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

Damjanović, Danijela; Katok, Anatole

doi:10.1090/S1079-6762-04-00139-8

ISSN 1079-6762

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

Authors: Danijela Damjanović and Anatole Katok
Journal: Electron. Res. Announc. Amer. Math. Soc. 10 (2004), 142-154
MSC (2000): Primary 37C85, 37C15, 58C15
DOI: https://doi.org/10.1090/S1079-6762-04-00139-8
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 $N\ge 6$.

Local rigidity of partially hyperbolic actions of $\mathbb {Z}^k$ and $\mathbb {R}^k,\,\,k\ge 2$. I. KAM method and actions on the torus

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