  We study chaotic orbits of conservative low--dimensional maps and present
numerical results showing that the probability density functions (pdfs) of the
sum of $N$ iterates in the large $N$ limit exhibit very interesting
time-evolving statistics. In some cases where the chaotic layers are thin and
the (positive) maximal Lyapunov exponent is small, long--lasting
quasi--stationary states (QSS) are found, whose pdfs appear to converge to
$q$--Gaussians associated with nonextensive statistical mechanics. More
generally, however, as $N$ increases, the pdfs describe a sequence of QSS that
pass from a $q$--Gaussian to an exponential shape and ultimately tend to a true
Gaussian, as orbits diffuse to larger chaotic domains and the phase space
dynamics becomes more uniformly ergodic.
