Central Limit Theorems for the Shrinking Target Problem

Suppose B _i :=B(p,r _i ) are nested balls of radius r _i about a point p in a dynamical system (T,X,μ). The question of whether T ⁱ x∈B _i infinitely often (i.o.) for μ 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 $lim_{ntoinfty} frac{1}{E_{n}} sum_{j=1}^{n} 1_{B_{i}} (T^{i} x) to1$ for μ a.e. x∈X. This is a self-norming type of strong law of large numbers. We establish self-norming central limit theorems (CLT) of the form $lim_{ ntoinfty} frac{1}{a_{n}} sum_{i=1}^{n} [1_{B_{i}} (T^{i} x)-mu(B_{i})] to N(0,1)$ (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.