Local Rigidity of Diophantine Translations in Higher-dimensional Tori Article Swipe
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Nikolaos Karaliolios
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YOU?
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· 2018
· Open Access
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· DOI: https://doi.org/10.1134/s1560354718010021
· OA: W2580726207
YOU?
·
· 2018
· Open Access
·
· DOI: https://doi.org/10.1134/s1560354718010021
· OA: W2580726207
We prove a theorem asserting that, given a Diophantine rotation $\\alpha $ in\na torus $\\T ^{d} \\equiv \\R ^{d} / \\Z ^{d}$, any perturbation, small enough in\nthe $C^{\\infty}$ topology, that does not destroy all orbits with rotation\nvector $\\alpha$ is actually smoothly conjugate to the rigid rotation. The proof\nrelies on a K.A.M. scheme (named after Kolmogorov-Arnol'd-Moser), where at each\nstep the existence of an invariant measure with rotation vector $\\alpha$\nassures that we can linearize the equations around the same rotation $\\alpha$.\nThe proof of the convergence of the scheme is carried out in the $C^{\\infty}$\ncategory.\n
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