A recurrence-type strong Borel–Cantelli lemma for Axiom A diffeomorphisms

Abstract

Let $(X,\mu ,T,d)$ be a metric measure-preserving dynamical system such that three-fold correlations decay exponentially for Lipschitz continuous observables. Given a sequence $(M_k)$ that converges to $0$ slowly enough, we obtain a strong dynamical Borel–Cantelli result for recurrence, that is, for $\mu $-almost every $x\in X$,

$$ \begin{align*} \lim_{n \to \infty}\frac{\sum_{k=1}^{n} \mathbf{1}_{B_k(x)}(T^{k}x)} {\sum_{k=1}^{n} \mu(B_k(x))} = 1, \end{align*} $$

where $\mu (B_k(x)) = M_k$. In particular, we show that this result holds for Axiom A diffeomorphisms and equilibrium states under certain assumptions.

1. Introduction

The Poincaré recurrence theorem is a fundamental result in dynamical systems. It states that if $(X, \mu , T)$ is a separable measure-preserving dynamical system with a Borel probability measure $\mu $ , then $\mu $ -almost every (a.e.) point $x \in X$ is recurrent, that is, there exists an increasing sequence $n_i$ such that $T^{n_i}x \to x$ . Equipping the system with a metric d such that $(X,d)$ is separable and d-open sets are measurable, we may restate the result as $\liminf _{n \to \infty } d(T^{n}x,x) = 0$ . A natural question to ask is if anything can be said about the rate of convergence. In [5], Boshernitzan proved that if the Hausdorff measure $\mathcal {H}_\alpha $ is $\sigma $ -finite on X for some $\alpha> 0$ , then for $\mu $ -a.e. $x \in X$ ,

$$ \begin{align*} \liminf_{n \to \infty} n^{1/\alpha} d(x,T^{n}x) < \infty. \end{align*} $$

Moreover, if $H_{\alpha }(X) = 0$ , then for $\mu $ -a.e. $x\in X$ ,

$$ \begin{align*} \liminf_{n \to \infty} n^{1/\alpha} d(x,T^{n}x) = 0. \end{align*} $$

We may state Boshernitzan’s result in a different way: for $\mu $ -a.e. $x\in X$ , there exists a constant $c> 0$ such that

$$ \begin{align*} \sum_{k=1}^{\infty} \mathbf{1}_{B(x,ck^{-1/\alpha})}(T^{k}x) = \infty, \end{align*} $$

where $B(x,r)$ denotes the open ball with centre x and radius r. Such a sum resembles those that appear in dynamical Borel–Cantelli lemmas for shrinking targets, $B_k = B(y_k,r_k)$ , where the centres $y_k$ do not depend on x. A dynamical Borel–Cantelli lemma is a zero-one law that gives conditions on the dynamical system and a sequence of sets $A_k$ such that $\sum _{k=1}^{\infty } \mathbf {1}_{A_k}(T^{k}x)$ converges or diverges for $\mu $ -a.e. x, depending on the convergence or divergence of $\sum _{k=1}^{\infty } \mu (A_k)$ . In some cases, it is possible to prove the stronger result, namely that if $\sum _{k=1}^{\infty } \mu (A_k) = \infty $ , then $\sum _{k=1}^{\infty } \mathbf {1}_{A_k}(T^{k}x) \sim \sum _{k=1}^{\infty } \mu (A_k)$ . Such results are called strong dynamical Borel–Cantelli lemmas.

Although the first dynamical Borel–Cantelli lemma for shrinking targets was proved in 1967 by Philipp [24], recurrence versions have only recently been studied. The added difficulty arises because of the dependence on x that the sets $B_{k}(x)$ have. Persson proved in [23] that one can obtain a recurrence version of a strong dynamical Borel–Cantelli lemma for a class of mixing dynamical systems on the unit interval. Other recent recurrence results include those of Hussain et al. [17], who obtained a zero-one law under the assumptions of mixing conformal maps and Ahlfors regular measures. Under similar assumptions, Kleinbock and Zheng [19] prove a zero-one law under Lipschitz twists, which combines recurrence and shrinking targets results. Both these papers put strong assumptions on the underlying measure of the dynamical system, assuming, for example, Ahlfors regularity. For other recent improvements of Boshernitzan’s result, see §6.1.

The main result of the paper, Theorem 2.2, is a strong dynamical Borel–Cantelli result for recurrence for dynamical systems satisfying some general conditions. The theorem extends the main theorem of Persson [23], which is stated in Theorem 2.1, concerning a class of mixing dynamical systems on the unit interval to a more general class, that includes, for example, some mixing systems on compact smooth manifolds. Since we consider shrinking balls $B_k(x)$ for which the radii converge to $0$ at various rates, Theorem 2.2 is an improvement of Boshernitzan’s result for the systems considered. For now, we state the following result, which is the main application of Theorem 2.2. Equip a smooth manifold with the induced metric.

Theorem 1.1. Consider a compact smooth N-dimensional manifold M. Suppose $f \colon M \to M$ is an Axiom A diffeomorphism that is topologically mixing on a basic set and $\mu $ is an equilibrium state corresponding to a Hölder continuous potential on the basic set. Suppose that there exist $c_1>0$ and $s> N - 1$ satisfying

$$ \begin{align*} \mu(B(x,r)) \leq c_1r^{s} \end{align*} $$

for all $x \in M$ and $r \geq 0$ . Assume further that $(M_n)$ is a sequence converging to $0$ satisfying

$$ \begin{align*} M_n \geq \frac{(\log n)^{4+\varepsilon}}{n} \end{align*} $$

for some $\varepsilon> 0$ and

$$ \begin{align*} \lim_{\alpha \to 1^{+}}\limsup_{n\to \infty} \frac{M_n}{M_{\lfloor \alpha n\rfloor}} = 1. \end{align*} $$

Define $B_n(x)$ to be the ball at x with $\mu (B_n(x)) = M_n$ . Then,

$$ \begin{align*} \lim_{n \to \infty}\frac{\sum^{n}_{k=1} \mathbf{1}_{B_k(x)}(f^{k}x)} {\sum^{n}_{k=1} \mu(B_k(x))} = 1 \end{align*} $$

for $\mu $ -a.e. $x \in M$ .

The above theorem applies to hyperbolic toral automorphisms with the Lebesgue measure as the equilibrium measure, which corresponds to the $0$ potential. It is clear that the Lebesgue measure satisfies the assumption. By perturbing the potential by a Hölder continuous function with small norm, we obtain a different equilibrium state for which the above assumption on the measure holds as well. Equilibrium states that are absolutely continuous with respect to the Lebesgue measure also satisfy the assumption.

For shrinking targets (non-recurrence), Chernov and Kleinbock [9, Theorem 2.4] obtain a strong Borel–Cantelli lemma when T is an Anosov diffeomorphism, $\mu $ is an equilibrium state given by a Hölder continuous potential, and the targets are eventually quasi-round rectangles.

1.1. Paper structure

In §2, we state the main result of the paper, namely Theorem 2.2, and give examples of systems satisfying the assumptions. Applications to return times and pointwise dimension are also given. In §3, we prove a series of lemmas establishing properties of the measure $\mu $ and functions $r_n \colon X \to [0,\infty )$ defined as the radius of the balls $B_n(x)$ . In §4, we obtain estimates for the measure and correlations of the sets $E_n = \{x\in X : T^{n}x \in B_n(x)\}$ . This is done in Propositions 4.1 and 4.2 using indicator functions and decay of correlations. Finally, in §5, we use the previous propositions together with Theorem 2.1 to prove Theorem 2.2. Theorem 1.1 then follows once we show that the assumptions of Theorem 2.2 are satisfied. We conclude with final remarks (§6).

2. Results

2.1. Setting and notation

We say that T preserves the measure $\mu $ if ${\mu (T^{-1}A) = \mu (A)}$ for all $\mu $ -measurable sets A. From now on, $(X,\mu ,T,d)$ will denote a metric measure-preserving system (m.m.p.s.) for which $(X,d)$ is compact, $\mu $ is a Borel probability measure and T is a measurable transformation. Given a sequence $(M_n)$ in $[0,1]$ and ignoring issues for now, define the open ball $B_n(x)$ around x by $\mu (B_n(x)) = M_n$ and define the functions $r_n \colon X \to [0,\infty )$ by $B(x,r_n(x)) = B_n(x)$ . For each $n \in \mathbb {N}$ , define

$$ \begin{align*} E_n = \{x \in X : T^{n}x \in B(x,r_n(x))\}. \end{align*} $$

For a set A, we denote the diameter, cardinality and closure of A by $\operatorname {\mathrm {diam}} A$ , $|A|$ and $\overline {A}$ , respectively. By the $\delta $ -neighbourhood of A, we mean the set ${A(\delta )\hspace{-0.9pt} =\hspace{-0.9pt} \{x\hspace{-0.9pt} \in\hspace{-0.9pt} X\hspace{-0.9pt} :\hspace{-0.9pt} \operatorname {\mathrm {dist}}(x,A)\hspace{-0.9pt} <\hspace{-0.9pt} \delta \}}$ . For $a, b \in \mathbb {R}$ , we write $a \lesssim b$ to mean that there exists a constant $c> 0$ depending only on $(X,\mu ,T,d)$ such that $a \leq cb$ . For $V\subset X$ and $\varepsilon> 0$ , let $P_{\varepsilon }(V)$ denote the packing number of V by balls B of radius $\varepsilon $ . This is the maximum number of pairwise disjoint balls of radius $\varepsilon $ with centres in V. Since X is compact, $P_{\varepsilon }(V)$ is finite for all $V\subset X$ and $\varepsilon> 0$ . For a compact smooth manifold M, let d denote the induced metric and let $\operatorname {\mathrm {Vol}}$ denote the induced volume measure. The injectivity radius of M, which is the largest radius for which the exponential map at every point is a diffeomorphism, is denoted by $\operatorname {\mathrm {inj}}_M$ . Since M is compact, $\operatorname {\mathrm {inj}}_M> 0$ (see for instance [8]).

Note that the open balls $B_n(x)$ and radii $r_n(x)$ may not exist for some $n \in \mathbb {N}$ and $x \in X$ . Sufficient conditions on $\mu $ , x and $(M_n)$ , for which $B_n(x)$ and $r_n(x)$ exist, are given in Lemma 3.1. Notice that for a zero-one law, one would prove that $\limsup _{n\to \infty }E_n$ has either zero or full measure. Furthermore, in the shrinking target case, when one considers fixed targets $B_n = B(y_n,r_n)$ , the corresponding sets are $\tilde {E}_n = \{ x \in X : T^{n}x \in B_n\} = T^{-n} B_n$ and we can use the invariance of the measure $\mu $ to conclude that $\mu (\tilde {E}_n) = M_n$ . However, in the recurrence case, when the targets depend on x, we instead settle for estimates on the measure and correlations of the sets $E_n$ . We do so in Proposition 4.3 using decay of multiple correlations as our main tool. We state the definition of decay of multiple correlations for Hölder continuous observables as it is the form most commonly found in the literature; however, for our purposes, we only require decay of correlations for Lipschitz continuous observables.

Definition 2.1. (r-Fold decay of correlations)

For an m.m.p.s $(X,\mu ,T,d)$ , $r \in \mathbb {N}$ and $\theta \in (0,1]$ , we say that r-fold correlations decay exponentially for $\theta $ -Hölder continuous observables if there exist constants $c_1>0$ and $\tau \in (0,1)$ such that for all $\theta $ -Hölder continuous functions $\varphi _k : X \to \mathbb {R}$ , $k=0,\dotsc ,r-1$ , and integers $0 = n_0 < n_1 < \cdots < n_{r-1}$ ,

$$ \begin{align*} \Bigl| \int \prod_{k=0}^{r-1} \varphi_k \circ T^{n_k} \mathop{}\!d\mu - \prod_{k=0}^{r-1} \int \varphi_k \mathop{}\!d\mu \Bigr| \leq c_1 e^{-\tau n} \prod_{k=0}^{r-1} \|\varphi_k\|_{\theta}, \end{align*} $$

where $n = \min \{n_{i+1} - n_{i}\}$ and

$$ \begin{align*} \|\varphi\|_{\theta} = \sup_{x\neq y} \frac{|\varphi(x) - \varphi(y)|}{d(x,y)^{\theta}} + \|\varphi\|_{\infty}. \end{align*} $$

It is well known that if M is a compact smooth manifold, f an Axiom A diffeomorphism that is topologically mixing when restricted to a basic set $\Omega $ , and $\varphi \colon \Omega \to \mathbb {R}$ Hölder continuous, then there exists a unique equilibrium state $\mu _\varphi $ for $\varphi $ such that two-fold correlations decay exponentially for $\theta $ -Hölder continuous observables for all $\theta \in (0,1]$ (see [6]). In the same setting, Kotani and Sunada essentially prove in [20, Proposition 3.1] that, in fact, r-fold correlations decay exponentially for $\theta $ -Hölder continuous observables for all $r \geq 1$ and $\theta \in (0,1]$ . A similar result is proven in [10, Theorem 7.41] for billiards with bounded horizon and no corners, and for dynamically Hölder continuous functions – a class of functions that includes Hölder continuous functions. This result can be extended to billiards with no corners and unbounded horizon, and billiards with some corners and bounded horizon. Dolgopyat [12] proves that multiple correlations decay exponentially for partially hyperbolic systems that are ‘strongly u-transitive with exponential rate’. Dolgopyat lists systems in §6 of his paper for which this property holds: some time one maps of Anosov flows, quasi-hyperbolic toral automorphisms, some translations on homogeneous spaces and mostly contracting diffeomorphisms on three-dimensional manifolds. A similar property referred to as ‘Property $(\mathcal {P}_t)$ ’ for real $t \geq 1$ is used by Pène in [22]. It is stated that when $t> 1$ , property $(\mathcal {P}_t)$ holds for dynamical systems to which one can apply Young’s method in [25].

We state some assumptions on the system $(X,\mu ,T,d)$ and the sequence $(M_n)$ that are used to obtain our result.

Assumption 1. There exists $\varepsilon> 0$ such that

$$ \begin{align*} M_n \geq \frac{(\log n)^{4 + \varepsilon}}{n} \end{align*} $$

and

$$ \begin{align*} \lim_{\alpha \to 1^{+}}\limsup_{n \to \infty} \frac{M_n}{M_{\lfloor \alpha n \rfloor}} = 1. \end{align*} $$

Assumption 2. There exists $s>0$ such that for all $x \in X$ and $r>0$ ,

$$ \begin{align*} \mu(B(x,r)) \lesssim r^{s}. \end{align*} $$

Assumption 3. There exists positive constants $\rho _0$ and $\alpha _0$ such that for all $x \in X$ and ${0 < \varepsilon < \rho \leq \rho _0}$ ,

$$ \begin{align*} \mu \{y : \rho \leq d(x,y) < \rho + \varepsilon\} \lesssim \varepsilon^{\alpha_0}. \end{align*} $$

Assumption 4. There exists positive constants K and $\varepsilon _0$ such that for any $\varepsilon \in (0,\varepsilon _0)$ , the packing number of X satisfies

$$ \begin{align*} P_{\varepsilon}(X) \lesssim \varepsilon^{-K}. \end{align*} $$

Assumption 1 is needed for technical reasons that are explicit in [23, Lemma 2]. Assumption 2 is a standard assumption. Assumption 3 is more restrictive; however, in some cases, it can be deduced from Assumption 2. Assumption 4 holds for very general spaces, e.g. all compact smooth manifolds. As we will see in the proof of Theorem 1.1, Assumptions 3 and 4 hold when X is a compact smooth N-dimensional manifold and $s> N - 1$ . Assumption 3 is also used in [15], stated as Assumption B, in which the authors outline spaces that satisfy the assumption. For instance, the assumption is satisfied by dispersing billiard systems, compact group extensions of Anosov systems, a class of Lozi maps and one-dimensional non-uniformly expanding interval maps with invariant probability measure $d\mu = h\mathop {}\!d\unicode{x3bb} $ , where $\unicode{x3bb} $ is the Lebesgue measure and $h \in L^{1+\delta }(\unicode{x3bb} )$ for some $\delta> 0$ .

Assumption 4 is used in the following construction of a partition. Let $\varepsilon < \varepsilon _0$ and let $\{B(x_k,\varepsilon )\}_{k=1}^{L}$ be a maximal $\varepsilon $ -packing of X, that is, $L = P_{\varepsilon }(X)$ . Then, $\{B(x_k,2\varepsilon )\}_{k=1}^{L}$ covers X. Indeed, for any $x \in X$ , we must have that $d(x,x_k) < 2\varepsilon $ for some k, for we could otherwise add $B(x,\varepsilon )$ to the packing, which would contradict our maximality assumption. Now, let $A_1 = B(x_1,2\varepsilon )$ and recursively define

(2.1) $$ \begin{align} A_k = B(x_k,2\varepsilon) \setminus \bigcup_{i=1}^{k-1} B(x_i,2\varepsilon) \end{align} $$

for $k = 2,\dotsc ,L$ . Then, $\{A_k\}_{k=1}^{L}$ partitions X and satisfies $\operatorname {\mathrm {diam}} A_k < 4\varepsilon $ . By Assumption 4, $L \lesssim \varepsilon ^{-K}$ .

As previously mentioned, Theorem 2.2 extends Persson’s result in [23] by relaxing the conditions on $(X,\mu ,T,d)$ . In [23], it is assumed that $X = [0,1]$ and that $T \colon [0,1] \to [0,1]$ preserves a measure $\mu $ for which (two-fold) correlations decay exponentially for $L_1$ against $BV$ functions. This is satisfied, for example, when T is a piecewise uniformly expanding map and $\mu $ is a Gibbs measure. That correlations decay exponentially for $L_1$ against $BV$ functions allows one to directly apply it to indicator functions. As established, when T is an Axiom A diffeomorphism defined on a manifold X, multiple correlations decay exponentially for Hölder continuous functions. Thus, we may not directly apply the result to indicator functions and must use an approximation argument, complicating the proof. That $\mu $ is non-atomic in [23] translates in Theorem 1.1 to $\mu $ satisfying Assumption 2 with $s> \dim M - 1$ . Furthermore, Assumption 2 for $\mu $ and Assumption 1 for $(M_n)$ remain unchanged. As it will be of use later, we restate the main result of [23] by combining Proposition 1 and the main theorem. We do so to have a more general statement suitable for our use.

Theorem 2.1. (Persson [23])

Let $(X, \mu , T, d)$ be an m.m.p.s., $(M_n)$ a sequence in $[0,1]$ satisfying Assumption 1 and $\mu $ a measure satisfying Assumption 2. Define $B_n(x)$ to be the ball around x such that $\mu (B_n(x)) = M_n$ and let $E_n = \{x \in X : T^{n}x \in B_n(x)\}$ . Suppose that there exists $C,\eta> 0$ such that for all $n,m \in \mathbb {N}$ ,

(2.2) $$ \begin{align} | \mu(E_n) - M_n | \leq C e^{-\eta n} \end{align} $$

and

(2.3) $$ \begin{align} \mu(E_{n+m} \cap E_{n}) \leq \mu(E_{n+m})\mu(E_n) + C(e^{-\eta n} + e^{-\eta m}). \end{align} $$

Then,

$$ \begin{align*} \lim_{n \to \infty}\frac{\sum^{n}_{k=1} \mathbf{1}_{B_k(x)}(T^{k}x)} {\sum^{n}_{k=1} \mu(B_k(x))} = 1 \end{align*} $$

for $\mu $ -a.e. $x \in X$ .

In this paper, we establish inequalities in equations (2.2) and (2.3) for systems satisfying multiple decorrelation for Lipschitz continuous observables and conclude a strong Borel–Cantelli lemma for recurrence.

2.2. Main result

The main result of the paper is the following theorem.

Theorem 2.2. Let $(X,\mu ,T,d)$ be an m.m.p.s. for which three-fold correlations decay exponentially for Lipschitz continuous observables and such that Assumptions 2–4 hold. Suppose further that $(M_n)$ is a sequence in $[0,1]$ converging to $0$ and satisfying Assumption 1. Then,

$$ \begin{align*} \lim_{n \to \infty}\frac{\sum^{n}_{k=1} \mathbf{1}_{B_k(x)}(T^{k}x)} {\sum^{n}_{k=1} \mu(B_k(x))} = 1 \end{align*} $$

for $\mu $ -a.e. $x \in X$ .

Remark 2.3. Although we consider open balls $B_k(x)$ for our results, we only require control over the measure of $B_k(x)$ and its $\delta $ -neighbourhoods. Thus, it is possible to substitute the open balls for closed balls or other neighbourhoods of x, as long as one retains similar control of the sets as stated in Assumptions 2 and 3. Furthermore, it is possible to relax the condition that $(M_n)$ converges to $0$ as long as one can ensure that $r_n \leq \rho _0$ on $\operatorname {\mathrm {supp}}\mu $ for large n.

After proving that Assumptions 3 and 4 hold when X is a compact smooth manifold and $\mu $ satisfies Assumption 2 for $s> N - 1$ , we apply the above result to Axiom A diffeomorphisms and obtain Theorem 1.1. In §6.2, we list some systems that satisfy Assumptions 2–4, but it remains to show that some of these systems satisfy three-fold decay of correlations.

2.3. Return times corollary

Define the hitting time of $x\in X$ into a set B by

$$ \begin{align*} \tau_{B}(x) = \inf \{k> 0 : T^{k}x \in B\}. \end{align*} $$

When $x \in B$ , we call $\tau _{B}(x)$ the return time of x into B. In 2007, Galatolo and Kim [13] showed that if a system satisfies a strong Borel–Cantelli lemma for shrinking targets with any centre, then for every $y\in X$ , the hitting times satisfy

$$ \begin{align*} \lim_{r \to 0} \frac{\log \tau_{\overline{B}(y,r)}(x)} {-\log \mu(\overline{B}(y,r))} = 1 \end{align*} $$

for $\mu $ -a.e. x. In our case, the proof translates directly to return times (see the corollary in [23] for a simplified proof).

Corollary 2.4. Under the assumptions of Theorem 2.2,

$$ \begin{align*} \lim_{r \to 0} \frac{\log \tau_{B(x,r)}(x)}{-\log \mu(B(x,r))} = 1 \end{align*} $$

for $\mu $ -a.e. x.

The above corollary has an application to the pointwise dimension of $\mu $ . For each $x \in X$ , define the lower and upper pointwise dimensions of $\mu $ at x by

$$ \begin{align*} \underline{d}_{\mu}(x) = \liminf_{r \to 0} \frac{\log \mu(B(x,r))}{\log r} \quad \text{and} \quad \overline{d}_{\mu}(x) = \limsup_{r \to 0} \frac{\log \mu(B(x,r))}{\log r}, \end{align*} $$

and the lower and upper recurrence rates of x by

$$ \begin{align*} \underline{R}(x) = \liminf_{r \to 0}\frac{\log\tau_{B(x,r)}(x)}{-\log r} \quad \text{and} \quad \overline{R}(x) = \limsup_{r \to 0}\frac{\log\tau_{B(x,r)}(x)}{-\log r}. \end{align*} $$

When T is a Borel measurable transformation that preserves a Borel probability measure $\mu $ and $X \subset \mathbb {R}^{N}$ is any measurable set, Barreira and Saussol [4, Theorem 1] proved that

$$ \begin{align*} \underline{R}(x) \leq \underline{d}_{\mu}(x) \quad \text{and} \quad \overline{R}(x) \leq \overline{d}_{\mu}(x) \end{align*} $$

for $\mu $ -a.e. $x\in X$ . Using Corollary 2.4, we obtain that for $\mu $ -a.e. $x\in X$ ,

(2.4) $$ \begin{align} \underline{R}(x) = \underline{d}_{\mu}(x) \quad \text{and} \quad \overline{R}(x) = \overline{d}_{\mu}(x), \end{align} $$

whenever the system satisfies a strong Borel-Cantelli lemma for recurrence. Contrast the above result with [4, Theorem 4], which states that equation (2.4) holds when $\mu $ has long return times, that is, for $\mu $ -a.e. $x\in X$ and sufficiently small $\varepsilon> 0$ ,

$$ \begin{align*} \liminf_{r \to 0} \frac{\log (\mu \{y\in B(x,r) : \tau_{B(x,r)}(y) \leq \mu(B(x,r))^{-1+\varepsilon}\})} {\log \mu(B(x,r))}> 1. \end{align*} $$

Furthermore, if X is a compact smooth manifold, then [4, Theorem 5] states that if the equilibrium measure $\mu $ is ergodic and supported on a locally maximum hyperbolic set of a $C^{1+\alpha }$ diffeomorphism, then $\mu $ has long return times and $\underline {R}(x) = \overline {R}(x) = \dim _{H}\mu $ for $\mu $ -a.e. x. However, Barreira, Pesin and Schmeling [3] prove that if f is a $C^{1+\alpha }$ diffeomorphism on a smooth compact manifold and $\mu $ a hyperbolic f-invariant compactly supported ergodic Borel probability measure, then $\underline {d}_{\mu }(x) = \overline {d}_\mu (x) = \dim _H\mu $ for $\mu $ -a.e. x.

3. Properties of $\mu $ and $r_n$

As previously mentioned, the functions $r_n \colon X \to [0,\infty )$ may not be well defined since it may be the case that the measure $\mu $ assigns positive measure to the boundary of an open ball. Lemma 3.1 gives sufficient conditions on x, $(M_n)$ and $\mu $ for $r_n(x)$ to be well defined. Lemma 3.2 gives conditions for which $r_n(x)$ converges uniformly to $0$ , which allows us to use Assumption 3. Lemma 3.3 shows that we can use Assumption 3 to control the $\delta $ -neighbourhood of the sets $A_k$ in equation (2.1). Recall that $(X,d)$ is a compact space and $(M_n)$ is a sequence in $[0,1]$ .

Lemma 3.1. If $\mu $ has no atoms and satisfies $\mu (B) = \mu (\overline {B})$ for all open balls $B \subset X$ , then for each $n \in \mathbb {N}$ and $x \in X$ , there exists $r_n(x) \in [0,\infty )$ such that

(3.1) $$ \begin{align} \mu(B(x,r_n(x))) = M_n. \end{align} $$

Moreover, for each $n \in \mathbb {N}$ , the function $r_n \colon X \to [0,\infty )$ is Lipschitz continuous with

$$ \begin{align*} | r_n(x) - r_n(y) | \leq d(x,y) \end{align*} $$

for all $x,y \in X$ .

Proof. Let $n \in \mathbb {N}$ and $x \in X$ . The infimum

$$ \begin{align*} r_n(x) = \inf \{r \geq 0 : \mu(B(x,r)) \geq M_n\} \end{align*} $$

exists since the set contains $\operatorname {\mathrm {diam}} X \in (0,\infty )$ and has a lower bound. If $r_n(x) = 0$ , then, since $\mu $ is non-atomic, it must be that $M_n = 0$ , in which case $\mu (B(x,r_n(x))) = M_n$ . If $r_n(x)> 0$ , then for all $k> 0$ , we have that $\mu (B(x,r_n(x) + {1}/{k})) \geq M_n$ and thus $\mu (\overline {B(x,r_n(x))}) \geq M_n$ . Similarly, $\mu (B(x,r_n(x))) \leq M_n$ . Hence, by using our assumption, we obtain that

$$ \begin{align*} M_n \leq \mu(\overline{B(x,r_n(x))}) = \mu(B(x,r_n(x))) \leq M_n, \end{align*} $$

which proves equation (3.1).

To show that $r_n$ is Lipschitz continuous, let $x, y \in X$ and let $\delta = d(x,y)$ . Then, $B(y,r_n(y)) \subset B(x,r_n(y) + \delta )$ and

$$ \begin{align*} M_n = \mu(B(y,r_n(y))) \leq \mu(B(x,r_n(y) + \delta)). \end{align*} $$

Hence, $r_n(x) \leq r_n(y) + \delta $ . We conclude that $|r_n(x) - r_n(y)| \leq \delta $ by symmetry.

The assumption $\mu (B) = \mu (\overline {B})$ for sufficiently small radii follows from Assumption 3.

Lemma 3.2. If $\lim _{n \to \infty }M_n = 0$ , then $\lim _{n \to \infty }r_n = 0$ uniformly on $\operatorname {\mathrm {supp}} \mu $ .

Proof. We first prove pointwise convergence on $\operatorname {\mathrm {supp}} \mu $ . Let $x\in X$ and suppose that $r_n(x)$ does not converge to $0$ as $n \to \infty $ for some $x \in X$ . Then there exists $\varepsilon> 0$ and an increasing sequence $(n_i)$ such that $r_{n_i}(x)> \varepsilon $ and

$$ \begin{align*} \mu( B(x,\varepsilon) ) \leq \mu( B(x,r_{n_i}(x)) ) = M_{n_i} \end{align*} $$

for all i. Taking the limit as $i \to \infty $ gives us $\mu ( B(x,\varepsilon ) ) = 0$ . Hence, $x \notin \operatorname {\mathrm {supp}} \mu $ .

Now suppose that $r_n$ does not converge to $0$ uniformly on $\operatorname {\mathrm {supp}} \mu $ . Then there exist $\varepsilon> 0$ , an increasing sequence $(n_i)$ and points $x_i \in \operatorname {\mathrm {supp}} \mu $ such that $r_{n_i}(x_i)> \varepsilon $ for all $i> 0$ . Since $\operatorname {\mathrm {supp}} \mu $ is compact, we may assume that $x_i$ converges to some $x \in \operatorname {\mathrm {supp}} \mu $ . By Lipschitz continuity,

$$ \begin{align*} r_{n_i}(x) \geq r_{n_i}(x_i) - d(x,x_i)> \varepsilon - d(x,x_i). \end{align*} $$

Thus,

$$ \begin{align*} \lim_{i \to \infty} r_{n_i}(x) \geq \varepsilon, \end{align*} $$

which contradicts the fact that we have pointwise convergence to $0$ on $\operatorname {\mathrm {supp}}\mu $ .

Lemma 3.3. Suppose that $\mu $ satisfies Assumption 3 and that $(X,d)$ satisfies Assumption 4. Let $\rho < \min \{\varepsilon _0, \rho _0\}$ and consider the partition $\{A_k\}_{k=1}^{L}$ given in equation (2.1) where $\varepsilon = {\rho }/{2}$ . Then for each k and $\delta < \rho $ ,

$$ \begin{align*} \mu(A_k(\delta) \setminus A_k) \lesssim \rho^{-K} \delta^{\alpha_0}. \end{align*} $$

Proof. Let $0 < \delta < \rho \leq \rho _0$ and fix k. It suffices to prove that $A_k(\delta ) \setminus A_k$ is contained in

$$ \begin{align*} (B(x_k,\rho+\delta) \setminus B(x_k,\rho)) \cup \bigcup_{i=1}^{k-1} (B(x_i,\rho) \setminus B(x_i,\rho - \delta)). \end{align*} $$

Indeed, since $\delta < \rho \leq \rho _0$ , it then follows by Assumption 3 that

$$ \begin{align*} \mu(A_k(\delta) \setminus A_k) \lesssim k\delta^{\alpha_0} \leq L\delta^{\alpha_0}. \end{align*} $$

Since $\rho < \varepsilon _0$ , we have that $L \lesssim \rho ^{-K}$ by Assumption 4 and we conclude. If $A_k(\delta ) \setminus A_k$ is empty, then there is nothing to prove. So suppose that $A_k(\delta ) \setminus A_k$ is non-empty and contains a point x. Suppose $x \notin B(x_k,\rho + \delta ) \setminus B(x_k,\rho )$ . Since $x \in A_k(\delta ) \subset B(x_k,\rho +\delta )$ , we must have $x\in B(x_k,\rho )$ . Furthermore, since $x\notin A_k$ , by the construction of $A_k$ , it must be that $x\in B(x_i,\rho )$ for some $i < k$ . If $x \in B(x_i,\rho -\delta )$ , then, since $x \in A_k(\delta )$ , there exists $y\in A_k$ such that $d(x,y) < \delta $ . Hence, $y \in B(x_i,\rho )$ and $y\notin A_k$ , which is a contradiction. Thus, $x\in B(x_i,\rho ) \setminus B(x_i, \rho - \delta )$ , which concludes the proof.

4. Measure and correlations of $E_n$

We give measure and correlations estimates of $E_n$ in Proposition 4.3, which will follow from Propositions 4.1 and 4.2. The proofs of those propositions are modifications of the proofs of [18, Lemmas 3.1 and 3.2]. The modifications account for the more difficult setting; that is, working with a general metric space $(X,d)$ , and for assuming decay of correlation for a smaller class of observables, that is, for Lipschitz continuous functions rather than $L_1$ and $BV$ functions.

Throughout the section, $(M_n)$ denotes a sequence in $[0,1]$ that converges to $0$ and satisfies Assumption 1. We also assume that the space $(X,\mu ,T,d)$ satisfies Assumptions 2–4. Hence, the functions $r_n$ defined by $\mu (B(x,r_n(x))) = M_n$ are well defined for all n by Lemma 3.1 and converge uniformly to $0$ by Lemma 3.2. Thus, for large enough n, the functions $r_n$ are uniformly bounded by $\rho _0$ on $\operatorname {\mathrm {supp}}\mu $ . For notational convenience, C will denote an arbitrary positive constant depending solely on $(X,\mu ,T,d)$ and may vary from equation to equation.

In the following propositions, it will be useful to partition the space into sets whose diameters are controlled. Since we wish to work with Lipschitz continuous functions to apply decay of correlations, we also approximate the indicator functions on the elements of the partition. Let $\rho < \min \{\varepsilon _0, \rho _0\}$ and recall the construction of the partition $\{A_k\}_{k=1}^{L}$ of X in equation (2.1), where $\varepsilon = {\rho }/{2}$ . For $\delta \in (0,\rho )$ , define the functions $\{h_k \colon X \to [0,1]\}_{k=1}^{L}$ by

(4.1) $$ \begin{align} h_k(x) = \min \{1, \delta^{-1} \operatorname{\mathrm{dist}}(x, X \setminus A_{k}(\delta))\}. \end{align} $$

Notice that the functions are Lipschitz continuous with Lipschitz constant $\delta ^{-1}$ .

Proposition 4.1. With the above assumptions on $(X,\mu ,T,d)$ and $(M_n)$ , suppose additionally that two-fold correlations decay exponentially for Lipschitz continuous observables. Let $n\in \mathbb {N}$ and consider the function $F \colon X \times X \to [0,1]$ given by

$$ \begin{align*} F(x,y) = \begin{cases} 1 & \text{if}\ x \in B(y, r_n(y)), \\ 0 & \text{otherwise}. \end{cases} \end{align*} $$

Then there exists constants $C,\eta> 0$ independent of n such that

(4.2) $$ \begin{align} \Bigl| \int F(T^{n}x, x) \mathop{}\!d\mu(x) - M_n \Bigr| \leq C e^{-\eta n}. \end{align} $$

Proof. We prove the upper bound part of equation (4.2) as the lower bound follows similarly.

Consider $Y = \{(x,y) : F(x,y) = 1\}$ . To establish an upper bound, define $\hat {F} \colon X \times X \to [0, 1]$ as follows. Let $\varepsilon _n \in (0,\rho _0]$ decay exponentially as $n\to \infty $ and define

$$ \begin{align*} \hat{F}(x,y) = \min \bigl\{1, \varepsilon_n^{-1} \operatorname{\mathrm{dist}}\bigl((x,y), (X\times X) \setminus Y(\varepsilon_n)\bigr) \bigr\}. \end{align*} $$

Note that it suffices to prove the result for sufficiently large n. We also assume that $\varepsilon _0,\rho _0 < 1$ , as we may replace them by smaller values otherwise. Therefore, by Lemma 3.2, we may assume that $r_n(x) \leq \rho _0$ for all $x\in \operatorname {\mathrm {supp}}\mu $ . If $M_n = 0$ , then $r_n = 0$ on $\operatorname {\mathrm {supp}}\mu $ and we find that we do not have any further restrictions on $\varepsilon _n$ . If $M_n> 0$ , then by Assumption 2,

$$ \begin{align*} r_n(x) \gtrsim M_n^{1/s}> 0. \end{align*} $$

Thus, by using Assumption 1 and the exponential decay of $\varepsilon _n$ , we may assume that $3\varepsilon _n < r_n(x)$ for all $x\in X$ . We continue with assuming that $M_n> 0$ since the arguments are significantly easier when $M_n = 0$ .

Now, $\hat {F}$ is Lipschitz continuous with Lipschitz constant $\varepsilon _n^{-1}$ . Furthermore,

(4.3) $$ \begin{align} F(x,y) \leq \hat{F}(x,y) \leq \mathbf{1}_{B(y,r_n(y) + 3\varepsilon_n)}(x). \end{align} $$

This is because if $\hat {F}(x,y)> 0$ , then $d((x,y),(x',y')) < \varepsilon _n$ for some $(x',y') \in Y$ , and so

$$ \begin{align*} d(x,y) &\leq d(x',y') + d(x,x') + d(y',y) \\ &< r_n(y') + 2\varepsilon_n \\ &\leq r_n(y) + 3\varepsilon_n, \end{align*} $$

where the last inequality follows from the Lipschitz continuity of $r_n$ . Hence,

$$ \begin{align*} \int F(T^{n}x,x) \mathop{}\!d\mu(x) \leq \int \hat{F}(T^{n}x,x) \mathop{}\!d\mu(x). \end{align*} $$

Using a partition of X, we construct an approximation H of $\hat {F}$ of the form

$$ \begin{align*} \sum \hat{F}(x,y_{k}) h_{k}(y), \end{align*} $$

where $h_{k}$ are Lipschitz continuous, for the purpose of leveraging two-fold decay of correlations. Let $\rho _n < \min \{\varepsilon _0, \rho _0\}$ and consider the partition $\{A_k\}_{k=1}^{L_n}$ of X given in equation (2.1) for $\varepsilon = \rho _n$ . Let $\delta _n < \rho _n$ and obtain the functions $h_k$ as in equation (4.1) for $\delta = \delta _n$ . Let

$$ \begin{align*} I = \{k \in [1,L_n] : A_k \cap \operatorname{\mathrm{supp}}\mu \neq \emptyset\} \end{align*} $$

and $y_k \in A_k \cap \operatorname {\mathrm {supp}}\mu $ for $k \in I$ . Note that $|I|\leq L_n \lesssim \rho _n^{-K}$ by Assumption 4, since $\rho _n < \varepsilon _0$ . Define $H \colon X \times X \to \mathbb {R}$ by

$$ \begin{align*} H(x,y) = \sum_{k\in I} \hat{F}(x,y_k)h_k(y). \end{align*} $$

For future reference, note that

$$ \begin{align*} \sum_{k \in I} h_k \leq \sum_{k \in I} (\mathbf{1}_{A_k} + \mathbf{1}_{A_k(\delta_n) \setminus A_k} ) \leq 1 + \sum_{k \in I} \mathbf{1}_{A_k(\delta_n) \setminus A_k}, \end{align*} $$

which implies that

$$ \begin{align*} \int \sum_{k \in I} h_k(x) \mathop{}\!d\mu(x) \leq 1 + \sum_{k \in I} \mu(A_k(\delta_n) \setminus A_k). \end{align*} $$

Using Lemma 3.3, we obtain that

(4.4) $$ \begin{align} \int \sum_{k \in I} h_k(x) \mathop{}\!d\mu(x) \leq 1 + C \rho_n^{-2K}\delta_n^{\alpha_0}. \end{align} $$

Now,

(4.5) $$ \begin{align} \int \hat{F}(T^{n}x,x) \mathop{}\!d\mu(x) \leq &\int H(T^{n}x,x) \mathop{}\!d\mu(x) \nonumber\\ &+ \int |\hat{F}(T^{n}x,x) - H(T^{n}x,x)| \mathop{}\!d\mu(x). \end{align} $$

We use decay of correlations to bound the first integral from above:

$$ \begin{align*} \int \hat{F}(T^{n}x,y_k)h_k(x) \mathop{}\!d\mu(x) \leq &\int \hat{F}(x,y_k)\mathop{}\!d\mu(x) \int h_k(x) \mathop{}\!d\mu(x) \\ &+ C\|\hat{F}\|_{\text{Lip}}\|h_k\|_{\text{Lip}}e^{-\tau n}. \end{align*} $$

Notice that by equation (4.3),

$$ \begin{align*} \int \hat{F}(x,y_k)\mathop{}\!d\mu(x) \leq \mu( B(y_k, r_n(y_k) + 3\varepsilon_n) ). \end{align*} $$

Since $y_k \in \operatorname {\mathrm {supp}} \mu $ , we have that $3\varepsilon _n < r_n(y_k) \leq \rho _0$ . Hence, by Assumption 3,

(4.6) $$ \begin{align} \mu( B(y_k, r_n(y_k) + 3\varepsilon_n) ) \leq M_n + C\varepsilon_n^{\alpha_0}. \end{align} $$

Furthermore, $\|\hat {F}\|_{\text {Lip}} \lesssim \varepsilon _n^{-1}$ and $\|h_k\|_{\text {Lip}} \lesssim \delta _n^{-1}$ . Hence,

$$ \begin{align*} \int \hat{F}(T^{n}x,y_k)h_k(x) \mathop{}\!d\mu(x) \leq (M_n + C\varepsilon_n^{\alpha_0}) \int h_k(x) \mathop{}\!d\mu(x) + C\varepsilon_n^{-1}\delta_n^{-1}e^{-\tau n} \end{align*} $$

and

$$ \begin{align*} \int H(T^{n}x,x) \mathop{}\!d\mu(x) \leq (M_n + C\varepsilon_n^{\alpha_0})\sum_{k \in I} \int h_k(x)\mathop{}\!d\mu(x) + C \sum_{k\in I} \varepsilon_n^{-1} \delta_n^{-1}e^{-\tau n}. \end{align*} $$

In combination with equation (4.4) and $|I| \lesssim \rho _n^{-K}$ , the former equation gives

(4.7) $$ \begin{align} \int H(T^{n}x,x) \mathop{}\!d\mu(x) \leq (M_n + C\varepsilon_n^{\alpha_0}) (1 + C\rho_n^{-2K}\delta_n^{\alpha_0}) + C \varepsilon_n^{-1} \delta_n^{-1}\rho_n^{-K}e^{-\tau n}. \end{align} $$

We now establish a bound on

$$ \begin{align*} \int | \hat{F}(T^{n}x,x) - H(T^{n}x,x) |\mathop{}\!d\mu(x) \end{align*} $$

in equation (4.5). By the triangle inequality,

$$ \begin{align*} |\hat{F}(T^{n}x,x) \mathbf{1}_{A_k}(x) - \hat{F}(T^{n}x,y_k) h_k(x)| \end{align*} $$

is bounded above by

$$ \begin{align*} | \hat{F}(T^{n}x,x) - \hat{F}(T^{n}x,y_k) | \mathbf{1}_{A_k}(x) + | \mathbf{1}_{A_k}(x) - h_k(x) | \hat{F}(T^{n}x,y_k). \end{align*} $$

Now, if $x \in A_k$ , then by Lipschitz continuity,

$$ \begin{align*} | \hat{F}(T^{n}x,x) - \hat{F}(T^{n}x,y_k) | \leq \varepsilon_n^{-1} d( (T^{n}x, x), (T^{n}x, y_k) ) \lesssim \rho_n \varepsilon_n^{-1}. \end{align*} $$

Also, using the fact that $\hat {F} \leq 1$ and $\mathbf {1}_{A_k} \leq h_k \leq \mathbf {1}_{A_k(\delta _n)}$ , we obtain

$$ \begin{align*} | \hat{F}(T^{n}x,x) \mathbf{1}_{A_k}(x) - \hat{F}(T^{n}x,y_k) h_k(x) | \leq C\rho_n \varepsilon_n^{-1} \mathbf{1}_{A_k}(x) + \mathbf{1}_{A_k(\delta_n) \setminus A_k}(x). \end{align*} $$

Hence, since $\hat {F}(x,y) = \sum _{k \in I} \hat {F}(x,y)\mathbf {1}_{A_k}(y)$ for $y \in \operatorname {\mathrm {supp}}\mu $ , we obtain that for $x \in \operatorname {\mathrm {supp}}\mu $ ,

$$ \begin{align*} | \hat{F}(T^{n}x,x) - H(T^{n}x,x) | \end{align*} $$

is bounded above by

$$ \begin{align*} &\sum_{k \in I} | \hat{F}(T^{n}x,x) \mathbf{1}_{A_k}(x) - \hat{F}(T^{n}x,y_k) h_k(x) | \\ &\qquad \leq \sum_{k \in I} ( C \varepsilon_n^{-1} \rho_n \mathbf{1}_{A_k}(x) + \mathbf{1}_{A_k(\delta_n) \setminus A_k}(x) ). \end{align*} $$

Now, $\sum _{k \in I} \mathbf {1}_{A_k} = 1$ on $\operatorname {\mathrm {supp}}\mu $ gives

(4.8) $$ \begin{align} |\hat{F}(T^{n}x,x) - H(T^{n}x,x)| \leq C \varepsilon_n^{-1} \rho_n + \sum_{k \in I} \mathbf{1}_{A_k(\delta_n) \setminus A_k}(x) \end{align} $$

for $x\in \operatorname {\mathrm {supp}}\mu $ and thus

$$ \begin{align*} \int | \hat{F}(T^{n}x,x) - H(T^{n}x,x) | \mathop{}\!d\mu(x) \leq C \varepsilon_n^{-1} \rho_n + \sum_{k \in I} \mu(A_k(\delta_n) \setminus A_k). \end{align*} $$

By Lemma 3.3,

(4.9) $$ \begin{align} \int | \hat{F}(T^{n}x,x) - H(T^{n}x,x) | \mathop{}\!d\mu(x) \lesssim \varepsilon_n^{-1} \rho_n + \rho_n^{-2K} \delta_n^{\alpha_0}. \end{align} $$

By combining equations (4.5), (4.7) and (4.9), we obtain that

$$ \begin{align*} \int \hat{F}(T^{n}x,x)\mathop{}\!d\mu(x) - M_n \lesssim \varepsilon_n^{\alpha_0} + \rho_n^{-2K}\delta_n^{\alpha_0} + \varepsilon_n^{-1}\rho_n + \varepsilon_n^{-1}\delta_n^{-1}\rho_n^{-2K}e^{-\tau n}. \end{align*} $$

Now, let

$$ \begin{align*} \varepsilon_n &= e^{-\gamma n}, \\ \rho_n &= \varepsilon_n^2, \\ \delta_n &= \varepsilon_n^{2 + 4K/\alpha_0}, \end{align*} $$

where $\gamma = \tfrac 12\bigl (3 + 4K + {4K}/{\alpha _0}\bigr )^{-1} \tau $ . Then, $\varepsilon _n$ decays exponentially as $n \to \infty $ and $\delta _n < \rho _n$ as required. For large enough n, we have $3\varepsilon _n < r_n(x)$ and $\rho _n < \min \{\varepsilon _0, \rho _0\}$ as we assumed. Finally,

$$ \begin{align*} \eta = \min \Bigl\{\frac{\tau}{2}, \gamma, \alpha_0\gamma\Bigr\} \end{align*} $$

gives us an upper bound

$$ \begin{align*} \int F(T^{n}x,x)\mathop{}\!d\mu(x) - M_n \lesssim e^{-\eta n}.\\[-34.5pt] \end{align*} $$

Proposition 4.2. With the above assumptions on $(X,\mu ,T,d)$ and $(M_n)$ , suppose additionally that three-fold correlations decay exponentially for Lipschitz continuous observables. Let $m,n \in \mathbb {N}$ and consider the function $F\colon X\times X\times X \to [0,1]$ given by

$$ \begin{align*} F(x,y,z) = \begin{cases} 1 & \text{if}\ x\in B(z, r_{n+m}(z))\ \text{and}\ y\in B(z, r_n(z)), \\ 0 & \text{otherwise}. \end{cases} \end{align*} $$

Then, there exists $C,\eta> 0$ independent of n and m such that

$$ \begin{align*} \int F( T^{n + m}x, T^{n}x, x) \mathop{}\!d\mu(x) \leq M_n M_{n+m} + C( e^{-\eta n} + e^{- \eta m} ). \end{align*} $$

The proof of Proposition 4.2 follows roughly that of Proposition 4.1 by making some modifications to account for the added dimension.

Proof. Again, we assume, without loss of generality, that n is sufficiently large and that $\varepsilon _0,\rho _0 < 1$ and $M_n> 0$ .

Notice that $F(x,y,z) = G_1(x,z) G_2(y,z)$ for

$$ \begin{align*} G_1(x,z) = \begin{cases} 1 & \text{if}\ x \in B(z, r_{n+m}(z)), \\ 0 & \text{otherwise}, \end{cases} \end{align*} $$

and

$$ \begin{align*} G_2(y,z) = \begin{cases} 1 & \text{if}\ y \in B(z, r_{n}(z)), \\ 0 & \text{otherwise}. \end{cases} \end{align*} $$

Let $\varepsilon _{m,n} < \rho _0$ converge exponentially to $0$ as $\min \{m,n\} \to \infty $ . Define the functions $\hat {G}_1, \hat {G}_2 \colon X \times X \to [0,1]$ similarly to how $\hat {F}$ was defined in Proposition 4.1. That is, $\hat {G}_1$ and $\hat {G}_2$ are Lipschitz continuous with Lipschitz constant $\varepsilon _{m,n}^{-1}$ and satisfy

$$ \begin{align*} G_1(x,y) \leq \hat{G}_1(x,y) \leq \mathbf{1}_{B(y, r_{n+m}(y) + 3\varepsilon_{m,n})}(x) \end{align*} $$

and

$$ \begin{align*} G_2(x,y) \leq \hat{G}_2(x,y) \leq \mathbf{1}_{B(y, r_{n}(y) + 3\varepsilon_{m,n})}(x). \end{align*} $$

Thus,

(4.10) $$ \begin{align} \int F( T^{n + m}x, T^{n}x, x) \mathop{}\!d\mu(x) \leq \int \hat{G}_1(T^{n + m}x,x) \hat{G}_2(T^{n}x, x) \mathop{}\!d\mu(x). \end{align} $$

Let $\rho _{m,n} < \min \{\varepsilon _0, \rho _0\}$ and consider the partition $\{A_k\}_{k=1}^{L_{m,n}}$ of X given in equation (2.1) for $\varepsilon = {\rho _{m,n}}/{2}$ . Let $\delta _{m,n} < \rho _{m,n}$ and obtain the functions $h_k$ as in equation (4.1) for $\delta = \delta _{m,n}$ . Let

$$ \begin{align*} I = \{k \in [1,L_{m,n}] : A_k \cap \operatorname{\mathrm{supp}}\mu \neq \emptyset\} \end{align*} $$

and $y_k \in A_k \cap \operatorname {\mathrm {supp}}\mu $ for $k \in I$ . Define the functions $H_1, H_2 \colon X \times X \to \mathbb {R}$ by

$$ \begin{align*} H_i(x,y) = \sum_{k \in I} \hat{G}_i(x, y_k) h_k(x) \end{align*} $$

for $i = 1,2$ . Using the triangle inequality, we bound the right-hand side of equation (4.10) by

(4.11) $$ \begin{align} &\int |\hat{G}_1(T^{n+m}x,x) \hat{G}_2(T^{n}x,x) - H_1(T^{n+m}x,x) H_2(T^{n}x,x)| \mathop{}\!d\mu(x) \nonumber\\ &\qquad + \int H_1(T^{n+m}x,x) H_2(T^{n}x,x) \mathop{}\!d\mu(x). \end{align} $$

We look to bound the first integral. Using the triangle inequality and the fact that $\hat {G}_1 \leq 1$ , we obtain that the integrand is bounded above by

$$ \begin{align*} | \hat{G}_2(T^{n}x,x) - H_2(T^{n}x,x) | + H_2(T^{n}x,x) | \hat{G}_1(T^{n+m}x,x) - H_1(T^{n+m}x,x) |. \end{align*} $$

As shown in equation (4.9),

$$ \begin{align*} \int | \hat{G}_2(T^{n}x,x) - H_2(T^{n}x,x) | \mathop{}\!d\mu(x) \lesssim \varepsilon_{m,n}^{-1} \rho_{m,n} + \rho_{m,n}^{-2K} \delta_{m,n}^{\alpha_0} \end{align*} $$

and by equation (4.8),

$$ \begin{align*} | \hat{G}_1(T^{n+m}x,x) - H_1(T^{n+m}x,x) | \leq C \varepsilon_{m,n}^{-1} \rho_{m,n} + \sum_{k \in I} \mathbf{1}_{A_k(\delta_{m,n}) \setminus A_k}(x) \end{align*} $$

and

$$ \begin{align*} H_2(T^{n}x,x) \leq 1 + \sum_{k \in I} \mathbf{1}_{A_k(\delta_{m,n}) \setminus A_k}(x). \end{align*} $$

Thus,

$$ \begin{align*} \int H_2(T^{n}x,x) | \hat{G}_1(T^{n+m}x,x) - H_1(T^{n+m}x,x) | \mathop{}\!d\mu(x) \end{align*} $$

is bounded above by

$$ \begin{align*} &C \varepsilon_{m,n}^{-1} \rho_{m,n} + C(1 + \varepsilon_{m,n}^{-1} \rho_{m,n}) \sum_{k \in I} \mathbf{1}_{A_k(\delta_{m,n}) \setminus A_k}(x) \\ &\qquad + \sum_{k \in I} \sum_{k' \in I} \mathbf{1}_{A_k(\delta_{m,n}) \setminus A_k}(x) \mathbf{1}_{A_{k'}(\delta_{m,n}) \setminus A_{k'}}(x). \end{align*} $$

Integrating and using Lemma 3.3 together with the trivial estimate $\mu (A \cap B) \leq \mu (A)$ , we obtain that

(4.12) $$ \begin{align} &\int H_2(T^{n}x,x) \bigl| \hat{G}_1(T^{n+m}x,x) - H_1(T^{n+m}x,x) \bigr| \mathop{}\!d\mu(x) \nonumber\\ &\qquad \lesssim \varepsilon_{m,n}^{-1} \rho_{m,n} + (1 + \varepsilon_{m,n}^{-1} \rho_{m,n}) \rho_{m,n}^{-2K} \delta_{m,n}^{\alpha_0} + \rho_{m,n}^{-3K} \delta_{m,n}^{\alpha_0}. \end{align} $$

We now estimate the second integral in equation (4.11) using decay of correlations:

$$ \begin{align*} \int H_1(T^{n+m}x,x) H_2(T^{n}x,x) \mathop{}\!d\mu(x) \end{align*} $$

is bounded above by

(4.13) $$ \begin{align} &\sum_{k \in I} \sum_{k' \in I} \|\hat{G}_1(\cdot, y_k)\|_{L^{1}} \|\hat{G}_2(\cdot, y_{k'})\|_{L^{1}} \|h_k h_{k'}\|_{L^{1}} \nonumber\\ &\qquad + \sum_{k \in I} \sum_{k' \in I} C \|\hat{G}_1\|_{\text{Lip}} \|\hat{G}_2\|_{\text{Lip}} \|h_k h_{k'}\|_{\text{Lip}} ( e^{-\tau n} + e^{-\tau m} ). \end{align} $$

Similarly to how equations (4.4) and (4.6) were established in Proposition 4.1, we obtain the bounds

$$ \begin{align*} \|\hat{G}_1(\cdot,y_k)\|_{L^{1}} &\leq M_{n+m} + C\varepsilon_{m,n}^{\alpha_0}, \\ \|\hat{G}_2(\cdot,y_k)\|_{L^{1}} &\leq M_{n} + C\varepsilon_{m,n}^{\alpha_0} \end{align*} $$

and

$$ \begin{align*} \sum_{k \in I} \sum_{k' \in I} \|h_k h_{k'}\|_{L^{1}} \leq 1 + C \rho_{m,n}^{-3K} \delta_{m,n}^{\alpha_0}. \end{align*} $$

Also,

$$ \begin{align*} \|\hat{G}_1\|_{\text{Lip}} = \|\hat{G}_2\|_{\text{Lip}} \lesssim \varepsilon_{m,n}^{-1} \end{align*} $$

and

$$ \begin{align*} \|h_k h_{k'}\|_{\text{Lip}} \lesssim \delta_{m,n}^{-1}. \end{align*} $$

Combining the previous equations with equation (4.13), we obtain the upper bound

(4.14) $$ \begin{align} \int H_1(T^{n+m}x,x) H_2(T^{n}x,x) \mathop{}\!d\mu(x) &\leq M_{n+m}M_n + C\varepsilon_{m,n}^{\alpha_0} + C\rho_{m,n}^{-3K}\delta_{m,n} ^{\alpha_0} \nonumber\\ &\quad + C\rho_{m,n}^{-2K}\varepsilon_{m,n}^{-2}\delta_{m,n}^{-1} (e^{-\tau n} + e^{-\tau m}). \end{align} $$

Combining and simplifying equations (4.12) and (4.14), we obtain that

$$ \begin{align*} &\int F(T^{n+m}x,T^{n}x,x) \mathop{}\!d\mu(x) - M_{n+m}M_n \lesssim \varepsilon_{m,n}^{-1} \rho_{m,n} + \varepsilon_{m,n}^{-1} \rho_{m,n}^{1-2K} \delta_{m,n}^{\alpha_0} \\ &\qquad+ \rho_{m,n}^{-3K} \delta_{m,n}^{\alpha_0} + \varepsilon_{m,n}^{\alpha_0} + \rho_{m,n}^{-2K}\varepsilon_{m,n}^{-2}\delta_{m,n}^{-1} (e^{-\tau n} + e^{-\tau m}). \end{align*} $$

Now, let

$$ \begin{align*} \varepsilon_{m,n} &= e^{-\gamma \min \{m,n\}}, \\ \rho_{m,n} &= \varepsilon_{m,n}^2, \\ \delta_{m,n} &= \varepsilon_{m,n}^{2 + 6K/\alpha_0}, \end{align*} $$

where $\gamma = \tfrac 12\bigl (4 + 4K + {6K}/{\alpha _0}\bigr )^{-1} \tau $ . Then, $\varepsilon _{m,n}$ decays exponentially as $n \to \infty $ and ${\delta _{m,n} < \rho _{m,n}}$ as required. For large enough n, we have $3\varepsilon _{m,n}< r_n(x)$ and $\rho _{m,n} < \min \{\varepsilon _0, \rho _0\}$ as we assumed. Finally,

$$ \begin{align*} \eta = \min \Bigl\{ \frac{\tau}{2}, \gamma, \alpha_0\gamma, (1 + 2\alpha_0 + 2K)\gamma \Bigr\} \end{align*} $$

gives us the upper bound

$$ \begin{align*} \int F(T^{n+m}x,T^{n}x,x) \mathop{}\!d\mu(x) - M_{n+m}M_n \lesssim e^{-\eta n} + e^{-\eta m}.\\[-39pt] \end{align*} $$

Proposition 4.3. With the above assumptions on $(X,\mu ,T,d)$ and $(M_n)$ , suppose additionally that three-fold correlations decay exponentially for Lipschitz continuous observables. There exists $C,\eta> 0$ such that for all $m,n \in \mathbb {N}$ ,

$$ \begin{align*} | \mu(E_n) - M_n | \leq C e^{-\eta n} \end{align*} $$

and

$$ \begin{align*} \mu(E_{n+m} \cap E_{n}) \leq \mu(E_{n+m})\mu(E_n) + C(e^{-\eta n} + e^{-\eta m}). \end{align*} $$

Proof. The first inequality follows directly from Proposition 4.1 by noticing that $\mu (E_n) = \int F(T^{n}x,x)\mathop {}\!d\mu (x)$ and the second inequality follows from the first in combination with Proposition 4.2 by noticing that $\mu (E_{n+m} \cap E_{n}) = \int F(T^{n+m}x,T^{n}x,x)\mathop {}\!d\mu (x)$ .

5. Proof of Theorems 1.1 and 2.2

We can now prove our main results.

Proof of Theorem 2.2

By Proposition 4.3, we see that the conditions of Theorem 2.1 are met and we conclude the proof.

Recall that M is a compact smooth N-dimensional manifold, and that d and $\operatorname {\mathrm {Vol}}$ are the induced metric and volume measure. The injectivity radius of M is denoted by $\operatorname {\mathrm {inj}}_M$ and is positive.

Proof of Theorem 1.1

By [20, Proposition 3.1], the system has three-fold decay of correlations for Lipschitz continuous observables. Thus, the theorem follows from Theorem 2.2 once we verify that Assumptions 3 and 4 hold.

For Assumption 4, let $\varepsilon \in (0,\operatorname {\mathrm {inj}}_M)$ and consider a maximal $\varepsilon $ -packing $\{B(x_k,\varepsilon )\}_{k=1}^{L}$ of M, that is, $L = P_\varepsilon (M)$ . Now,

$$ \begin{align*} L \min_{k} \operatorname{\mathrm{Vol}}( B(x_k,\varepsilon) ) &\leq \sum_{k=1}^{L} \operatorname{\mathrm{Vol}}( B(x_k,\varepsilon) ) \\ &= \operatorname{\mathrm{Vol}}\Bigl( \bigcup_{k=1}^{L} B(x_k,\varepsilon) \Bigr) \\ &\leq \operatorname{\mathrm{Vol}}(M). \end{align*} $$

By [11, Proposition 14], there exists $C_1>0$ such that

(5.1) $$ \begin{align} \operatorname{\mathrm{Vol}}(B(y,\varepsilon)) \geq C_1 \varepsilon^{N} \end{align} $$

for all $y\in M$ . Hence,

$$ \begin{align*} L \leq \frac{\operatorname{\mathrm{Vol}}(M)}{\min_{k}\operatorname{\mathrm{Vol}}( B(x_k,\varepsilon) )} \lesssim \varepsilon^{-N}. \end{align*} $$

Thus, Assumption 4 holds for $K = N$ and $\varepsilon _0 = \operatorname {\mathrm {inj}}_M$ .

For Assumption 3, let $x\in M$ and $0 < \varepsilon < \rho \leq \operatorname {\mathrm {inj}}_M$ . For convenience, let $\varepsilon ' = {\varepsilon }/{2}$ . Consider a maximal $\varepsilon '$ -packing $\{B(x_k,\varepsilon ')\}_{k=1}^{L}$ of the annulus $\{y : \rho \leq d(x,y) < \rho + \varepsilon \}$ . Then, $\{B(x_k,\varepsilon ')\}_{k=1}^{L}$ is contained in the set $\{y : \rho - \varepsilon ' \leq d(x,y) < \rho + 3\varepsilon '\}$ . Thus,

$$ \begin{align*} L\min_k \operatorname{\mathrm{Vol}}(B(x_k, \varepsilon')) &\leq \sum_{i=1}^{n} \operatorname{\mathrm{Vol}}(B(x_k, \varepsilon')) \\ &= \operatorname{\mathrm{Vol}} \Bigl( \bigcup_{i=1}^{n} B(x_k, \varepsilon') \Bigr) \\ &\leq \operatorname{\mathrm{Vol}} \{y \in M : \rho - \varepsilon' \leq d(x,y) < \rho + 3\varepsilon'\} \\ &= \operatorname{\mathrm{Vol}}(B(x, \rho + 3\varepsilon')) - \operatorname{\mathrm{Vol}}(B(x, \rho - \varepsilon')). \end{align*} $$

Since M is compact, there exists $C_2> 0$ such that for all $0 < s < r$ and $y \in M$ ,

$$ \begin{align*} \operatorname{\mathrm{Vol}} ( B(y, r) ) - \operatorname{\mathrm{Vol}} ( B(y, s) ) \leq C_2(r-s) \end{align*} $$

(see for instance [8, p. 127]). Therefore,

$$ \begin{align*} L\min_k \operatorname{\mathrm{Vol}}(B(x_k, \varepsilon')) \lesssim 4\varepsilon' \lesssim \varepsilon. \end{align*} $$

Using equation (5.1) again and noting that $\varepsilon ' < \varepsilon < \operatorname {\mathrm {inj}}_M$ , we obtain that

$$ \begin{align*} L \lesssim \frac{\varepsilon}{\varepsilon^{N}} = \varepsilon^{1-N}. \end{align*} $$

Now, $\{B(x_k,\varepsilon )\}_{k=1}^{L}$ covers $\{y \in M : \rho \leq d(x,y) < \rho + \varepsilon \}$ . Hence,

$$ \begin{align*} \mu \{y \in M : \rho \leq d(x,y) < \rho + \varepsilon\} &\leq \mu\Bigl(\bigcup_{k=1}^{L}B(x_k,\varepsilon)\Bigr) \\ &\leq L\max_{k} \mu(B(x_k,\varepsilon)) \\ &\lesssim \varepsilon^{1-N} \varepsilon^{s} \\ &= \varepsilon^{1+s-N}, \end{align*} $$

where in the third inequality, we have used that $\mu (B(x_k,\varepsilon )) \lesssim \varepsilon ^{s}$ . Thus, Assumption 3 holds for $\rho _0 = \operatorname {\mathrm {inj}}_M$ and $\alpha _0 = 1 + s - N$ . Note that $\alpha _0> 0$ as we assumed that ${s> N - 1}$ .

6. Final remarks

6.1. Improvements of Boshernitzan’s result

By restricting the underlying system, one can improve Boshernitzan’s general result. Pawelec [21] proved that

$$ \begin{align*} \liminf_{n \to \infty} (n \log \log n)^{1/\beta} d(T^{n}x,x) = 0 \end{align*} $$

for $\mu $ -a.e. x under the assumptions of exponential decay of correlations and some assumptions on $\mu $ related to $\beta $ . For self-similar sets, Chang, Wu and Wu [7] and Baker and Farmer [2] obtained a dichotomy result for the set

$$ \begin{align*} R = R(T, (r_n)) = \{x \in X : d(T^{n}x,x) < r_n \text{ for infinitely many}\ n \in \mathbb{N}\} \end{align*} $$

to have zero or full measure depending on the convergence or divergence of a series involving $(r_n)$ . For mixing systems on $[0,1]$ , Kirsebom, Kunde and Persson [18] obtained several results, with varying conditions on $(r_n)$ , on the measure of R and related sets. For integer matrices on the N-dimensional torus such that no eigenvalue is a root of unity, they obtain a condition for R to have zero or full measure, again depending on the convergence of a series involving $(r_n)$ . He and Liao [16] extend Kirsebom, Kunde and Persson’s results to non-integer matrices and to targets that are hyperrectangles or hyperboloids instead of open balls. Allen, Baker and Bárány [1] give sufficient conditions for the set R, adapted to subshifts of finite type, to have measure $1$ and sufficient conditions for the set to have measure $0$ . They also obtain a critical threshold for which the measure of R transitions from $0$ to $1$ .

6.2. Spaces satisfying the assumptions

If $\mu $ is absolutely continuous with respect to the Lebesgue measure $\unicode{x3bb} $ with density in $L^{p}$ , then $\mu $ satisfies Assumptions 2 and 3. Namely, if $\mathop {}\!d\mu = h \mathop {}\!d\unicode{x3bb} $ , then

$$ \begin{align*} \mu(A) = \int_{A} h \mathop{}\!d\unicode{x3bb} \leq \|h\|_{L^{p}} \unicode{x3bb}(A)^{1/q}, \end{align*} $$

where q is the harmonic conjugate of p. Since $\unicode{x3bb} (B(x,\rho + \varepsilon ) \setminus B(x,\rho ) ) \leq \varepsilon ^{\alpha _0}$ and $\unicode{x3bb} (B(x,\rho )) \leq \rho ^{s}$ for some $\alpha _0$ and s, we see that $\mu $ also satisfies the assumptions. Gupta, Holland and Nicol [14] consider planar dispersing billiards. These systems have an absolutely continuous invariant measure and so Assumptions 2 and 3 are satisfied. Furthermore, some of the systems have exponential decay of two-fold correlations for Hölder observables as shown by Young [25] and exponential decay of multiple decorrelation as shown in [10, Theorem 7.41]. Thus, our results hold for such systems. The authors in [14] also consider Lozi maps, for which a broad class satisfies Assumption 3 and have exponential decay of two-fold correlations for Hölder observables. For our result to apply to these systems, one would have to show that three-fold correlations decay exponentially as well.

6.3. Improvements of the result

The main limitation of the method of proof is the reliance on Assumption 3, namely that we require control over the measure of $\delta $ -neighbourhoods of sets. It is reasonable to think that the result should hold for measures that do not satisfy this assumption; however, in that case, instead of considering open balls, one would need to consider other shrinking neighbourhoods. The reason for this is that for a general measure $\mu $ , there exists $M_n \in [0,1]$ and $x\in X$ such that there is no open ball $B_n(x)$ with $\mu (B_n(x)) = M_n$ . Furthermore, one would like to improve the condition $M_n \geq n^{-1} (\log n)^{4+\varepsilon }$ in Assumption 1 to one closer to the critical decay rate $M_n \geq n^{-1}$ .