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Nikolaos Karaliolios

London, SW7 2AZ, UK
Imperial College London South Kensington Campus


Karaliolios N.
Local Rigidity of Diophantine Translations in Higher-dimensional Tori
2018, vol. 23, no. 1, pp.  12-25
We prove a theorem asserting that, given a Diophantine rotation $\alpha $ in a torus $\mathbb{T} ^{d} \equiv \mathbb{R} ^{d} / \mathbb{Z} ^{d}$, any perturbation, small enough in the $C^{\infty}$ topology, that does not destroy all orbits with rotation vector $\alpha$ is actually smoothly conjugate to the rigid rotation. The proof relies on a KAM scheme (named after Kolmogorov–Arnol'd–Moser), where at each step the existence of an invariant measure with rotation vector $\alpha$ assures that we can linearize the equations around the same rotation $\alpha$. The proof of the convergence of the scheme is carried out in the $C^{\infty}$ category.
Keywords: KAM theory, quasi-periodic dynamics, Diophantine translations, local rigidity
Citation: Karaliolios N.,  Local Rigidity of Diophantine Translations in Higher-dimensional Tori, Regular and Chaotic Dynamics, 2018, vol. 23, no. 1, pp. 12-25

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