Journal of Modern Dynamics (JMD)

Differentiable rigidity for quasiperiodic cocycles in compact Lie groups

Pages: 125 - 142, Volume 11, 2017      doi:10.3934/jmd.2017006

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Nikolaos Karaliolios - Imperial College London, South Kensington Campus, 180 Queen’s Gate, Huxley Building, London SW7 2AZ, United Kingdom (email)

Abstract: We study close-to-constants quasiperiodic cocycles in $\mathbb{T}^{d} \times G$, where $d \in \mathbb{N}^{*} $ and $G$ is a compact Lie group, under the assumption that the rotation in the basis satisfies a Diophantine condition. We prove differentiable rigidity for such cocycles: if such a cocycle is measurably conjugate to a constant one satisfying a Diophantine condition with respect to the rotation, then it is $C^{\infty}$-conjugate to it, and the KAM scheme actually produces a conjugation. We also derive a global differentiable rigidity theorem, assuming the convergence of the renormalization scheme for such dynamical systems.

Keywords:  Rigidity, quasi-periodic cocycles, compact Lie groups, KAM theory.
Mathematics Subject Classification:  Primary: 37C55.

Received: November 2014;      Revised: June 2016;      Available Online: January 2017.