Journal of Modern Dynamics (JMD)

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

Pages: 203 - 235, Issue 2, April 2011      doi:10.3934/jmd.2011.5.203

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Danijela Damjanovic - Department of Mathematics, Rice University, 6100 Main St., Houston, TX 77005, United States (email)
Anatole Katok - Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, United States (email)

Abstract: 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, homogeneous actions, KAM method.
Mathematics Subject Classification:  37C85.

Received: March 2010;      Revised: April 2011;      Available Online: July 2011.