**Author:**
Alberto Berretti, Guido Gentile

**Title:** *
Periodic and quasi-periodic orbits for the standard map
*

**Abstract: **
We consider both periodic and quasi-periodic solutions for the
standard map, and we study the corresponding conjugating functions,
i.e. the functions conjugating the motions to trivial rotations.
We compare the invariant curves with rotation numbers
satisfying the Bryuno condition and the sequences of periodic
orbits with rotation numbers given by their convergents
We prove the following results:
(1) for rotation numbers corresponding to the convergents
we study the radius of convergence of the conjugating functions and
we find lower bounds on them, which tend to a limit which
is a lower bound on the corresponding quantity for the Bryuno
rotation number;
(2) the periodic orbits consist of points which are
more and more close to the invariant curve with rotation number
to which the convergents tend;
(3) such orbits lie on analytical curves which tend uniformly to the
invariant curve.

**Keywords:**
Standard map, perturbation theory,
resonances, periodic orbits, invariant curves

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