Dimension and hitting time in rapidly mixing systems
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. \[ \lim_{r\to 0}\;\frac{\log \tau_r(x, x_0)}{-\log r}=d_\mu(x_0). \] This result is obtained by proving a kind of dynamical Borel-Cantelli lemma which holds also in systems having polynomial decay of correlations.
systems having polynomial decay of correlations
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