Exponential mixing, KAM and smooth local rigidity

Abstract

Consider actions of ${\mathbb{Z}}^r$ by ergodic automorphisms on a compact nilmanifolds for $r\ge 2$. We show that small $C^k$ perturbations of such higher rank partially hyperbolic actions are smoothly conjugate to the original action, using a KAM scheme. The driving force for convergence of this iteration is the exponential mixing of the original action.

The proof of Lemma 3.1 has a gap. While there is exponential mixing for Holder functions, the rate of the mixing depends on the Holder exponent of the function. This leads to a vicious circle

References in corpus (1)

• Global rigidity of higher rank Anosov algebraic actions