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Herman’s last geometric theorem. (English) Zbl 1175.37062
Authors’ abstract: We present a proof of Herman’s Last Geometric Theorem asserting that if \(F \) is a smooth diffeomorphism of the annulus having the intersection property, then any given \(F \)-invariant smooth curve on which the rotation number of \(F \) is Diophantine is accumulated by a positive measure set of smooth invariant curves on which \(F \) is smoothly conjugated to rotation maps. This implies in particular that a Diophantine elliptic fixed point of an area preserving diffeomorphism of the plane is stable. The remarkable feature of this theorem is that it does not require any twist assumption.

MSC:
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
28D05 Measure-preserving transformations
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