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Local rigidity of actions of higher rank abelian groups and
KAM method
Danijela Damjanovic and Anatole Katok
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 $N\ge 6$.
Copyright 2004 Danijela Damjanovic and Anatole Katok
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Article Info
- ERA Amer. Math. Soc. 10 (2004), pp. 142-154
- Publisher Identifier: S 1079-6762(04)00139-8
- 2000 Mathematics Subject Classification. Primary 37C85, 37C15, 58C15
- Key words and phrases. Local rigidity, group actions, KAM method, torus
- Received by editors September 19, 2004
- Posted on December 10, 2004
- Communicated by Gregory Margulis
- Comments (When Available)
Danijela Damjanovic
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
E-mail address: damjanov@math.psu.edu
Anatole Katok
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802
katok_a@math.psu.edu
Anatole Katok was partially supported by NSF grant DMS 0071339
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