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

Danijela Damjanović; Anatole Katok

doi:http://doi.org/10.4007/annals.2010.172.1805

Abstract

We show $\Bbb C^\infty$ local rigidity for $\mathbb{Z}^k$ $(k\ge 2)$ higher rank partially hyperbolic actions by toral automorphisms, using a generalization of the KAM (Kolmogorov-Arnold-Moser) iterative scheme. We also prove the existence of irreducible genuinely partially hyperbolic higher rank actions on any torus $\mathbb{T}^N$ for any even $N\ge 6$.

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@article{ZJW1, author={Wang, Z. J.}, TITLE={Local rigidity of partially hyperbolic actions}, JOURNAL={J. Mod. Dyn.}, VOLUME={4}, YEAR={2010}, PAGES={271--327}, ZBLNUMBER={pre05790308}, }
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@article{ZJW2, author={Wang, Z. J.}, TITLE={New cases of differentiable rigidity for partially hyperbolic actions: symplectic groups and resonance directions}, JOURNAL={J. Mod. Dyn.}, NOTE={to appear}, }

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

Abstract

Authors