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The action spectrum near positive definite invariant tori. (English) Zbl 1053.37035
The author proves that the Birkhoff normal form near a positive definite KAM torus is given by Mather’s \(\alpha\)-function, which is viewed here as an averaged energy or action. In dimension two this was shown by K. F. Siburg [Isr. J. Math. 113, 285–304 (1999; Zbl 0996.37051) and Comment. Math. Helv. 75, 681–700 (2000; Zbl 0985.37054)], and the present result for \(2n\) dimensions relies on a simplified proof. In particular, variational techniques allow one to circumvent KAM theory for the proof, whose main step is the existence of a family of periodic orbits near the invariant torus. Thus, the connection between Birkhoff invariants and the action spectrum can be derived essentially independent from KAM theory.

37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
37J50 Action-minimizing orbits and measures (MSC2010)
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
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