Chamber flows on $SL(n,\RR)/\Gamma$,, , ().   Google Scholar [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).   Google Scholar [5] D. Damjanovic, Central extensions of simple Lie groups and rigidity of some abelian partially hyperbolic algebraic actions,, J. Modern Dyn., 1 (2007), 665.   Google Scholar [6] Vinay V. Deodhar, On central extensions of rational points of algebraic groups,, Amer. J. Math., 100 (1978), 303.  doi: doi:10.2307/2373853.  Google Scholar [7] A. J. Hahn and O. T. O'Meara, The classical groups and K-theory,, Springer Verlag, (1980), 55.   Google Scholar [8] M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds. Lecture Notes in Mathematics,", 583, (1977).   Google Scholar [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.   Google Scholar [10] G. A. Margulis, "Discrete Subgroups of Semisimple Lie Groups,", Springer-Verlag, (1991).   Google Scholar [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.  Google Scholar [12] R. Steinberg, Generateurs, relations et revetements de groupes algebriques,, Colloque de Bruxelles, (1962), 113.   Google Scholar
 [1] Laurent Di Menza, Virginie Joanne-Fabre. An age group model for the study of a population of trees. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020464 [2] Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448 [3] Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319 [4] Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382

2019 Impact Factor: 0.5