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

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

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

Danijela Damjanovic ^1, and Anatole Katok ^2,

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

Abstract
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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.

Keywords: Local rigidity, KAM method., homogeneous actions.

Mathematics Subject Classification: 37C8.

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

References:

[1]	E. J. Benveniste, Rigidity of isometric lattice actions on compact Riemannian manifolds, Geom. Funct. Anal., 10 (2000), 516-542. doi: 10.1007/PL00001627.
[2]	D. Damjanović, Central extensions of simple Lie groups and rigidity of some abelian partially hyperbolic algebraic actions, J. Mod. Dyn., 1 (2007), 665-688. doi: 10.3934/jmd.2007.1.665.
[3]	D. Damjanović and A. Katok, Local rigidity of restrictions of Weyl chamber flows, C. R. Math. Acad. Sci. Paris, 344 (2007), 503-508.
[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.
[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), 142-154.
[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), 1805-1858.
[7]	D. Fisher, First cohomology and local rigidity of group actions,, Ann. of Math., ().
[8]	L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows, Duke Math. J., 119 (2003), 465-526.
[9]	R. S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 65-222.
[10]	B. Hasselblatt and A. Katok, Principal structures, in "Handbook in Dynamical Systems," 1A, North-Holland, Amsterdam, (2002), 1-203.
[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), 287-314.
[12]	D. Mieczkowski, The first cohomology of parabolic actions for some higher-rank abelian groups and representation theory, J. Mod. Dyn., 1 (2007), 61-92. doi: 10.3934/jmd.2007.1.61.
[13]	D. Mieczkowski, "The Cohomological Equation and Representation Theory," Ph.D thesis, The Pennsylvania State University, 2006.
[14]	J. Moser, On commuting circle mappings and simoultaneous Diophantine approximations, Math. Z., 205 (1990), 105-121. doi: 10.1007/BF02571227.
[15]	F. Ramirez, Cocycles over higher-rank abelian actions on quotients of semisimple Lie groups, J. Mod. Dyn., 3 (2009), 335-357. doi: 10.3934/jmd.2009.3.335.
[16]	E. Zehnder, Generalized implicit-function theorems with applications to some small divisor problems. I, Comm. Pure Appl. Math., 28 (1975), 91-140.
[17]	Z. Wang, Local rigidity of partially hyperbolic actions, J. Mod. Dyn., 4 (2010), 271-327. doi: 10.3934/jmd.2010.4.271.
[18]	Z. Wang, New cases of differentiable rigidity for partially hyperbolic actions: Symplectic groups and resonance directions, J. Mod. Dyn., 4 (2010), 585-608. doi: 10.3934/jmd.2010.4.585.

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References:

[1]	E. J. Benveniste, Rigidity of isometric lattice actions on compact Riemannian manifolds, Geom. Funct. Anal., 10 (2000), 516-542. doi: 10.1007/PL00001627.
[2]	D. Damjanović, Central extensions of simple Lie groups and rigidity of some abelian partially hyperbolic algebraic actions, J. Mod. Dyn., 1 (2007), 665-688. doi: 10.3934/jmd.2007.1.665.
[3]	D. Damjanović and A. Katok, Local rigidity of restrictions of Weyl chamber flows, C. R. Math. Acad. Sci. Paris, 344 (2007), 503-508.
[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.
[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), 142-154.
[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), 1805-1858.
[7]	D. Fisher, First cohomology and local rigidity of group actions,, Ann. of Math., ().
[8]	L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows, Duke Math. J., 119 (2003), 465-526.
[9]	R. S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 65-222.
[10]	B. Hasselblatt and A. Katok, Principal structures, in "Handbook in Dynamical Systems," 1A, North-Holland, Amsterdam, (2002), 1-203.
[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), 287-314.
[12]	D. Mieczkowski, The first cohomology of parabolic actions for some higher-rank abelian groups and representation theory, J. Mod. Dyn., 1 (2007), 61-92. doi: 10.3934/jmd.2007.1.61.
[13]	D. Mieczkowski, "The Cohomological Equation and Representation Theory," Ph.D thesis, The Pennsylvania State University, 2006.
[14]	J. Moser, On commuting circle mappings and simoultaneous Diophantine approximations, Math. Z., 205 (1990), 105-121. doi: 10.1007/BF02571227.
[15]	F. Ramirez, Cocycles over higher-rank abelian actions on quotients of semisimple Lie groups, J. Mod. Dyn., 3 (2009), 335-357. doi: 10.3934/jmd.2009.3.335.
[16]	E. Zehnder, Generalized implicit-function theorems with applications to some small divisor problems. I, Comm. Pure Appl. Math., 28 (1975), 91-140.
[17]	Z. Wang, Local rigidity of partially hyperbolic actions, J. Mod. Dyn., 4 (2010), 271-327. doi: 10.3934/jmd.2010.4.271.
[18]	Z. Wang, New cases of differentiable rigidity for partially hyperbolic actions: Symplectic groups and resonance directions, J. Mod. Dyn., 4 (2010), 585-608. doi: 10.3934/jmd.2010.4.585.

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