Dimension and waiting time in rapidly mixing systems

TL;DR

This paper establishes that in systems with superpolynomial decay of correlations, the first entry time into small neighborhoods scales with the local dimension, linking dynamical mixing properties to geometric measures.

Contribution

It proves a precise relation between decay of correlations and the scaling of entry times with local dimension in rapidly mixing systems.

Findings

First entry time scales with local dimension for superpolynomial decay systems.

Decay of correlations faster than any power law implies a specific entry time behavior.

Connects statistical mixing rates with geometric properties of the system.

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. $$\underset{r\to 0}{\lim}\frac{\log \tau_{r}(x,x_{0})}{-\log r}=d_{\mu }(x_{0}).$$