FM 00-16 ; mp_arc 01-406

Author: Alberto Berretti, Guido Gentile

Title: Renormalization Group and Field Theoretic Techniques for the Analysis of the Lindstedt Series
 
Abstract: The Lindstedt series were introduced in the XIX century in Astronomy to study perturbatively quasi-periodic motions in Celestial Mechanics. In Mathematical Physics, after getting the attention of Poincare', who studied them widely by pursuing to all orders the analysis of Lindstedt and Newcomb, their use was somehow superseded by other methods usually referred to as KAM theory. Only recently, after Eliasson's work, they have been reconsidered as a tool to prove KAM-type results, in a spirit close to that of the Renormalization Group in quantum field theory. Following this new approach we discuss here the use of the Lindstedt series in the context of some model problems, like the standard map and natural generalizations, with particular attention to the properties of analyticity in the perturbative parameter.

Keywords: Standard map, semistandard map and generalizations, perturbation theory, renormalization group, trees, resonances, Bryuno function, invariant curves, scaling properties, Lindstedt series

Alberto Berretti
Dipartimento di Matematica
II Universita` di Roma (Tor Vergata)
Via della Ricerca Scientifica, 00133 Roma, Italy
e-mail: berretti@mat.uniroma2.it

Guido Gentile
Dipartimento di Matematica
Universita` di Roma Tre
Largo San Leonardo Murialdo 1, 00146 Roma, Italy
e-mail: gentile@matrm3.mat.uniroma3.it