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

TL;DR

This paper establishes a precise relationship between hitting times and measure scaling in dynamical systems with superpolynomial decay of correlations, extending previous results from balls to more general sets.

Contribution

It generalizes the logarithm law for hitting times from balls to sublevel sets of Lipschitz functions in systems with superpolynomial decay of correlations.

Findings

Hitting time scales as inverse measure of the set for systems with superpolynomial decay.

The result extends previous logarithm laws from geometric balls to Lipschitz sublevel sets.

Applications include observed systems and geodesic flows on negatively curved manifolds.

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 $S_{r}=\{x:f(x)\leq r\}$ 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 \tau (x,S_{r})}{-\log r}=\underset{r\to 0}{\lim}\frac{\log \mu (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.