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

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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Authors

Danijela Damjanović

Department of Mathematics
Rice University, 6100 Main Street, Houston, TX 77005
United States

Anatole Katok

Department of Mathematics
The Pennsylvania State University
University Park
State College, 16802
United States