Hitting time in regular sets and logarithm law for rapidly mixing dynamical systems

We prove that if a system has superpolynomial (faster than any power law) decay of correlations (with respect to Lipschitz observables), then the time $\tau (x,S_{r})$ is 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 function $f$ scales as $\frac{1}{\mu (S_{r})}$ i.e.,

$$\underset{r\rightarrow 0}{\lim }\frac{\log \tau (x,S_{r})}{-\log r}=\underset {r\rightarrow 0}{\lim }\frac{\log \mu (S_{r})}{\log r}.$$

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 in negatively curved manifolds.