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,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 $\lim_{n\to\infty} \frac{1}{E_{n}} \sum_{j=1}^{n} 1_{B_{i}} (T^{i} x) \to1$ 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 $\lim_{ n\to\infty} \frac{1}{a_{n}} \sum_{i=1}^{n} [1_{B_{i}} (T^{i} x)-\mu(B_{i})] \to N(0,1)$ (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.

Appendices

Appendix A: Gal-Koksma Theorem

We recall the following result of Gal and Koksma as formulated by W. Schmidt [27, 28] and stated by Sprindzuk [30]:

Proposition A.1

Let $(\varOmega,\mathcal{B},\mu)$ be a probability space and let f _k (ω), (k=1,2,…) be a sequence of non-negative μ measurable functions and g _k , h _k be sequences of real numbers such that 0≤g _k ≤h _k ≤1, (k=1,2,…,). Suppose there exists C>0 such that

$$ \int\biggl(\sum_{m<k\le n} \bigl( f_k (\omega) - g_k\bigr) \biggr)^2\,d \mu\le C \sum_{m<k \le n} h_k $$

(∗)

for arbitrary integers m<n. Then for any ϵ>0

$$\sum_{1\le k \le n} f_k (\omega) =\sum _{1\le k\le n} g_k + O \bigl(\varTheta^{1/2} (n) \log^{3/2+\epsilon} \varTheta(n) \bigr) $$

for μ a.e. ω∈Ω, where Θ(n)=∑_1≤k≤n h _k .

Appendix B: Assumption (C) for Expanding Systems

In this appendix we show that if the invariant measure μ has a density ρ(x) with respect to Lebesgue measure m then Assumption (C) is valid. Recall we define

Lemma B.1

Let B _i (x) denote a decreasing sequence of balls with center x and suppose $\mu(B_{i} (x)) \le\frac{C_{2}}{i^{\gamma}}$ for some constants C ₂ and 0<γ≤1. Suppose μ has a density ρ with respect to Lebesgue measure m with support X and there exists C ₇>0, δ>0 such that for all k, ϵ,

Then for μ a.e. p∈X there exists η(p)∈(0,1) and κ>1 such that for all i sufficiently large

$$\mu\bigl(B_i (p) \cap T^{-r} B_i (p) \bigr)\le \mu\bigl(B_i (p)\bigr)^{1+\eta} $$

for all r=1,…,log^κ i.

Proof

Let $\rho(x)=\frac{d\mu}{dm} (x)$ be the density of μ with respect to Lebesgue measure m.

Let σ≥1 and γ>σ. We choose ϵ _k so that for all x a ball of radius ϵ _k about x, denoted B(x,ϵ _k ), satisfies c ₁/k ^σ≤m(B(x,ϵ _k ))≤c ₂/k ^σ, so ϵ _k ≃k ^−σ/D where D is the dimension of X and c ₁,c ₂>0.

Let κ>1 and define A _k :={x:d(T ^j x,x)≤ϵ _k for some 1≤j≤log^κ k}. Evidently . By the estimate on for all large k, $\mu(A_{k}) \le c_{3} \epsilon_{k}^{\tau}$ where τ<δ. Let

$$F_k:=\bigl\{ x: \mu\bigl(B(x,\epsilon_k) \cap A_k\bigr) \ge1/k^{\gamma}\bigr\} $$

and define the Hardy-Littlewood maximal function M _k for $\phi(x)= 1_{A_{k}} (x)\rho(x)$ by

$$M_k(x):=\sup_{a>0}\frac{1}{m(B_a(x))}\int _{B_a(x)} 1_{A_k}(y)\rho(y)\,dm(y). $$

If x∈F _k then $M_{k}> c_{2}^{-1} k^{\sigma-\gamma}$.

A theorem of Hardy and Littlewood ([7] Theorem 3.17) states that

$$m\bigl( |M_k|>C\bigr)\le c_4 \frac{\|1_{A_k} \rho\|_1}{C} $$

for some constant c ₄, where ∥⋅∥₁ is the norm with respect to m. Hence

$$m(F_k)\le m\bigl(M_k > c_2^{-1} k^{\sigma-\gamma}\bigr) \le c_4 \mu(A_{k}) c_2 k^{\gamma-\sigma} \le k^{\gamma-\sigma(1+\tilde{\tau})} $$

where $0<\tilde{\tau} < \tau/D$. We need to alter τ/D to $\tilde {\tau}$ to take into account the fact that a ball of radius ϵ has measure roughly ϵ ^D.

Choosing σ large enough that $\sigma\tilde{\tau}>1$ and then taking $\sigma<\gamma< \sigma-1+\sigma\tilde{\tau}$ the series ∑ _k m(F _k ) converges.

So for m a.e. x ₀ there exists an N(x ₀) such that x ₀∉F _k for all k>N(x ₀). Hence for k>N(x ₀), μ(B(x,ϵ _k )∩A _k )≤1/k ^γ, thus μ(B(x,ϵ _k )∩A _k )≤m(B(x,ϵ _k ))^1+η for some η>0 (recall $m(B(x,\epsilon_{k}))\simeq\frac{1}{k^{\sigma}}$ and γ>σ).

Furthermore by the Lebesgue differentiation theorem for m a.e. x

$$\lim_{\epsilon\to0} \frac{1}{m(B_{\epsilon} (x))} \int _{B_{\epsilon} (x)} \rho(y) dm=\rho(x) $$

and for μ a.e. x, ρ(x)>0 as X is the support of μ. Hence for m a.e. x ₀ there exists an $\tilde{N}(x_{0})$ and $\tilde{\eta}>0$ such that for all $k>\tilde{N} (x_{0})$ we have $\mu(B(x,\epsilon_{k}) \cap A_{k}) \le\mu(B(x,\epsilon _{k}))^{1+\tilde{\eta}}$.

As κ was arbitrary by interpolating between the sequence ϵ _k we have that for μ a.e. x∈X there exists η′>0 , κ′>1 such that

$$\mu\bigl(B_i(x) \cap T^{-r} B_i(x) \bigr)\le \mu\bigl(B_i (x)\bigr)^{1+\eta'} $$

for 1≤r≤log^κ′ i. This is Assumption (C). □