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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
The copyright for this article reverts to public domain after 28 years from publication.

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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