  We prove that if a system has superpolynomial (faster than any power law)
decay of correlations (with respect to Lipschitz observables) then the time
$\tau (x,S_{r})$ needed for a typical point $x$ to enter for the first time a
set $S_{r}=\{x:f(x)\leq r\}$ which is a sublevel of a Lipschitz funcion $f$
scales as $\frac{1}{\mu (S_{r})}$ i.e. \begin{equation*} \underset{r\to 0}{\lim
}\frac{\log \tau (x,S_{r})}{-\log r}=\underset{r\to 0}{\lim}\frac{\log \mu
(S_{r})}{\log (r)}. \end{equation*} This generalizes a previous result obtained
for balls. We will also consider relations with the return time distributions,
an application to observed systems and to the geodesic flow of negatively
curved manifolds.
