On the linearization of analytic diffeomorphisms of the torus

We provide an arithmetic condition weaker then the Bryuno condition for which it is possible to apply a KAM scheme in dimension greater then one. The KAM scheme will be provided in the setting of linearization of analytic diffeomorphisms of the torus that are close to a rotation.

💡 Research Summary

The paper addresses the problem of analytically linearizing diffeomorphisms of the torus (T^{d}) that are close to a rotation. In one dimension the optimal arithmetic condition for analytic linearization is the Bryuno condition, which is known to be both necessary and sufficient. In higher dimensions the situation is far less clear: while Herman showed that a global KAM theory cannot exist for (d\ge2), it remained open whether the Bryuno condition is still optimal for local linearization.

The authors introduce a new, weaker arithmetic condition—called the “weak‑Bryuno” condition—tailored to the multi‑dimensional setting. The definition relies on a directional decomposition of the frequency space. For each integer vector (\ell\in\mathbb Z^{d}), each unit direction (\beta\in S^{d-1}), and a scale‑dependent parameter (\delta), they define
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