Differentiable Rigidity for quasiperiodic cocycles in compact Lie groups

Abstract

We study close-to-constants quasiperiodic cocycles in ${\mathbb{T}}^d\times G$, where $d\in {\mathbb{N}}^{\ast }$ 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 K.A.M. scheme actually produces a conjugation. We also derive a global differentiable rigidity theorem, assuming the convergence of the renormalization scheme for such dynamical systems.

16 pages. arXiv admin note: substantial text overlap with arXiv:1407.4763

References in corpus (2)

• Global aspects of the reducibility of quasiperiodic cocycles in semisimple compact Lie groups
• Continuous spectrum or measurable reducibility for quasiperiodic cocycles in ${\mathbb{T}}^d\times SU(2)$

Cited by in corpus (4)

• Continuous spectrum or measurable reducibility for quasiperiodic cocycles in ${\mathbb{T}}^d\times SU(2)$
• Fibered rotation vector and hypoellipticity for quasiperiodic cocycles in compact Lie groups
• Anosov-Katok constructions for quasi-periodic $\mathrm{SL}(2,R)$ cocycles
• Cohomological rigidity and the Anosov-Katok construction