  In this paper, we succeed in integrating Strichartz estimates (encoding the
dispersive effects of the equations) in Birkhoff normal form techniques. As a
consequence, we deduce a result on the long time behavior of quintic NLS
solutions on the circle for small but very irregular initial data (in $H^s$ for
$s > 2/5$). Note that since $2/5 < 1$, we cannot claim conservation of energy
and, more importantly, since $2/5 < 1/2$, we must dispense with the algebra
property of $H^s$. This is the first dynamical result where we use the
dispersive properties of NLS in a context of Birkhoff normal form.
