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

Abstract

We prove that if a system has superpolynomial (faster than any power law) decay of correlations (with respect to Lipschitz observables) then the time $\tau (x,S_r)$ needed for a typical point $x$ to enter for the first time a set $S_r=\{x:f(x)\le r\}$ which is a sublevel of a Lipschitz funcion $f$ scales as $\frac{1}{\mu (S_r)}$ i.e. \begin{equation*} \underset{r\to 0}{\lim }\frac{\log τ(x,S_{r})}{-\log r}=\underset{r\to 0}{\lim}\frac{\log μ(S_{r})}{\log (r)}. \end{equation*} 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 of negatively curved manifolds.

References in corpus (3)

• The dynamical Borel-Cantelli lemma and the waiting time problems
• Long hitting time, slow decay of correlations and arithmetical properties
• Poincaré recurrence for observations