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. $$\underset{r\to 0}{\lim}\frac{\log \tau_{r}(x,x_{0})}{-\log r}=d_{\mu }(x_{0}).$$ This result is obtained by proving a kind of dynamical Borel-Cantelli lemma wich holds also in systems having polinomial decay of correlations.

Dimension and hitting time in rapidly mixing systems

GALATOLO, STEFANO
2007-01-01

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. $$\underset{r\to 0}{\lim}\frac{\log \tau_{r}(x,x_{0})}{-\log r}=d_{\mu }(x_{0}).$$ This result is obtained by proving a kind of dynamical Borel-Cantelli lemma wich holds also in systems having polinomial decay of correlations.
2007
Galatolo, Stefano
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11568/112208
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