Central Limit Theorems for the Shrinking Target Problem

Abstract

Suppose $B_i:= B(p,r_i)$ are nested balls of radius $r_i$ about a point $p$ in a dynamical system $(T,X,\mu)$. The question of whether $T^i x\in B_i$ infinitely often (i. o.) for $\mu$ a.e.\ $x$ is often called the shrinking target problem. In many dynamical settings it has been shown that if $E_n:=\sum_{i=1}^n \mu (B_i)$ diverges then there is a quantitative rate of entry and $\lim_{n\to \infty} \frac{1}{E_n} \sum_{j=1}^{n} 1_{B_i} (T^i x) \to 1$ for $\mu$ a.e. $x\in X$. This is a self-norming type of strong law of large numbers. We establish self-norming central limit theorems (CLT) of the form $\lim_{n\to \infty} \frac{1}{a_n} \sum_{i=1}^{n} [1_{B_i} (T^i x)-\mu(B_i)] \to N(0,1)$ (in distribution) for a variety of hyperbolic and non-uniformly hyperbolic dynamical systems, the normalization constants are $a^2_n \sim E [\sum_{i=1}^n 1_{B_i} (T^i x)-\mu(B_i)]^2$. Dynamical systems to which our results apply include smooth expanding maps of the interval, Rychlik type maps, Gibbs-Markov maps, rational maps and, in higher dimensions, piecewise expanding maps. For such central limit theorems the main difficulty is to prove that the non-stationary variance has a limit in probability.