Dimension and hitting time in rapidly mixing systems

Abstract

We prove that if a system has superpolynomial (faster than any power law) decay of correlations then the time $\tau _{r}(x,x_{0})$ needed for a typical point $x$ to enter for the first time a ball $B(x_{0},r)$ centered in $x_{0},$ with small radius $r$ scales as the local dimension at $x_{0},$ i.e.% \begin{equation*} \underset{r\rightarrow 0}{\lim }\frac{\log \tau _{r}(x,x_{0})}{-\log r}% =d_{\mu }(x_{0}). \end{equation*} This result is obtained by proving a kind of dynamical Borel-Cantelli lemma wich holds also in systems having polinomial decay of correlations.