95-151 Gallavotti, G., Gentile, G., Mastropietro, V.

Field theory and KAM tori (52K, TeX) Mar 20, 95

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Abstract. The parametric equations of KAM tori for a $l$ degrees of freedom quasi integrable system, are shown to be one point Schwinger functions of a suitable euclidean quantum field theory on the $l$ dimensional torus. KAM theorem is equivalent to a ultraviolet stability theorem. A renormalization group treatment of the field theory leads to a resummation of the formal pertubation series and to an expansion in terms of $l^2$ new parameters forming a $l\times l$ matrix $\s_\e$ (identified as a family of renormalization constants). The matrix $\s_\e$ is an analytic function of the coupling $\e$ at small $\e$: the breakdown of the tori at large $\e$ is speculated to be related to the crossing by $\s_\e$ of a ``critical" surface at a value $\e=\e_c$ where the function $\s_\e$ is still finite. A mechanism for the possible universality of the singularities of parametric equations for the invariant tori, in their parameter dependence as well as in the $\e_c-\e$ dependence, is proposed.}

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