Exponential mixing, KAM and smooth local rigidity

Ralf Spatzier; Lei Yang

Exponential mixing, KAM and smooth local rigidity

Dynamical Systems 2022-01-19 v2

Authors: Ralf Spatzier , Lei Yang

Abstract

Consider actions of $\Z ^r$ by ergodic automorphisms on a compact nilmanifolds for $r \geq 2$ . 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.

Keywords

ergodic theory group actions

Cite

@article{arxiv.2106.01585,
  title  = {Exponential mixing, KAM and smooth local rigidity},
  author = {Ralf Spatzier and Lei Yang},
  journal= {arXiv preprint arXiv:2106.01585},
  year   = {2022}
}

Comments

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

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