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 τr(x, x0) needed for a typical point x to enter for the first time a ball B(x0, r) centered in x0, with small radius r scales as the local dimension at x0, i.e. lim r → 0 log τr(x, x 0)/- log r = dμ(x0). This result is obtained by proving a kind of dynamical Borel-Cantelli lemma wich holds also in systems having polinomial decay of correlations. © International Press 2007.