# American Institute of Mathematical Sciences

April  2011, 5(2): 203-235. doi: 10.3934/jmd.2011.5.203

## Local rigidity of homogeneous parabolic actions: I. A model case

 1 Department of Mathematics, Rice University, 6100 Main St., Houston, TX 77005, United States 2 Department of Mathematics, The Pennsylvania State University, University Park, PA 16802

Received  March 2010 Revised  April 2011 Published  July 2011

We show a weak form of local differentiable rigidity for the rank $2$ abelian action of upper unipotents on $SL(2,R)$ $\times$ $SL(2,R)$ $/\Gamma$. Namely, for a $2$-parameter family of sufficiently small perturbations of the action, satisfying certain transversality conditions, there exists a parameter for which the perturbation is smoothly conjugate to the action up to an automorphism of the acting group. This weak form of rigidity for the parabolic action in question is optimal since the action lives in a family of dynamically different actions. The method of proof is based on a KAM-type iteration and we discuss in the paper several other potential applications of our approach.
Citation: Danijela Damjanovic, Anatole Katok. Local rigidity of homogeneous parabolic actions: I. A model case. Journal of Modern Dynamics, 2011, 5 (2) : 203-235. doi: 10.3934/jmd.2011.5.203
 [1] E. J. Benveniste, Rigidity of isometric lattice actions on compact Riemannian manifolds,, Geom. Funct. Anal., 10 (2000), 516.  doi: 10.1007/PL00001627.  Google Scholar [2] D. Damjanović, Central extensions of simple Lie groups and rigidity of some abelian partially hyperbolic algebraic actions,, J. Mod. Dyn., 1 (2007), 665.  doi: 10.3934/jmd.2007.1.665.  Google Scholar [3] D. Damjanović and A. Katok, Local rigidity of restrictions of Weyl chamber flows,, C. R. Math. Acad. Sci. Paris, 344 (2007), 503.   Google Scholar [4] 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)$$/\Gamma,, Int. Math. Res. Notes, (2010). doi: 10.1093/imrn/rnq252. Google Scholar [5] D. Damjanović and A. Katok, Local rigidity of actions of higher-rank abelian groups and KAM method,, Electron. Res. Announc. Amer. Math. Soc., 10 (2004), 142. Google Scholar [6] D. Damjanović and A. Katok, Local rigidity of partially hyperbolic actions I. KAM method and \Z^k actions on the torus,, Ann. of Math. (2), 172 (2010), 1805. Google Scholar [7] D. Fisher, First cohomology and local rigidity of group actions,, Ann. of Math., (). Google Scholar [8] L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows,, Duke Math. J., 119 (2003), 465. Google Scholar [9] R. S. Hamilton, The inverse function theorem of Nash and Moser,, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 65. Google Scholar [10] B. Hasselblatt and A. Katok, Principal structures,, in, 1A (2002), 1. Google Scholar [11] 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 [12] D. Mieczkowski, The first cohomology of parabolic actions for some higher-rank abelian groups and representation theory,, J. Mod. Dyn., 1 (2007), 61. doi: 10.3934/jmd.2007.1.61. Google Scholar [13] D. Mieczkowski, "The Cohomological Equation and Representation Theory,", Ph.D thesis, (2006). Google Scholar [14] J. Moser, On commuting circle mappings and simoultaneous Diophantine approximations,, Math. Z., 205 (1990), 105. doi: 10.1007/BF02571227. Google Scholar [15] F. Ramirez, Cocycles over higher-rank abelian actions on quotients of semisimple Lie groups,, J. Mod. Dyn., 3 (2009), 335. doi: 10.3934/jmd.2009.3.335. Google Scholar [16] E. Zehnder, Generalized implicit-function theorems with applications to some small divisor problems. I,, Comm. Pure Appl. Math., 28 (1975), 91. Google Scholar [17] Z. Wang, Local rigidity of partially hyperbolic actions,, J. Mod. Dyn., 4 (2010), 271. doi: 10.3934/jmd.2010.4.271. Google Scholar [18] Z. Wang, New cases of differentiable rigidity for partially hyperbolic actions: Symplectic groups and resonance directions,, J. Mod. Dyn., 4 (2010), 585. doi: 10.3934/jmd.2010.4.585. Google Scholar show all references ##### References:  [1] E. J. Benveniste, Rigidity of isometric lattice actions on compact Riemannian manifolds,, Geom. Funct. Anal., 10 (2000), 516. doi: 10.1007/PL00001627. Google Scholar [2] D. Damjanović, Central extensions of simple Lie groups and rigidity of some abelian partially hyperbolic algebraic actions,, J. Mod. Dyn., 1 (2007), 665. doi: 10.3934/jmd.2007.1.665. Google Scholar [3] D. Damjanović and A. Katok, Local rigidity of restrictions of Weyl chamber flows,, C. R. Math. Acad. Sci. Paris, 344 (2007), 503. Google Scholar [4] 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)$$/\Gamma$,, Int. Math. Res. Notes, (2010).  doi: 10.1093/imrn/rnq252.  Google Scholar [5] D. Damjanović and A. Katok, Local rigidity of actions of higher-rank abelian groups and KAM method,, Electron. Res. Announc. Amer. Math. Soc., 10 (2004), 142.   Google Scholar [6] D. Damjanović and A. Katok, Local rigidity of partially hyperbolic actions I. KAM method and $\Z^k$ actions on the torus,, Ann. of Math. (2), 172 (2010), 1805.   Google Scholar [7] D. Fisher, First cohomology and local rigidity of group actions,, Ann. of Math., ().   Google Scholar [8] L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows,, Duke Math. J., 119 (2003), 465.   Google Scholar [9] R. S. Hamilton, The inverse function theorem of Nash and Moser,, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 65.   Google Scholar [10] B. Hasselblatt and A. Katok, Principal structures,, in, 1A (2002), 1.   Google Scholar [11] 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 [12] D. Mieczkowski, The first cohomology of parabolic actions for some higher-rank abelian groups and representation theory,, J. Mod. Dyn., 1 (2007), 61.  doi: 10.3934/jmd.2007.1.61.  Google Scholar [13] D. Mieczkowski, "The Cohomological Equation and Representation Theory,", Ph.D thesis, (2006).   Google Scholar [14] J. Moser, On commuting circle mappings and simoultaneous Diophantine approximations,, Math. Z., 205 (1990), 105.  doi: 10.1007/BF02571227.  Google Scholar [15] F. Ramirez, Cocycles over higher-rank abelian actions on quotients of semisimple Lie groups,, J. Mod. Dyn., 3 (2009), 335.  doi: 10.3934/jmd.2009.3.335.  Google Scholar [16] E. Zehnder, Generalized implicit-function theorems with applications to some small divisor problems. I,, Comm. Pure Appl. Math., 28 (1975), 91.   Google Scholar [17] Z. Wang, Local rigidity of partially hyperbolic actions,, J. Mod. Dyn., 4 (2010), 271.  doi: 10.3934/jmd.2010.4.271.  Google Scholar [18] Z. Wang, New cases of differentiable rigidity for partially hyperbolic actions: Symplectic groups and resonance directions,, J. Mod. Dyn., 4 (2010), 585.  doi: 10.3934/jmd.2010.4.585.  Google Scholar
 [1] Danijela Damjanovic and Anatole Katok. Local rigidity of actions of higher rank abelian groups and KAM method. Electronic Research Announcements, 2004, 10: 142-154. [2] Masayuki Asaoka. Local rigidity of homogeneous actions of parabolic subgroups of rank-one Lie groups. Journal of Modern Dynamics, 2015, 9: 191-201. doi: 10.3934/jmd.2015.9.191 [3] Zhenqi Jenny Wang. Local rigidity of partially hyperbolic actions. Journal of Modern Dynamics, 2010, 4 (2) : 271-327. doi: 10.3934/jmd.2010.4.271 [4] Zhenqi Jenny Wang. Local rigidity of partially hyperbolic actions. Electronic Research Announcements, 2010, 17: 68-79. doi: 10.3934/era.2010.17.68 [5] Qiao Liu. Local rigidity of certain solvable group actions on tori. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020269 [6] Woochul Jung, Keonhee Lee, Carlos Morales, Jumi Oh. Rigidity of random group actions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (12) : 6845-6854. doi: 10.3934/dcds.2020130 [7] A. Katok and R. J. Spatzier. Nonstationary normal forms and rigidity of group actions. Electronic Research Announcements, 1996, 2: 124-133. [8] Danijela Damjanović, James Tanis. Cocycle rigidity and splitting for some discrete parabolic actions. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 5211-5227. doi: 10.3934/dcds.2014.34.5211 [9] James Tanis, Zhenqi Jenny Wang. Cohomological equation and cocycle rigidity of discrete parabolic actions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3969-4000. doi: 10.3934/dcds.2019160 [10] Manfred Einsiedler and Elon Lindenstrauss. Rigidity properties of \zd-actions on tori and solenoids. Electronic Research Announcements, 2003, 9: 99-110. [11] Andrei Török. Rigidity of partially hyperbolic actions of property (T) groups. Discrete & Continuous Dynamical Systems - A, 2003, 9 (1) : 193-208. doi: 10.3934/dcds.2003.9.193 [12] Federico Rodriguez Hertz. Global rigidity of certain Abelian actions by toral automorphisms. Journal of Modern Dynamics, 2007, 1 (3) : 425-442. doi: 10.3934/jmd.2007.1.425 [13] Danijela Damjanovic, James Tanis, Zhenqi Jenny Wang. On globally hypoelliptic abelian actions and their existence on homogeneous spaces. Discrete & Continuous Dynamical Systems - A, 2020, 40 (12) : 6747-6766. doi: 10.3934/dcds.2020164 [14] Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020184 [15] Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020196 [16] Boris Kalinin, Anatole Katok. Measure rigidity beyond uniform hyperbolicity: invariant measures for cartan actions on tori. Journal of Modern Dynamics, 2007, 1 (1) : 123-146. doi: 10.3934/jmd.2007.1.123 [17] Boris Kalinin, Anatole Katok, Federico Rodriguez Hertz. Errata to "Measure rigidity beyond uniform hyperbolicity: Invariant measures for Cartan actions on tori" and "Uniqueness of large invariant measures for $\Zk$ actions with Cartan homotopy data". Journal of Modern Dynamics, 2010, 4 (1) : 207-209. doi: 10.3934/jmd.2010.4.207 [18] Xuanji Hou, Lei Jiao. On local rigidity of reducibility of analytic quasi-periodic cocycles on $U(n)$. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3125-3152. doi: 10.3934/dcds.2016.36.3125 [19] Xuanji Hou, Jiangong You. Local rigidity of reducibility of analytic quasi-periodic cocycles on $U(n)$. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 441-454. doi: 10.3934/dcds.2009.24.441 [20] Zhenqi Jenny Wang. New cases of differentiable rigidity for partially hyperbolic actions: Symplectic groups and resonance directions. Journal of Modern Dynamics, 2010, 4 (4) : 585-608. doi: 10.3934/jmd.2010.4.585

2019 Impact Factor: 0.465