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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.

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