**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