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

Damjanović, Danijela^1, 2 ; Katok, Anatole¹

¹ Department of Mathematics, The Pennsylvania State University, University Park, PA 16802
² Erwin Schroedinger Institute, Boltzmanngasse 9, A-1090 Vienna, Austria

Electronic research announcements of the American Mathematical Society, Tome 10 (2004), pp. 142-154.

Résumé

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 $N\ge 6$.

Détail

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

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