FM 00-14 ; mp_arc 00-252

Author: Alberto Berretti, Guido Gentile

Title: Non-universal behaviour of scaling properties for generalized semistandard and standard maps
 
Abstract: We consider two-dimensional maps generalizing the semistandard map by allowing more general analytic nonlinear terms having only Fourier components $f_{\nu}$ with positive label $\nu$, and study the solutions corresponding to homotopically nontrivial invariant curves with complex rotation number. Then we show that, if the perturbation parameter is suitably rescaled, when the rotation number tends to a rational value non-tangentially to the real axis, the limit of the conjugating function is a well defined analytic function. The rescaling depends not only on the limit value of the rotation number, but also on the map, and it is obtainable by the solution of a Diophantine problem: so no universality property is exhibited. We show also that the rescaling can be different from that of the corresponding generalized standard maps, i.e. of the maps having also the Fourier components $f_{-\nu}=f_{\nu}$. The results allow us to give quantitative bounds, from above and from below, on the radius of convergence of the limit function for generalized standard maps in the case of nonlinear terms which are trigonometric polynomials, solving a problem left open in a previous work of ours.

Keywords: Generalized standard and semistandard maps, perturbation theory, resonances, invariant tori, scaling properties, genericity

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