Differentiable Rigidity for quasiperiodic cocycles in compact Lie groups Article Swipe

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Quasiperiodic function Mathematics Rigidity (electromagnetism) Diophantine equation Lie group Differentiable function Pure mathematics Rotation number Kolmogorov–Arnold–Moser theorem Conjugate Mathematical analysis Rotation (mathematics) Hamiltonian system Geometry Physics Quantum mechanics

Nikolaos Karaliolios ·

YOU? · · 2017 · Open Access · · DOI: https://doi.org/10.3934/jmd.2017006 · OA: W2273319438

We study close-to-constants quasiperiodic cocycles in $\\mathbb{T} ^{d} \\times\nG$, where $d \\in \\mathbb{N} ^{*} $ and $G$ is a compact Lie group, under the\nassumption that the rotation in the basis satisfies a Diophantine condition. We\nprove differentiable rigidity for such cocycles: if such a cocycle is\nmeasurably conjugate to a constant one satisfying a Diophantine condition with\nrespect to the rotation, then it is $C^{\\infty}$-conjugate to it, and the\nK.A.M. scheme actually produces a conjugation. We also derive a global\ndifferentiable rigidity theorem, assuming the convergence of the\nrenormalization scheme for such dynamical systems.\n