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

Danijela Damjanovic; Anatole Katok

doi:10.3934/jmd.2011.5.203

Article Contents

2011, Volume 5, Issue 2: 203-235. Doi: 10.3934/jmd.2011.5.203

This issue Previous Article Next Article Measures invariant under horospherical subgroups in positive characteristic

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
2.
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802

Received: February 28, 2010

Revised: March 31, 2011

Published: June 2011

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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.

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Mathematics Subject Classification: 37C85.

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