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ISSN 1079-6762

   

 

Local rigidity of actions of higher rank abelian groups and KAM method


Authors: Danijela Damjanovic and Anatole Katok
Journal: Electron. Res. Announc. Amer. Math. Soc. 10 (2004), 142-154
MSC (2000): Primary 37C85, 37C15, 58C15
Published electronically: December 10, 2004
MathSciNet review: 2119035
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Abstract: We develop a new method for proving local differentiable rigidity for actions of higher rank abelian groups. Unlike earlier methods it does not require previous knowledge of structural stability and instead uses a version of the KAM (Kolmogorov-Arnold-Moser) iterative scheme. As an application we show $\mathcal{C}^\infty$ local rigidity for $\mathbb{Z} ^k (k\ge 2)$ partially hyperbolic actions by toral automorphisms. We also prove the existence of irreducible genuinely partially hyperbolic higher rank actions by automorphisms on any torus $\mathbb{T} ^N$ for any even $N\ge 6$.


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Additional Information

Danijela Damjanovic
Affiliation: Department of Mathematics, The Pennsylvania State University, University Park, PA 16802
Address at time of publication: Erwin Schroedinger Institute, Boltzmanngasse 9, A-1090 Vienna, Austria
Email: damjanov@math.psu.edu

Anatole Katok
Affiliation: Department of Mathematics, The Pennsylvania State University, University Park, PA 16802
Email: katok_a@math.psu.edu

DOI: http://dx.doi.org/10.1090/S1079-6762-04-00139-8
PII: S 1079-6762(04)00139-8
Keywords: Local rigidity, group actions, KAM method, torus
Received by editor(s): September 19, 2004
Published electronically: December 10, 2004
Additional Notes: Anatole Katok was partially supported by NSF grant DMS 0071339
Communicated by: Gregory Margulis
Article copyright: © Copyright 2004 Danijela Damjanovic and Anatole Katok
The copyright for this article reverts to public domain 28 years after publication.