1.2 Main definitions

To define $X_{\beta }^d$ and $X_{\beta ^+}^d$ more carefully, we recall some basic notions from symbolic dynamics and from combinatorics on words. Let A be a finite set (the alphabet), $A^{\mathbb {Z}}$ the set of all bi-infinite strings of symbols in A, and $\sigma \colon A^{\mathbb {Z}} \to A^{\mathbb {Z}}$ the left shift map defined by $\sigma (x)_n = x_{n+1}$ . Equipping $A^{\mathbb {Z}}$ with the product topology, a (two-sided) shift space is a closed set $X \subset A^{\mathbb {Z}}$ that is shift-invariant in the sense that $\sigma (X) = X$ .

A (finite) word is an element $w\in A^n$ for some $n\in \mathbb {N} \cup \{0\}$ ; we will write $|w| = n$ to denote the length of $w$ . The set of all finite words is $A^* := \bigcup _{n\geq 0} A^n$ . Given a shift space $X \subset A^{\mathbb {Z}}$ , the collection of words

$$ \begin{align*} \mathcal{L} = \mathcal{L}(X) := \{ x_i x_{i+1} \cdots x_j : x\in X,\ i,j\in \mathbb{Z} \} \end{align*} $$

is the language of X and we will write $\mathcal {L}_n := \mathcal {L} \cap A^n$ for the set of words of length n in the language. The language $\mathcal {L}$ of a shift space is factorial (closed under passing to subwords) and extendable (given $w\in \mathcal {L}$ , there exist $a,b\in A$ such that $awb\in \mathcal {L}$ ).

It is often convenient to describe a shift space and its language via a set of forbidden words: given $\mathcal {F} \subset A^+ := \bigcup _{n\geq 1} A^n$ , the associated shift space is

$$ \begin{align*} X = X(\mathcal{F}) = \{ x\in A^{\mathbb{Z}} : x_i x_{i+1} \cdots x_j \notin \mathcal{F} \text{ for all } i,j\in \mathbb{Z} \text{ with } i\leq j \}. \end{align*} $$

That is, $X(\mathcal {F})$ is the set of all infinite sequences over A that have no subword in $\mathcal {F}$ .

Given $n\in \mathbb {N}$ , a word $w\in A^n$ , and a rational number $\alpha> 1$ such that $\alpha n\in \mathbb {N}$ , the $\alpha $ -power of $w$ is the word $w^\alpha $ formed by the first $\alpha n$ symbols of $w^\infty = wwww\cdots $ . For example, we have

$$ \begin{align*} (\mathsf{do})^{2} = \mathsf{dodo}, \quad \mathsf{b(an)^{5/2} = banana}, \quad \mathsf{(alf)^{7/3} = alfalfa}. \end{align*} $$

Definition 1.1. For a natural number $d\geq 2$ and a real number $\beta>1$ , the d-ary $\beta $ -free shift space $X_{\beta }^d$ is the shift space defined by the set of forbidden words,

(1.2) $$ \begin{align} \mathcal{F}_\beta := \{ v^\alpha : v\in A^+,\ \alpha \in [\beta,\infty) \cap \mathbb{Q} \}. \end{align} $$

In other words, $X_{\beta }^d = X(\mathcal {F}_\beta )$ consists of all $x\in A^{\mathbb {Z}}$ that remain after we remove all strings containing an $\alpha $ -power for any $\alpha \geq \beta $ . Replacing $[\beta ,\infty )$ with $(\beta ,\infty )$ in (1.2) gives a set $\mathcal {F}_{\beta ^+}$ of forbidden words that determines a subshift $X_{\beta ^+}^d$ .

If $\beta \in \mathbb {R} \setminus \mathbb {Q}$ , then $X_{\beta ^+}^d = X_{\beta }^d$ . From now on, $\beta $ will denote either a real number or a ‘rational number with a $+$ ’. The set of possible $\beta $ carries the obvious order, so we could write Theorem 1.1 as: ‘For every $d\geq 2$ and $\beta \geq 12^+$ , the shift space $X_{\beta }^d$ has a unique MME’. (In fact, for $d\geq 3$ , we prove the result for $\beta = 12$ as well; see Theorem 2.1.)

1.3 Overview of paper

See §2.1 for a discussion of entropy in the context of symbolic dynamics, and for two results that together imply Theorems 1.1 and 1.2:

• • Theorem 2.1 formulates a weak analog of Bowen’s specification property that is satisfied by $X_{\beta }^d$ and which quickly leads to the counting estimates that establish Theorem 1.2;

• • Theorem 2.3 uses this specification-type property to obtain uniqueness of the MME, proving Theorem 1.1.

Theorems 2.1 and 2.3 are proved in §§3 and 4, respectively.

See §2.2 for a review of some prior work on power-free languages and a description of the set of $(\beta ,d)$ for which $X_{\beta }^d$ is known or conjectured to have positive entropy; as indicated in Remark 1.3, this set is somewhat larger than the set of parameters to which Theorems 1.1 and 1.2 apply. See §2.3 for a discussion of this issue and some other open problems. With this in mind, both of these main results should be regarded in some sense as ‘proofs of concept’ demonstrating the utility of specification-type techniques in this setting, rather than as attempts to wring out the strongest possible results from these ideas: to present the main ideas as efficiently and readably as possible, no serious attempt has been made here to push these techniques to the fullest extent.

2.1 Entropy

We describe the main concepts that we will need from thermodynamic formalism, as they appear in symbolic dynamics. See [Reference WaltersWal82] for an introduction to entropy and the variational principle, and [Reference Climenhaga and ThompsonCT21] for a survey of the development of specification-type properties in the study of thermodynamic formalism.

Given a finite alphabet A and a shift space $X \subset A^{\mathbb {Z}}$ , recall that $\mathcal {L} \subset A^*$ denotes the language of X, with $\mathcal {L}_n = \mathcal {L} \cap A^n$ denoting the set of words of length n in the language. Equivalently, $\mathcal {L}_n$ is the set of words $w\in A^n$ for which the cylinder

$$ \begin{align*} [w] := \{ x\in X : x_1 \cdots x_n = w \} \end{align*} $$

is non-empty. The topological entropy of X is

(2.1) $$ \begin{align} h = h(X) := \lim_{n\to\infty} \frac 1n \log \#\mathcal{L}_n, \end{align} $$

where the limit exists by Fekete’s lemma since $\#\mathcal {L}_{n+k} \leq (\#\mathcal {L}_n)(\#\mathcal {L}_k)$ .

An invariant measure on X is a Borel probability measure $\mu $ such that for every Borel set $E\subset X$ , we have $\mu (\sigma ^{-1}E) = \mu (E)$ . The measure-theoretic entropy of $\mu $ is

(2.2) $$ \begin{align} h_\mu := \lim_{n\to\infty} \frac 1n H_n(\mu) \quad\text{where } H_n(\mu) := \sum_{w\in \mathcal{L}_n} -\mu([w]) \log \mu([w]); \end{align} $$

the existence of the limit is again a consequence of Fekete’s lemma and the subadditivity property $H_{n+k}(\mu ) \leq H_n(\mu ) + H_k(\mu )$ . Writing $\mathcal {M}_\sigma (X)$ for the space of invariant measures, the variational principle says that

(2.3) $$ \begin{align} h(X) = \sup \{ h_\mu : \mu \in \mathcal{M}_\sigma(X) \}. \end{align} $$

A measure achieving the supremum is called a measure of maximal entropy (MME). Every shift space on a finite alphabet has an MME because the function $\mu \mapsto h_\mu $ is upper semi-continuous in the weak* topology, in which $\mathcal {M}_\sigma (X)$ is compact.

It is an important question in ergodic theory and thermodynamic formalism to understand for which classes of shift spaces the MME is unique, and to describe its properties; see §2.3 for more. There are examples of shifts with multiple MMEs, but uniqueness is known to hold for every topologically transitive subshift of finite type (including the full shift), every transitive sofic shift, and many other examples.

We will focus our attention on a variant of Rufus Bowen’s specification property. For a shift space, this property reduces to

(2.4) $$ \begin{align} \begin{aligned} &\text{there exists } \tau \in \mathbb{N} \text{ such that for every } k\in \mathbb{N} \text{ and } w^{(1)},\ldots, w^{(k)} \in \mathcal{L}, \\ &\text{there exist } u^{(1)},\ldots, u^{(k-1)} \in \mathcal{L}_\tau \text{ such that } w^{(1)} u^{(1)} w^{(2)} \cdots u^{(k-1)} w^{(k)} \in \mathcal{L}. \end{aligned} \end{align} $$

(Here, we use the notation $w^{(j)}$ for an element of an indexed set of words, reserving the notation $w^j$ for the power of $w$ obtained by concatenating j copies, and the notation $w_j$ for the jth symbol of a word $w$ . We point out that there are many versions of the specification property in the literature; see [Reference YamamotoYam09, Reference Kwietniak, Ła̧cka and OprochaKLO16] for overviews. The property in (2.4) is a uniform specification property that imposes no a priori requirement on existence of a periodic shadowing orbit.) It can quickly be seen that this is equivalent to

(2.5) $$ \begin{align} \begin{aligned} &\text{there exists } \tau\in \mathbb{N} \text{ such that for every } v,w\in \mathcal{L}, \\ &\text{there exists } u\in \mathcal{L}_\tau \text{ such that } vuw\in \mathcal{L}. \end{aligned} \end{align} $$

Every topologically mixing sofic shift (in particular, every topologically mixing subshift of finite type) has specification. Bowen proved [Reference BowenBow75b] that every shift space with specification has a unique MME.

Specification produces periodic orbits: writing $P_n := \{ x\in X : \sigma ^n(x) = x\}$ , one can prove that $(1/n) \log \# P_n \to h$ whenever (2.5) is satisfied. (To see this, use [Reference BertrandBer88] to get a synchronizing word $v$ and then for each $w\in \mathcal {L}_n$ , use (2.4) to get $u,u' \in \mathcal {L}_\tau $ such that $vuwu'v \in \mathcal {L}$ , which implies $x := (vuwu')^\infty \in P_{n+2\tau +|v|}$ .) Since the power-free shifts $X_{\beta }^d$ have no periodic orbits, they cannot satisfy Bowen’s specification property. To formulate a weaker property that they do satisfy, recall that a prefix of $w\in A^n$ is a word of the form $w_{[1,k]} := w_1 w_2 \cdots w_k$ for some $k \in \{1,\ldots , n\}$ , and a suffix of $w$ is a word of the form $w_{[k,n]}$ . The following theorem will be proved in §3.

Theorem 2.1. Fix $(d,\beta )$ such that $d\geq 3$ and $\beta \geq 12$ , or $d=2$ and $\beta \geq 12^+$ . Let ${X = X_{\beta }^d}$ be the corresponding d-ary $\beta $ -free shift and $\mathcal {L} \subset \{1,\ldots ,d\}^*$ its language. Define $\mathcal {C}^p,\mathcal {G},\mathcal {C}^s \subset \mathcal {L}$ as

(2.6) $$ \begin{align} \begin{aligned} \mathcal{G} &:= \{ w \in \mathcal{L} : \text{no prefix or suffix of } w \text{ is a } 4-\text{power} \}, \\ \mathcal{C}^p &= \, \mathcal{C}^s := \mathcal{A}^* \cap \mathcal{L} \quad\text{where } \mathcal{A} := \{ v^4 : v\in \mathcal{L}\}. \end{aligned} \end{align} $$

(Here, $\mathcal {A}^*$ uses the Kleene star to denote arbitrary finite (possibly empty) concatenations of words from $\mathcal {A}$ , just as $A^*$ denoted arbitrary concatenations of symbols from the alphabet.) That is, $\mathcal {C}^p = \mathcal {C}^s$ is the set of words in $\mathcal {L}$ that can be written as a concatenation of $4$ -powers. Then, the following are true.

1. (I) (Decomposition): for every $w\in \mathcal {L}$ , there exist $v\in \mathcal {G}$ and $u^{p,s} \in \mathcal {C}^{p,s}$ such that ${w=u^p v u^s}$ .

2. (II) (Entropy gap): $h(\mathcal {C}^p \cup \mathcal {C}^s) := \varlimsup _{n\to \infty } ( 1/n) \log \#(\mathcal {C}^p \cup \mathcal {C}^s)_n < h(X)$ .

3. (III) (Variable-length specification): there exists $T\in \mathbb {N}$ such that for all $u,v,w,x\in \mathcal {G}$ , there are $p,q,r \in \mathcal {L}_{T}$ such that $upvqwrx\in \mathcal {L}$ .

4. (IV) (Same-length specification): there exists $\tau \in \mathbb {N}$ such that for every $v,w\in \mathcal {G}$ with $|v| = |w|$ , there exists $u\in \mathcal {L}_\tau $ such that $vuw\in \mathcal {G}$ .

Moreover, $\#\mathcal {L}_n$ satisfies the bounds in (1.1).

Conditions (I) and (II) require a decomposition of the language in which the prefix and suffix collections $\mathcal {C}^p$ and $\mathcal {C}^s$ do not carry full entropy. Conditions (III) and (IV) are weakenings of the specification properties (2.4) and (2.5). In particular, shifts with specification satisfy Conditions (I)–(IV) by taking $\mathcal {C}^p=\mathcal {C}^s=\emptyset $ and $\mathcal {G} = \mathcal {L}$ .

We will prove that every shift space satisfying Conditions (I)–(IV) has a unique MME. Our argument will use a slightly weaker version of Condition (IV): the following is proved in §3.3.2 by iterating Condition (IV).

Our general uniqueness result will also use the following consequence of Condition (II) instead of Condition (II) itself.

1. (II′) The quantity $Q := (\sum _{i=0}^\infty \#\mathcal {C}^p_i e^{-ih})(\sum _{k=0}^\infty \#\mathcal {C}^s_k e^{-kh})$ is finite.

The following theorem is proved in §4. Together with Theorem 2.1 and Lemma 2.2, it completes the proof of Theorem 1.1.

Remark 2.4. (Other non-uniform specification results)

Conditions analogous to those appearing in Theorem 2.3 were introduced by the author and Dan Thompson in [Reference Climenhaga and ThompsonCT12]. In that paper, Conditions (III) and (IV $^{\prime }$ ) were replaced by the stronger (but simpler to formulate) condition that ‘the specification property is satisfied for words in $\mathcal {G}$ ’, in the sense that

(2.7) $$ \begin{align} \begin{aligned} &\text{for every } k\in \mathbb{N} \text{ and } w^{(1)},\ldots, w^{(k)} \in \mathcal{G}, \\ &\text{there exist } u^{(1)},\ldots, u^{(k-1)} \in \mathcal{L}_\tau \text{ such that } w^{(1)} u^{(1)} w^{(2)} \cdots u^{(k-1)} w^{(k)} \in \mathcal{L}. \end{aligned} \end{align} $$

This condition, together with Conditions (I) and (II $^{\prime }$ ), implies uniqueness of the MME. This was proved in [Reference Climenhaga and ThompsonCT12, Reference Climenhaga and ThompsonCT13] modulo an extra assumption on ‘extending to words in $\mathcal {G}$ ’. This extra condition was ultimately proved to be unnecessary by Pacifico et al [Reference Pacifico, Yang and YangPYY22]. Theorem 2.3 is another generalization of the result from [Reference Climenhaga and ThompsonCT12]: observe that Conditions (III) and (IV $^{\prime }$ ) are weakened versions of (2.7), obtained by restricting to the case $k=4$ in Condition (III), and to words of the same length in Condition (IV $^{\prime }$ ).

It is also worth comparing Theorem 2.3 with [Reference PavlovPav22, Theorem 3.2], where Pavlov uses ideas from his earlier work in [Reference PavlovPav19] to prove uniqueness of the MME under the following conditions:

• • there exists $C \geq 1$ such that $\#\mathcal {L}_n \leq C e^{nh}$ for all n;

• • there exists $\mathcal {G} \subset \mathcal {L}$ satisfying a weaker version of Condition (III), which only requires the ability to glue two words (instead of four);

• • the collection $\mathcal {G}$ can also be chosen to have the property that for every ergodic MME $\mu $ , there exists a syndetic set $S \subset \mathbb {N}$ such that $\inf _{n\in S} \mu (\mathcal {G}_n)> 0$ .

Pavlov’s argument for uniqueness is rather different from the one given here; his proof uses the maximal ergodic theorem and does not rely on establishing any sort of Gibbs property. Indeed, he proves a Gibbs bound after proving uniqueness, whereas we proceed in the opposite order. It would be interesting to understand whether his techniques could be used to sharpen the results here.

2.2 Growth in power-free languages

Power-free languages have been well studied. For a full introduction to these languages and other topics in combinatorics on words, we refer to [Reference Lothaire, Perrin, Berstel, Choffrut, Cori, Foata, Pin, Pirillo, Reutenauer, Schützenberger, Sakarovitch and SimonLot83, Reference Allouche and ShallitAS03]. We will focus on the growth properties of power-free languages; overviews of the literature on this topic can be found in the surveys by Berstel [Reference BerstelBer05] and Shur [Reference ShurShu11, Reference ShurShu12].

2.2.1 Repetition threshold and Dejean’s conjecture

A natural first question is to ask when $X_\beta ^d$ is non-empty. One quickly sees that it is empty when $d=2$ and $\beta \leq 2$ : the only non-empty words over $A = \{\mathsf {0,1}\}$ that do not contain a square are

$$ \begin{align*} \mathsf{0},\quad \mathsf{01},\quad \mathsf{010},\quad\text{and}\quad \mathsf{1},\quad\mathsf{10},\quad\mathsf{101}. \end{align*} $$

However, the Thue–Morse sequence (this sequence is $\mathsf {0 1 10 1001 10010110\cdots }$ , where the first $2^n$ symbols determine the next $2^n$ symbols via the morphism $\mathsf {0\leftrightarrow 1}$ ) does not contain any subword that is an $\alpha $ -power for any $\alpha> 2$ (it is overlap-free), so $X_{2^+}^2 \neq \emptyset $ . Thus, $\beta = 2$ is the repetition threshold for the alphabet size $d=2$ : writing

$$ \begin{align*} \mathsf{RT}(d) := \inf \{ \beta> 1 : X_\beta^d \neq \emptyset \}, \end{align*} $$

we have $\mathsf {RT}(2) = 2$ . In 1972, Dejean proved that $\mathsf {RT}(3) = \frac 74$ and conjectured that

$$ \begin{align*} \mathsf{RT}(4) = \frac 75{,} \quad\mathsf{RT}(d) = \frac{d}{d-1} \quad \text{for all } d\geq 5. \end{align*} $$

This was eventually proved for all d through a combination of work by Pansiot [Reference PansiotPan84], Moulin-Ollagnier [Reference OllagnierMO92], Mohammad-Noori and Currie [Reference Mohammad-Noori and CurrieMNC07], Carpi [Reference CarpiCar07], Currie and Rampersad [Reference Currie and RampersadCR11], and Rao [Reference RaoRao11].

2.2.2 Entropy and the exponential conjecture

With the repetition thresholds known, it is natural to ask for which values of $(\beta ,d)$ with $\beta> \mathsf {RT}(d)$ the topological entropy of $X_\beta ^d$ is positive. (The literature in combinatorics on words tends to use the term exponential complexity rather than positive topological entropy, but the meaning is the same.) That is, we want to determine the set

$$ \begin{align*} \mathcal{P} := \{ (\beta,d) \in (1,\infty) \times \{2,3,\ldots\} : h(X_\beta^d)> 0 \}. \end{align*} $$

Early results in this direction were given by Brandenburg, who proved that $X_3^2$ and $X_2^3$ have positive topological entropy [Reference BrandenburgBra83].

In the other direction, Restivo and Salemi [Reference Restivo and SalemiRS85] proved that $h(X_{2^+}^2) = 0$ by characterizing elements of this shift space via the Thue–Morse morphism; this was extended to $X_\beta ^2$ for $2 < \beta \leq \frac 73$ by Karhumäki and Shallit [Reference Karhumäki and ShallitKS04]. (Both [Reference Restivo and SalemiRS85, Reference Karhumäki and ShallitKS04] actually obtained more explicit polynomial bounds on $\#\mathcal {L}_n(X_\beta ^2)$ , but the subexponential nature of the growth is the main phenomenon in which we are interested here.)

This turns out to be sharp: Karhumäki and Shallit also proved that $h(X_\beta ^2)> 0$ for all $\beta> \frac 73$ . When $d\geq 3$ , no analog of the ‘polynomial plateau’ $(2,\frac 73] \times \{2\}$ has been observed. Ochem proved that for $d=3,4$ , every non-empty $X_\beta ^d$ has positive entropy, and conjectured that this is true for all $d\geq 3$ [Reference OchemOch06, Conjecture 4.4]. An equivalent formulation of this exponential conjecture is that, writing $\beta = \mathsf {RT}(d)$ , the shift space $X_{\beta ^+}^d$ has positive topological entropy. This has now been proved for all $d\geq 3$ except for the even numbers from $12$ to $26$ [Reference Kolpakov and RaoKR11, Reference Tunev and ShurTS12, Reference Currie, Mol and RampersadCMR20].

2.2.3 Uniqueness of the MME

The discussion so far is summarized in Figure 1, which illustrates how Theorem 1.1 can be viewed as a step towards answering the following question.

Question 2.6. Does every power-free shift space $X_\beta ^d$ with positive topological entropy have a unique MME?

Figure 1

A map of parameter space for $X_{\beta }^d$ (color online).

Figure 1 Long description

The diagram uses a horizontal x-axis labeled beta and a vertical y-axis labeled d with integer markings from 2 to 5.

* The leftmost region is white and labeled with a null symbol, indicating an empty set.

* Moving right, a stepped boundary defined by R T (d) transitions into a large blue shaded region. A red arrow points to this area with the text: h > 0, uniqueness of M M E unknown.

* At the bottom of this blue region, between beta values 2 and 7 over 3, there is a small orange rectangular area. A red arrow points to this labeled: polynomial plateau.

* A thick vertical black line at beta equals 12 marks the transition to the final region.

* The rightmost region is shaded green. A red arrow points into this area with the text: h > 0, unique M M E.

As mentioned in Remark 1.3 and §1.3, the arguments given in §3 have been written down with brevity and readability in mind, rather than with the goal of finding the largest set of $(\beta ,d)$ to which these specification-type techniques can be applied.

The conclusions of Theorems 1.1 and 1.2 hold for many smaller values of $\beta $ , at least when the alphabet is sufficiently large. For example, Pavlov has shown that if X is a subshift on a d-letter alphabet A defined by a set of forbidden words that contains at most $b_n$ words of length n, and if

(2.8) $$ \begin{align} \sum_{n=1}^\infty n^2 b_n (3d^{-1})^{n/3} < \frac 1{36}, \end{align} $$

then X has a unique MME, which has the K-property [Reference PavlovPav22, Theorem 7.2]. The brief argument establishing the following consequence of this result was provided to the author by Ronnie Pavlov.

Theorem 2.7. (Pavlov)

For every $\beta>3$ , there exists $d_0\in \mathbb {N}$ such that for all $d\geq d_0$ , both $X_{\beta }^d$ and $X_{\beta ^+}^d$ have a unique MME, and this measure has the K-property. Moreover, $\#\mathcal {L}_n \leq 4e^{nh}$ for all sufficiently large n.

Proof. First, observe that given any $N\in \mathbb {N}$ , the number of non-empty words with length $\leq N$ is at most $\sum _{\ell =1}^N d^\ell = d (({d^N - 1})/({d-1})) \leq 2d^N$ . Now, given $n\in \mathbb {N}$ , a forbidden word of length n must have the form $w^{p/q}$ , where $ p/q \leq \beta $ , and thus $|w| \leq n/\beta $ . Thus, we can take $b_n = 2d^{n/\beta }$ and the sum in (2.8) becomes

$$ \begin{align*} \sum_{n=1}^\infty n^2 2d^{n/\beta} (3d^{-1})^{n/3} = 2\sum_{n=1}^\infty n^2 \unicode{x3bb}_d^n = \frac{2\unicode{x3bb}_d(1+\unicode{x3bb}_d)}{(1-\unicode{x3bb}_d)^3} \quad\text{where } \unicode{x3bb}_d := 3^{1/3} d^{ (1/\beta) - (1/3)}, \end{align*} $$

provided $\unicode{x3bb} _d \in (0,1)$ . Since $\beta>3$ , we have $\unicode{x3bb} _d \to 0$ as $d\to \infty $ , so (2.8) is satisfied for all sufficiently large d. Together with [Reference PavlovPav22, Theorems 4.3 and 7.2], this proves the result.

Remark 2.8. There are many similarities between the arguments in [Reference PavlovPav22] and the arguments here. In both cases, the overall approach is to formulate abstract conditions that are satisfied by the class of subshifts to be studied: subshifts with slow forbidden word growth in [Reference PavlovPav22] and power-free shifts here. The similarity between the abstract conditions was discussed in Remark 1.3. The arguments that Pavlov uses to verify his conditions for subshifts with slow forbidden word growth are based on ideas of Miller [Reference MillerMil12] and his own prior work on coded systems [Reference PavlovPav20]. In the specific case of power-free shifts, there are strong parallels to the approach here. The following observation is indicative: if we consider the case $\beta =12$ , so that $\mathcal {F} = \{ v^{12} : v\in \mathcal {L}\}$ , then the collection of words $\mathcal {A}$ defined in (2.6) is precisely the set of heavy subwords in the sense of [Reference PavlovPav22, Definition 5.1], and the collection $\mathcal {G}$ in (2.6) agrees with the set of words G from [Reference PavlovPav22, Definition 5.2].

Regarding the range of parameters that one could hope to treat using Theorem 2.3, we make the following observations. For $d\geq 3$ and $\beta \geq 16$ , taking $\mathcal {A} = \{v^4 : v\in \mathcal {L}\}$ will let us verify Conditions (III) and (IV) with transition times $T=0$ and $\tau =1$ , while going all the way to $d=2$ and $\beta = 12^+$ requires $T=\tau =2$ . It seems likely that the same choice of $\mathcal {A}$ could be made to work for at least some smaller values of d and $\beta $ at the cost of increasing the transition times and the lengths of the proofs, and one might also try to go further by replacing $v^4$ with a smaller power in the definition of $\mathcal {A}$ .

For purposes of the uniqueness argument, the exact values of the constants in the counting bounds (1.1) are not important; the crucial thing is that such constants exist. One might hope to improve these bounds using the approach here by making more careful estimates of $\#(\mathcal {A}^*)_n$ and $\#\mathcal {G}_n$ . It deserves to be mentioned that improved bounds can also be obtained by other methods: this was done by Bui and Rosenfeld [Reference Bui and RosenfeldBR25], using combinatorial techniques introduced in [Reference RosenfeldRos25]. The crucial ingredient is an explicit estimate of a constant $C \in (0,1)$ such that $\#\mathcal {L}_{n+m} \geq C \#\mathcal {L}_n \#\mathcal {L}_m$ for all $n,m\in \mathbb {N}$ [Reference Bui and RosenfeldBR25, Theorems 10 and 11]. These results apply to a broader range of parameters than Theorems 1.2 or 2.7, and it would be interesting to see if they could be used to study uniqueness of the MME for these parameters as well.

2.3 Related questions

The surveys [Reference BerstelBer05, Reference ShurShu11, Reference ShurShu12] describe related results and questions concerning growth in power-free languages. These include techniques for numerically approximating $h(X_\beta ^d)$ ; see [Reference ShurShu12, §§3–5] for a discussion of these and of the continuity properties of $\beta \mapsto h(X_\beta ^d)$ . It would be interesting to study the continuity properties not just of the topological entropy, but also of the dependence of the MME on the parameter $\beta $ .

Another direction that has been studied is to consider repetition-free languages and shift spaces defined by variants of the power-free condition. For example, one could allow powers of short words but not of longer words [Reference Entringer, Jackson and SchatzEJS74, Reference DekkingDek76, Reference Fraenkel and SimpsonFS95, Reference Rampersad, Shallit and WangRSW05], or could impose a restriction that we avoid patterns other than pure powers [Reference DekkingDek79, Reference Chiniforooshan, Kari and ZhiCKX12]. It seems likely that the methods developed in this paper could be useful in at least some such examples.

We conclude this section by mentioning some further questions drawn from the theory of uniformly hyperbolic dynamical systems, where thermodynamic formalism was developed and where the specification-type properties used in Theorem 2.3 assume their strongest form. In this setting, uniqueness of the MME is only the beginning of the story. For example, given a shift space X with the specification property (2.4), the following are true.

• • The unique MME $\mu $ is fully supported: it gives positive weight to every open set [Reference BowenBow75b]. Moreover, the system $(X,\sigma ,\mu )$ is measure-theoretically isomorphic to a Bernoulli shift [Reference BowenBow75a, Reference DaonDao13, Reference ClimenhagaCli18]: in particular, it has the K property [Reference LedrappierLed77] and is mixing. (The K (Kolmogorov) property follows from Bernoullicity and implies mixing; it is equivalent to completely positive entropy, meaning that $(X,\sigma ,\mu )$ has positive entropy over any partition.)

• • For every Hölder continuous function $\varphi \colon X\to \mathbb {R}$ , there is a unique equilibrium state $\mu $ that maximizes the quantity $h_\mu + \int \varphi \,d\mu $ . This equilibrium state is fully supported and Bernoulli.

• • The space of invariant measures $\mathcal {M}_\sigma (X)$ is very rich: its set of extreme points (ergodic measures) is entropy-dense [Reference Eizenberg, Kifer and WeissEKW94, Reference Pfister and SullivanPS05], which implies that $\mathcal {M}_\sigma (X)$ is linearly isomorphic to the Poulsen simplex; moreover, it contains measures isomorphic to every ergodic aperiodic transformation with entropy less than h [Reference KriegerKri75, Reference Quas and SooQS16].

The question of full support is closely related to what Shallit and Shur call the Restivo–Salemi property [Reference Shallit and ShurSS19, §4], after a question posed in [Reference Restivo and SalemiRS85, Problem 4]: this is the condition that given any $u,v\in \mathcal {L}(X_\beta ^d)$ , there exists a word $w$ such that $uwv \in \mathcal {L}(X_\beta ^d)$ . In the terminology of dynamical systems, this is the property that $(X_\beta ^d,\sigma )$ is topologically transitive, which must be true whenever there exists a fully supported ergodic measure; in particular, a positive answer to Question 2.9(a) would imply the Restivo–Salemi property. Numerical evidence suggests that the Restivo–Salemi property should hold for every $X_\beta ^d$ ; see [Reference ShurShu09, Conjecture 1] and [Reference Shallit and ShurSS19, Conjecture 42]. However, the author is not aware of a proof of this property for any $X_\beta ^d$ with positive entropy.

For the parameters $d,\beta $ in Theorem 2.7, Pavlov’s result from [Reference PavlovPav22] establishes the K property for the unique MME and one may hope to obtain a similar result for the parameters in Theorem 1.1. It is less clear to the author whether to expect the Bernoulli property to hold. As mentioned in Remark 1.3, Bernoullicity of the MME in hyperbolic dynamics is generally proved using local product structure. This same local product structure is also associated to the abundance of periodic orbits in hyperbolic dynamics. Since these are absent in $X_{\beta }^d$ and there is no obvious product structure to use, it is not clear how to approach the question of Bernoullicity.

Moving beyond the MME, the results mentioned previously for uniformly hyperbolic systems suggest the following questions.

3.1 Periods of words

Given $w=w_1 \cdots w_n \in A^n$ , an integer $\ell $ is a period of $w$ if ${1\leq \ell \leq n}$ and $w_i = w_{\ell +i}$ for all $1\leq i\leq n-\ell $ ; equivalently, if $w = (w_1 \cdots w_\ell )^{n/\ell }$ . The set of all periods of a word $w$ will be denoted $\Pi (w)$ and the smallest element of $\Pi (w)$ will be called the period of $w$ .

Given $w\in A^*$ and $\ell \in \mathbb {N}$ , let $\mathbf {p}_\ell (w)$ be the number of times the word $w_{[1,\ell ]}$ is repeated as a prefix of $w$ . That is, $\mathbf {p}_\ell (w) := k/\ell $ , where k is maximal with the property that $w_{[1,k]}$ is a prefix of $(w_{[1,\ell ]})^\infty $ . Similarly, let $\mathbf {s}_\ell (w)$ be the number of times the length- $\ell $ suffix of $w$ is repeated.

Lemma 3.5. Given $v,w\in A^*$ , there exist $a_1,a_2 \in A$ such that for every $\tau \in \mathbb {N}$ and every $u\in A^\tau $ satisfying $u_1 \neq a_1$ and $u_\tau \neq a_2$ , we have

$$ \begin{align*} \mathbf{p}_\ell(vuw) = \mathbf{p}_\ell(v) &\text{ for every } \ell \in \{1,\ldots, \lceil |v|/2\rceil \}, \\ \mathbf{s}_\ell(vuw) = \mathbf{s}_\ell(w) &\text{ for every } \ell \in \{1,\ldots, \lceil |w|/2\rceil \}. \end{align*} $$

3.2 Specification-type properties

The following lemmas verify Conditions (III) and (IV) for all values of $\beta $ and d given in Theorem 2.1; in fact, we will see that Condition (IV) holds for some smaller values of $\beta $ as well. The lemmas will also show how T and $\tau $ can be chosen smaller when $\beta $ and d are larger.

We will prove Condition (III) for the smaller values of $\beta $ after first showing how Lemma 3.5 can be used to ‘glue words in $\mathcal {G}$ together’.

Lemma 3.7 does not establish Property (IV) when $d=2$ , since in this case, the set ${A \setminus \{a_1,a_2\}}$ might be empty. Thus, we need to allow the use of a $2$ -symbol word to ‘disrupt the shorter periods’. This extra symbol requires us to exclude the case $\beta = 8$ .

Now, we extend Lemma 3.6 to smaller values of $\beta $ by increasing T.

Proof. First, suppose that $d\geq 3$ and $\beta \geq 12$ . Given $u,v,w,x\in \mathcal {G}$ , apply Lemma 3.5 to $u,v$ to get $a_1,a_2 \in A$ such that fixing $a\in A \setminus \{a_1,a_2\}$ , we have

$$ \begin{align*} \mathbf{p}_\ell(uav) = \mathbf{p}_\ell(u) < 4 &\text{ for every } \ell \in \{1,\ldots, \lceil |u|/2\rceil \}, \\ \mathbf{s}_\ell(uav) = \mathbf{s}_\ell(v) < 4 &\text{ for every } \ell \in \{1,\ldots, \lceil |v|/2\rceil \}. \end{align*} $$

Obtain $b = b(v,w)$ and $c = c(w,x)$ similarly. Now, suppose $s = t^\alpha $ is a subword of $uavbwcx$ . If s is a subword of any of $u,v,w,x$ , then $\alpha < \beta $ since each of these words lies in $\mathcal {L}$ . So, we can restrict our attention to the case when s includes either the beginning or the end of every word in $\{u,v,w,x\}$ that it intersects. Let z denote any of these four words that is intersected by s; then, one of the following occurs:

• • $|t| \leq \lceil |z|/2\rceil $ , in which case Lemma 3.5 guarantees that s does not extend past both endpoints of z; or

• • $|t|> \lceil |z|/2\rceil $ , in which case $|t| \geq ({|z|}/2) + 1$ , so $|z| \leq 2(|t|-1)$ .

The first of these two cases can occur for at most two distinct z and for each such z, the word s crosses at most $4|t|-1$ of the symbols in z. We conclude that

$$ \begin{align*} |s| \leq (4|t|-1) + 1 + (2|t|-2) + 1 + (2|t|-2) + 1 + (4|t|-1) = 12|t| - 3 < 12|t|, \end{align*} $$

so $\alpha < 12\leq \beta $ , which completes the proof of the lemma when $d\geq 3$ . For the case $d=2$ , we replace the symbols $a,b,c$ with words $p,q,r\in A^2$ once again provided by Lemma 3.5 in just the same way (following the procedure in Lemma 3.8). The rest of the argument is the same, except that in the final inequality, we obtain $|s| \leq 12|t|$ instead of ${|s| \leq 12|t|-3}$ .

3.3 An entropy gap

3.3.1 Upper bound on prefixes and suffixes

Since $\mathcal {C}^p = \mathcal {C}^s = \mathcal {A}^*$ , we must obtain an upper bound for $\#(\mathcal {A}^*)_n$ to prove Condition (II). Every element of $\mathcal {A}$ is a $4$ -power, so $\#(\mathcal {A}^*)_n = 0$ if n is not a multiple of $4$ . Given $w\in (\mathcal {A}^*)_{4k}$ for some $k\in \mathbb {N}$ , we can write $w = (v^{(1)})^4 \cdots (v^{(\ell )})^4$ , where $v^{(1)}, \ldots , v^{(\ell )} \in \mathcal {L}$ and $\sum _{i=1}^\ell |v^{(i)}| = k$ . Writing

$$ \begin{align*} \mathbb{J}_\ell := \{ \mathbf{k} = (k_1,\ldots, k_\ell) \in \mathbb{N}^\ell : k_1 + \cdots + k_\ell = k \} \end{align*} $$

and using the bound $\#\mathcal {L}_{k_i} \leq d^{k_i}$ , we have

$$ \begin{align*} (\mathcal{A}^*)_{4k} \subset \bigcup_{\ell=1}^k \bigcup_{\mathbf{k} \in \mathbb{J}_\ell} \prod_{i=1}^\ell \mathcal{L}_{k_i} \quad\Rightarrow\quad \#(\mathcal{A}^*)_{4k} \leq \sum_{\ell=1}^k \sum_{\mathbf{k} \in \mathbb{J}_\ell} \prod_{i=1}^\ell \#\mathcal{L}_{k_i} \leq \sum_{\ell=1}^k (\#\mathbb{J}_\ell) d^k. \end{align*} $$

There is a bijection between $\mathbb {J}_\ell $ and subsets of $\{1,\ldots , k-1\}$ with $\ell -1$ elements, obtained by mapping $\mathbf {k}$ to $\{s_1,\ldots , s_{\ell -1}\}$ , where $s_j = \sum _{i=1}^j k_i$ . Thus, $\#\mathbb {J}_\ell = \binom {k-1}{\ell -1}$ and $\sum _{\ell =1}^k \#\mathbb {J}_\ell = 2^{k-1}$ , which gives the bound

(3.1) $$ \begin{align} \#(\mathcal{A}^*)_{4k} \leq \frac 12 (2d)^k \quad\Rightarrow\quad h(\mathcal{A}^*) := \varlimsup_{n\to\infty} \frac 1n \log \#(\mathcal{A}^*)_{n} \leq \frac 14 \log(2d). \end{align} $$

3.3.2 General results on specification and growth

To prove a lower bound on $h(X)$ , which we do in §3.3.3, we first prove some general results about shift spaces. The first of these is Lemma 2.2, which stated that the ‘one-step’ specification Property (IV) implies the ‘multi-step’ Property (IV $^{\prime }$ ). Following this, we prove that Property (IV $^{\prime }$ ) gives a uniform relationship between $\#\mathcal {G}_n$ and $e^{nh}$ , which can be extended to $\#\mathcal {L}_n$ using Conditions (I) and (II $^{\prime }$ ).

Proof of Lemma 2.2.

Given $k\in \mathbb {N}$ and $\mathcal {D} \subset \mathcal {L}$ , let $\mathbf {S}(k,\mathcal {D})$ be the statement that:

• • for every $n\in \mathbb {N}$ and $w^{(1)},\ldots , w^{(k)} \in \mathcal {G}_n$ , there exist $u^{(1)},\ldots , u^{(k-1)} \in \mathcal {L}_\tau $ such that $w^{(1)}u^{(1)}w^{(2)}\cdots u^{(k-1)} w^{(k)} \in \mathcal {D}$ .

Then, the hypothesis of the lemma is that $\mathbf {S}(2,\mathcal {G})$ is true and (IV $^{\prime }$ ) is the condition that $\mathbf {S}(k,\mathcal {L})$ is true for every $k\in \mathbb {N}$ .

First, observe that if $\mathbf {S}(k,\mathcal {L})$ is true, then $\mathbf {S}(j,\mathcal {L})$ is true for every $1\leq j\leq k$ , since given any $w^{(1)}, \ldots , w^{(j)} \in \mathcal {G}_n$ , we can simply add $k-j$ copies of $w^{(j)}$ to the list, apply $\mathbf {S}(k,\mathcal {L})$ to obtain $w^{(1)} u^{(1)} w^{(2)} \cdots u^{(j-1)} w^{(j)} u^{(j)} w^{(j+1)} \cdots u^{(k-1)} w^{(k)} \in \mathcal {L}$ , and then truncate to obtain the desired word.

To prove Condition (IV $^{\prime }$ ), it thus suffices to prove that $\mathbf {S}(2^\ell ,\mathcal {L})$ is true for every $\ell \in \mathbb {N}$ . We will prove the stronger statement $\mathbf {S}(2^\ell ,\mathcal {G})$ by induction. The base case $\ell =1$ is given. Suppose that $\mathbf {S}(2^\ell ,\mathcal {G})$ is true for some $\ell \in \mathbb {N}$ . Given $w^{(1)},\ldots , w^{(k)} \in \mathcal {G}_n$ for $k = 2^{\ell +1}$ , apply $\mathbf {S}(2^\ell ,\mathcal {G})$ twice:

• • first, to $w^{(1)},\ldots , w^{(2^\ell )}$ , getting connecting words $u^{(1)},\ldots , u^{(2^\ell -1)} \in \mathcal {L}_\tau $ such that $v^{(1)} := w^{(1)} u^{(1)} w^{(2)} \cdots u^{(2^\ell -1)} w^{(2^\ell )} \in \mathcal {G}$ ;

• • second, to $w^{(2^\ell +1)},\ldots , w^{(2^{\ell +1})}$ , getting $u^{(2^\ell +1)},\ldots , u^{(2^{\ell +1}-1)} \in \mathcal {L}_\tau $ such that ${v^{(2)} := w^{(1)} u^{(1)} w^{(2)} \cdots u^{(2^{\ell +1}-1)} w^{(2^{\ell +1})} \in \mathcal {G}}$ .

Then, apply $\mathbf {S}(2,\mathcal {G})$ to the words $v^{(1)},v^{(2)} \in \mathcal {G}_{2^\ell n + (2^\ell - 1)\tau }$ , obtaining $u^{(2^\ell )} \in \mathcal {L}_\tau $ such that $v^{(1)} u^{(2^\ell )} v^{(2)} \in \mathcal {G}$ , which proves that $\mathbf {S}(2^{\ell +1},\mathcal {G})$ is true.

Lemma 3.10. Let $\mathcal {L}$ be the language of a shift space and h its topological entropy. If $\mathcal {G} \subset \mathcal {L}$ satisfies Condition (IV $^{\prime }$ ), then writing $C_1 = e^{\tau h}$ , we have

(3.2) $$ \begin{align} \#\mathcal{G}_n \leq C_1 e^{nh} \quad\text{for all } n\in \mathbb{N}. \end{align} $$

Proof. Fix $n{\kern-1pt}\in{\kern-1pt} \mathbb {N}$ . Given $k{\kern-1pt}\in{\kern-1pt} \mathbb {N}$ , use Condition (IV $^{\prime }$ ) to define a map $(\mathcal {G}_n)^k \to \mathcal {L}_{kn+(k-1)\tau }$ by $(w^{(1)},\ldots , w^{(k)}) {\kern-1pt}\mapsto{\kern-1pt} w^{(1)} u^{(1)} w^{(2)} \cdots u^{(k-1)} w^{(k)}$ , where the connecting words ${u^{(i)} {\kern-1.5pt}\in{\kern-1pt} \mathcal {L}_\tau }$ depend on the words $w^{(i)}$ . This map is injective, so

$$ \begin{align*} \#\mathcal{L}_{k(n+\tau)} \geq \#\mathcal{L}_{kn+(k-1)\tau} \geq (\#\mathcal{G}_n)^k. \end{align*} $$

Taking logs and dividing by k, we get

$$ \begin{align*} \log \#\mathcal{G}_n \leq \frac 1k \log \#\mathcal{L}_{k(n+\tau)} = \frac{(n+\tau)}{k(n+\tau)} \log \#\mathcal{L}_{k(n+\tau)} \xrightarrow{k\to\infty} (n+\tau) h. \end{align*} $$

Taking exponentials proves the lemma.

For the next lemma, we recall from Condition (II $^{\prime }$ ) that $Q := (\sum _{i=0}^\infty \#\mathcal {C}^p_i e^{-ih})(\sum _{k=0}^\infty \#\mathcal {C}^s_k e^{-kh})$ . By convention, we include the empty word in both $\mathcal {C}^p$ and $\mathcal {C}^s$ , so $\#\mathcal {C}^p_0 {\kern-1pt}={\kern-1pt} \#\mathcal {C}^s_0 {\kern-1pt}={\kern-1pt} 1$ , and one can also write

(3.3) $$ \begin{align} Q = \bigg( 1 + \sum_{i=1}^\infty \#\mathcal{C}^p_i e^{-ih} \bigg) \bigg( 1 + \sum_{k=1}^\infty \#\mathcal{C}^s_k e^{-kh} \bigg). \end{align} $$

Lemma 3.11. Let $\mathcal {L}$ be the language of a shift space and h its topological entropy. Suppose $\mathcal {C}^p,\mathcal {G},\mathcal {C}^s \subset \mathcal {L}$ satisfy Conditions (I) and (II $^{\prime }$ ), so that $Q<\infty $ . If $\mathcal {G}$ satisfies (3.2), then we have

(3.4) $$ \begin{align} \#\mathcal{L}_n \leq C_1 Q e^{nh} \quad \text{for all } n\in \mathbb{N}. \end{align} $$

Proof. By Condition (I), we have $\mathcal {L}_n \subset \bigcup _{i=0}^n \bigcup _{k=0}^{n-i} \mathcal {C}^p_i \mathcal {G}_{n-(i+k)} \mathcal {C}^s_k$ , so

$$ \begin{align*} \begin{aligned} \#\mathcal{L}_n \leq \sum_{i=0}^n \sum_{k=0}^{n-i} \#\mathcal{C}^p_i \#\mathcal{G}_{n-(i+k)} \#\mathcal{C}^s_k \leq \sum_{i=0}^n \sum_{k=0}^{n-i} \#\mathcal{C}^p_i (C_1 e^{nh} e^{-ih} e^{-kh}) \#\mathcal{C}^s_k. \end{aligned}\\[-47pt] \end{align*} $$

3.3.3 Lower bounds on growth

Returning to the shift spaces $X_{\beta }^d$ , fix $(\beta ,d)$ such that $d\geq 3$ and $\beta \geq 8$ , or $d=2$ and $\beta \geq 8^+$ . For these parameter values, we saw in Lemmas 3.7 and 3.8 that $\mathcal {G}$ satisfies Condition (IV) with $\tau =1$ when $d\geq 3$ , and $\tau =2$ when $d=2$ . (We did not prove Condition (III) for $\beta < 12$ , but that will not matter for the results in this section.) By Lemma 2.2, $\mathcal {G}$ satisfies Condition (IV $^{\prime }$ ), so Lemma 3.10 gives

(3.5) $$ \begin{align} \#\mathcal{G}_n \leq \begin{cases} e^{(n+2) h} &\text{if } d=2, \\ e^{(n+1) h} &\text{if } d\geq 3. \end{cases} \end{align} $$

Observe that

$$ \begin{align*} A^4 \setminus \mathcal{G}_4 = \{ a^4 : a\in A \} \quad\Rightarrow\quad \#\mathcal{G}_4 = d^4 - d. \end{align*} $$

Thus, when $d\geq 3$ , we can use (3.5) to get

(3.6) $$ \begin{align} d^4 - d \leq \#\mathcal{G}_4 \leq e^{4h} e^h \leq d e^{4h} \quad\Rightarrow\quad d^3 - 1 \leq e^{4h}. \end{align} $$

Since $d^3 - 1> 2d$ for all $d\geq 3$ , we can deduce from (3.1) and (3.6) that

$$ \begin{align*} h(\mathcal{A}^*) \leq \dfrac 14 \log(2d) < \dfrac 14 \log(d^3 - 1) \leq h(X), \end{align*} $$

which verifies Condition (II) when $d\geq 3$ . We can also prove the second upper bound in (1.1) using Lemma 3.11 and the fact that (3.1) and (3.6) give

(3.7) $$ \begin{align} \sum_{n=1}^\infty \#(\mathcal{A}^*)_n e^{-nh} &= \sum_{k=1}^\infty \#(\mathcal{A}^*)_{4k} e^{-4kh} \leq \sum_{k=1}^\infty \frac 12 \frac{(2d)^k}{(d^3-1)^k} \nonumber\\ &= \dfrac 12 \cdot \dfrac {{2d}/({d^3-1})}{1- {2d}/({d^3-1})} = \dfrac{d}{d^3-2d-1}. \end{align} $$

It remains to verify Condition (II) when $d=2$ . Here, using $\mathcal {G}_4$ as in (3.6) produces the weaker bound $d^4 - d \leq d^2 e^{4h}$ because of the weaker inequality in (3.5), and it turns out that this is not enough to establish Condition (II). Thus, we work instead with $\mathcal {G}_8$ , observing that

$$ \begin{align*} A^8 \setminus \mathcal{G}_8 = \{ a^4 w : a\in A, w\in A^4\} \cup \{ vb^4 : v\in A^4, b\in A\} \cup \{u^4 : u\in A^2\}, \end{align*} $$

from which we conclude that

$$ \begin{align*} \#\mathcal{G}_8 \geq d^8 - 2d^5 - d^2 = 256 - 64 - 4 = 188. \end{align*} $$

Using this together with (3.5) and the fact that $e^{2h} \leq d^2 = 4$ gives

$$ \begin{align*} e^{8h} \geq e^{-2h} \#\mathcal{G}_8 \geq 47, \end{align*} $$

so we can establish Condition (II) using (3.1) to write

$$ \begin{align*} h(\mathcal{A}^*) \leq \frac 14 \log(4) = \frac 18 \log(16) < \frac 18 \log(47) \leq h. \end{align*} $$

Now, the first upper bound in (1.1) follows from Lemma 3.11 and

(3.8) $$ \begin{align} \sum_{k=1}^\infty \#(\mathcal{A}^*)_{4k} e^{-4kh} \leq \sum_{k=1}^\infty \frac 12 \cdot \frac{4^k}{(47)^{k/2}} = \frac 12 \cdot \dfrac {{4}/{\sqrt{47}}}{1- {4}/{\sqrt{47}}} = \frac{2}{\sqrt{47}-4}, \end{align} $$

since $(1+ {2}/({\sqrt {47}-4}))^2 \leq 2.892$ .

4.1 Structure of the proof

We will prove Theorem 2.3 via three propositions, which we state in the following and prove in §§4.2–4.4. These propositions use different combinations of the Conditions (I), (II $^{\prime }$ ), (III), and (IV $^{\prime }$ ). If all four conditions hold, then all three propositions apply and combine to establish Theorem 2.3.

Observe that Lemmas 3.10 and 3.11 already proved the bounds on $\#\mathcal {L}_n$ claimed in the statement of Theorem 2.3. Writing $C_2 := Q e^{\tau h}$ , we will record these as

(4.1) $$ \begin{align} e^{nh} \leq \#\mathcal{L}_n \leq C_2 e^{nh} \quad\text{for all } n\in \mathbb{N}. \end{align} $$

Given $i,k,n\in \mathbb {N}$ such that $i+k \leq n$ , define a map $\varphi _{i,k} \colon \mathcal {L}_n \to \mathcal {L}_{n-(i+k)}$ by

(4.2) $$ \begin{align} \varphi_{i,k}(w) = w_{i+1} w_{i+2} \cdots w_{n-k-1}. \end{align} $$

That is, $\varphi _{i,k}$ truncates $w$ by removing the first i and last k symbols. Also, given a collection of words $\mathcal {D}_n \subset \mathcal {L}_n$ , we will write $[\mathcal {D}_n] = \bigcup _{w\in \mathcal {D}_n} [w] \subset X$ for the union of the corresponding cylinders.

Proposition 4.1. Let X be a shift space with topological entropy h whose language $\mathcal {L}$ contains collections $\mathcal {C}^p,\mathcal {G},\mathcal {C}^s$ satisfying Conditions (I) and (II $^{\prime }$ ), and for which (4.1) holds.

Then, there exist $\theta>0$ and $M\in \mathbb {N}$ such that given any collection of words $\mathcal {D}_n \subset \mathcal {L}_n$ satisfying $\nu [\mathcal {D}_n] \geq \frac 12$ for some $n> 2M$ and some MME $\nu $ , there exist $i,k \in \{0,1,\ldots , M\}$ such that

(4.3) $$ \begin{align} \#(\varphi_{i,k}(\mathcal{D}_n) \cap \mathcal{G}) \geq \theta e^{nh}. \end{align} $$

In particular, there exists $j\in [n-2M, n] \cap \mathbb {N}$ such that

(4.4) $$ \begin{align} \#\mathcal{G}_j \geq \theta e^{nh}. \end{align} $$

The next proposition does not require Condition (I) or (II $^{\prime }$ ), but does require Condition (III).

Proposition 4.2. Let X be a shift space with topological entropy h whose language $\mathcal {L}$ satisfies (4.1) and contains a collection $\mathcal {G}$ satisfying Condition (III). Suppose that $\mathcal {G}$ satisfies (4.4) in the following sense: there are $\theta>0$ and $M \in \mathbb {N}$ such that for every $n\in \mathbb {N}$ , there exists $j\in [n-2M, n] \cap \mathbb {N}$ such that $\#\mathcal {G}_j \geq \theta e^{nh}$ .

Then, there exists an MME $\mu $ on X that has the following Gibbs-type property: there is $c>0$ such that

(4.5) $$ \begin{align} \mu[w] \geq ce^{-|w| h} \quad \text{for all } w\in \mathcal{G}, \end{align} $$

and such that with $T\in \mathbb {N}$ as in Condition (III), we have

(4.6) $$ \begin{align} \mu([u] \cap \sigma^{-(|u| + T)}[v]) \geq c e^{-(|u| + |v|) h} \quad \text{for all } u,v\in \mathcal{G}. \end{align} $$

4.2 Uniform counting bounds

In this section, we prove Proposition 4.1.

Proof. We use the following general facts about probability vectors [Reference WaltersWal82, Corollary 4.2.1 and Theorem 4.3(ii)]: first, if $\mathbf {p} := (p_1,\ldots , p_k)$ is a probability vector, then

(4.7) $$ \begin{align} H(\mathbf{p}) := \sum_{i=1}^k -p_i \log p_i \leq \log k; \end{align} $$

and second, if given $I \subset \{1,\ldots , k\}$ , we write $p_I := \sum _{i\in I} p_i$ and $\mathbf {p}_I = (p_i / p_I)_{i\in I}$ for the corresponding probability vector of conditional probabilities, then we have

(4.8) $$ \begin{align} H(\mathbf{p}) = H(p_I, p_{I^c}) + p_I H(\mathbf{p}_I) + p_{I^c} H(\mathbf{p}_{I^c}), \end{align} $$

where $I^c = \{1,\ldots , k\} \setminus I$ . In the setting of the proposition, the partition into n-cylinders is generating for $\sigma ^n$ , so we have

$$ \begin{align*} nh &= nh_\nu(\sigma) = h_\nu(\sigma^n) \leq \sum_{w\in \mathcal{L}_n} -\nu[w] \log\nu[w] \\ &\leq \log 2 + \nu[\mathcal{D}_n] \sum_{w\in \mathcal{D}_n} -\frac{\nu[w]}{\nu[\mathcal{D}_n]} \log \frac{\nu[w]}{\nu[\mathcal{D}_n]} + \nu[\mathcal{D}_n^c] \sum_{w\in \mathcal{D}_n^c} -\frac{\nu[w]}{\nu[\mathcal{D}_n^c]} \log \frac{\nu[w]}{\nu[\mathcal{D}_n^c]}, \end{align*} $$

where the last inequality uses (4.8) and (4.7) with $k=2$ . Applying (4.7) to the two sums and using the fact that $\#\mathcal {D}_n^c \leq \#\mathcal {L}_n$ , we get

$$ \begin{align*} nh &\leq \log 2 + \nu[\mathcal{D}_n] \log \#\mathcal{D}_n + (1-\nu[\mathcal{D}_n]) \log \#\mathcal{L}_n \\ &\leq \log 2 + \nu[\mathcal{D}_n] \log \#\mathcal{D}_n + nh + \log C_2 - \nu[\mathcal{D}_n] \log \#\mathcal{L}_n, \end{align*} $$

where the last inequality uses (4.1). This implies that

$$ \begin{align*} -\log 2 - \log C_2 \leq \nu[\mathcal{D}_n] \log (\#\mathcal{D}_n / \#\mathcal{L}_n) \leq \gamma \log (\#\mathcal{D}_n / \#\mathcal{L}_n), \end{align*} $$

since $\nu [\mathcal {D}_n] \geq \gamma $ and the logarithm is non-positive. We obtain

$$ \begin{align*} (\#\mathcal{D}_n/\#\mathcal{L}_n)^\gamma \geq (2C_2)^{-1} \quad\Rightarrow\quad \#\mathcal{D}_n \geq (2C_2)^{-1/\gamma} \#\mathcal{L}_n, \end{align*} $$

and since $\#\mathcal {L}_n \geq e^{nh}$ by (4.1), this proves Proposition 4.4 with $\alpha = (2C_2)^{-1/\gamma }$ .

Taking $\alpha $ corresponding to $\gamma = \frac 12$ in Proposition 4.4, the proof of Proposition 4.1 is completed by the following lemma.

Proof. Given $M\in \mathbb {N}$ , let $\mathbb {I}(M) := \{ (i,k) \in (\mathbb {N} \cup \{0\})^2 : \max (i,k)> M\}$ , and observe that by Condition (II $^{\prime }$ ), we have $\delta (M) := \sum _{(i,k) \in \mathbb {I}(M)} (\#\mathcal {C}_i^p e^{-ih}) (\#\mathcal {C}^s_k e^{-kh}) \to 0$ as ${M \to \infty }$ . Also, consider for each $n>2M$ the quantity

(4.9) $$ \begin{align} r_n(M) := \max \{ \#(\varphi_{i,k}(\mathcal{D}_n) \cap \mathcal{G}) : 0\leq i,k\leq M \}. \end{align} $$

We must prove that there exists $M\in \mathbb {N}$ and $\theta>0$ such that $r_n(M) \geq \theta e^{nh}$ for all n. As in the proof of Lemma 3.11, we use Condition (I) to get

$$ \begin{align*} \mathcal{D}_n \subset \bigcup_{i=0}^n \bigcup_{k=0}^{n-i} \mathcal{C}^p_i ( \varphi_{i,k}(\mathcal{D}_n) \cap \mathcal{G}) \mathcal{C}^s_k. \end{align*} $$

Writing $\mathbb {I}_n(M) := \{ (i,k) \in \mathbb {I}(M) : i+k \leq n\}$ and using the upper bound on $\#\mathcal {G}_j$ , we obtain for each $M\in \mathbb {N}$ the inequality

$$ \begin{align*} \#\mathcal{D}_n &\leq \sum_{i=0}^M \sum_{k=0}^M (\#\mathcal{C}^p_i) \#(\varphi_{i,k}(\mathcal{D}_n) \cap \mathcal{G})(\#\mathcal{C}^s_k) + \sum_{(i,k) \in \mathbb{I}_n(M)} (\#\mathcal{C}^p_i)(\#\mathcal{G}_{n-(i+k)})(\#\mathcal{C}^s_k) \\ &\leq \sum_{i=0}^M \sum_{k=0}^M (\#\mathcal{C}^p_i)(\#\mathcal{C}^s_k) r_n(M) + \sum_{(i,k) \in \mathbb{I}_n(M)} (\#\mathcal{C}^p_i) C_1 e^{nh} e^{-ih} e^{-kh} (\#\mathcal{C}^s_k). \end{align*} $$

Writing $S(M) = (\sum _{i=0}^M \#\mathcal {C}^p_i)(\sum _{k=0}^M \#\mathcal {C}^s_k)$ , we use the inequality $\#\mathcal {D}_n \geq \alpha e^{nh}$ to get

$$ \begin{align*} \alpha e^{nh} \leq S(M) r_n(M) + \delta(M) C_1 e^{nh}. \end{align*} $$

Since $\delta (M) \to 0$ as $M\to \infty $ , we can take M sufficiently large that $\delta (M) C_1 < \alpha $ and obtain

$$ \begin{align*} r_n(M) \geq S(M)^{-1}(\alpha - \delta(M) C_1) e^{nh}, \end{align*} $$

which proves the lemma.

4.3 A Gibbs-type measure

In this section, we prove Proposition 4.2 by constructing an MME that has the Gibbs-type properties (4.5) and (4.6). Note that the first of these is a special case of the second, by taking $v$ to be the empty word, so we will only give the proof for (4.6).

The construction of $\mu $ , as well as the proof that it is invariant and is an MME, follows the usual argument from Misiurewicz’s proof of the variational principle: for each $n\in \mathbb {N}$ , let $\nu _n$ be a Borel probability measure on X with the property that $\nu _n[w] = 1/\#\mathcal {L}_n$ for every $w\in \mathcal {L}_n$ ; then let $\mu _n := (1/n) \sum _{k=0}^{n-1} \sigma _*^k \nu _n$ and let $\mu $ be any weak*-limit point of the sequence $\mu _n$ . See [Reference WaltersWal82, Theorem 8.6] for the argument that $\mu $ is invariant and is an MME.

It remains to prove (4.6). The following argument can be extracted from [Reference Climenhaga and ThompsonCT12, Lemma 5.15], but since the hypotheses are stated differently here, we give the details. Figure 2 illustrates the bookkeeping involved in the following computations by showing which indices are included in which words.

Figure 2

Getting the Gibbs bound from Property (III) (color online).

Figure 2 Long description

The diagram is bounded by a horizontal line from 0 on the far left to n on the far right.

From left to right, the segments are organized as follows.

First segment. A red double-headed arrow labeled k spans the top. Below it is a green block labeled w. Underneath w, there is a small gray block y on the left, a large yellow block w prime in the center, and a small gray block q on the right. A red double-headed arrow labeled k prime spans the width of w prime. A blue double-headed arrow labeled T spans the width of q.

Second segment. A solid pink block labeled u.

Third segment. A blue double-headed arrow labeled T spans a gap at the top. Below the gap is a gray block labeled p.

Fourth segment. A solid purple block labeled v.

Fifth segment. A red double-headed arrow labeled j spans the top. Below it is a green block labeled x. Underneath x, there is a small gray block r on the left, a large yellow block x prime in the center, and a small gray block z on the right. A blue double-headed arrow labeled T spans the width of r. A red double-headed arrow labeled j prime spans the width of x prime.

Observe that given $u,v\in \mathcal {G}$ and $n\in \mathbb {N}$ , for each $0 \leq k < n-|u|-T-|v|$ , we can write $j = n - k - |u| - T - |v|$ and obtain

(4.10) $$ \begin{align} \sigma_*^k \nu_n ([u] \cap \sigma^{-(|u|+T)}[v]) = \frac{ \#\{ (w,p,x) \in \mathcal{L}_k \times \mathcal{L}_T\times \mathcal{L}_j : wupvx \in \mathcal{L} \}} {\#\mathcal{L}_n}. \end{align} $$

If $k,j\geq 2M$ , then (4.4) gives $k' \in [k-T -2M, k-T]$ and $j' \in [j-T-2M, j-T]$ such that

(4.11) $$ \begin{align} \#\mathcal{G}_{k'} \geq \theta e^{(k-T)h} \quad\text{and}\quad \#\mathcal{G}_{j'} \geq \theta e^{(j-T)h}. \end{align} $$

For every $w'\in \mathcal {G}_{k'}$ and $x' \in \mathcal {G}_{j'}$ , Condition (III) gives $q,p,r \in \mathcal {L}_T$ such that $w'qupvrx' \in \mathcal {L}$ . Thus, there exist $y,z\in \mathcal {L}$ such that

$$ \begin{align*} w := yw'q \in \mathcal{L}_k,\quad x := rx'z \in \mathcal{L}_j,\quad\text{and}\quad wupvx = yw'qupvrx'z \in \mathcal{L}_n. \end{align*} $$

Together with (4.10), (4.11), and the bound $\#\mathcal {L}_n \leq C_2 e^{nh}$ from (4.1), this gives

$$ \begin{align*} \sigma_*^k \nu_n ([u] \cap \sigma^{-(|u|+T)}[v]) \geq \theta^2 e^{-2Th} e^{jh+kh} C_2^{-1} e^{-nh} = c e^{-(|u|+|v|)h}, \end{align*} $$

where $c := \theta ^2 e^{-3Th} C_2^{-1}$ . This bound holds for all $2M \leq k < n - |u| - T - |v| - 2M$ , so upon averaging over all $0\leq k < n$ , we get

$$ \begin{align*} \mu_n([u] \cap \sigma^{-(|u|+T)}[v]) \geq \frac{n-|u|-T-|v|-4M}{n} \cdot c e^{-(|u|+|v|)h}. \end{align*} $$

Sending $n\to \infty $ along an appropriate subsequence proves (4.6).

4.4 Uniqueness

In this section, we prove Proposition 4.3. This is where the new ideas from [Reference Pacifico, Yang and YangPYY22] become essential; the arguments in §§4.2 and 4.3 closely mirror those in [Reference Climenhaga and ThompsonCT12] (although the map $\varphi _{i,k}$ and the estimate (4.3) did not appear there), but to prove that $\mu $ is ergodic and that there is no other MME, the arguments in [Reference Climenhaga and ThompsonCT12] required an extra condition. Thus, we now follow [Reference Pacifico, Yang and YangPYY22], although with substantial simplifications, since that paper is written for flows on compact metric spaces and deals with equilibrium states for a broader class of potential functions.

The key innovation leading to the improvement over [Reference Climenhaga and ThompsonCT12] is the next lemma, which follows [Reference Pacifico, Yang and YangPYY22, Lemma 7.3] and holds in any shift space.

Proof. Since $\nu _1 \perp \nu _2$ , there are disjoint Borel sets $P_1,P_2 \subset X$ such that $\nu _\ell (P_\ell )=1$ for $\ell =1,2$ . Fix compact sets $K_\ell \subset P_\ell $ such that $\nu _\ell (K_\ell ) \geq 1-\zeta $ . The distance between $K_1$ and $K_2$ is positive, so there exists $n_0\in \mathbb {N}$ such that given any $u\in \mathcal {L}_{2n_0+1}$ , the centered cylinder $\sigma ^{n_0} [u]$ intersects at most one of $K_1$ and $K_2$ .

We claim that $\mathcal {U}_n = \{ w \in \mathcal {L}_n : [w] \cap \sigma ^{-\lfloor n/2 \rfloor } K_1 \neq \emptyset \}$ satisfies the conclusions of the lemma. Clearly, $[\mathcal {U}_n] \supset \sigma ^{-\lfloor n/2 \rfloor }K_1$ and since $\nu _1$ is invariant, we get

$$ \begin{align*} \nu_1[\mathcal{U}_n] \geq \nu_1(\sigma^{-\lfloor n/2 \rfloor} K_1) = \nu_1(K_1) \geq 1-\zeta, \end{align*} $$

which proves the first claim. For the second claim, we write $k = \lfloor n/2 \rfloor $ , and start by showing that given any $i,j \leq k - n_0$ and any $w\in \mathcal {U}_n$ , we have

(4.12) $$ \begin{align} \sigma^{k-i} [\varphi_{i,j} (w)] \cap K_2 = \emptyset. \end{align} $$

Writing $v = \varphi _{i,j}(w)$ and $u = w_{[k-n_0,k+n_0]}$ , as shown in Figure 3, we have

$$ \begin{align*} \sigma^k[w] \subset \sigma^{k-i}[v] \subset \sigma^{n_0} [u]. \end{align*} $$

Since $\sigma ^k[w]$ intersects $K_1$ , we see that $\sigma ^{n_0}[u]$ does as well, so $\sigma ^{n_0}[u] \cap K_2 = \emptyset $ by our choice of $n_0$ . This proves (4.12), which gives $\sigma ^{k-i}[\varphi _{i,j}\mathcal {U}_n] \subset X \setminus K_2$ . By our choice of $K_2$ and invariance of $\nu _2$ , we now have

$$ \begin{align*} \begin{aligned} \nu_2([\varphi_{i,j}\mathcal{U}_n]) = \nu_2(\sigma^{k-i}[\varphi_{i,j}\mathcal{U}_n]) \leq \nu_2(X\setminus K_2) < \zeta. \end{aligned} \\[-58pt] \end{align*} $$

Figure 3

Words in $\mathcal {U}_n$ miss $K_2$ (color online).

Now, we prove uniqueness of the MME $\mu $ by showing that there is no MME $\nu \perp \mu $ and that $\mu $ is ergodic (so the only MME $\nu \ll \mu $ is $\mu $ itself). The first of these follows [Reference Pacifico, Yang and YangPYY22, Proposition 7.2]. Recall that c is the constant in the lower Gibbs bounds (4.5) and (4.6), and $\theta ,M$ are given by Proposition 4.1.

Proof. Suppose for a contradiction that $\nu \perp \mu $ is an MME. Fix $\zeta < \min (\frac 12, c\theta )$ , and apply Lemma 4.6 with $\nu _1 = \nu $ and $\nu _2 = \mu $ to get $\mathcal {U}\subset \mathcal {L}$ and $n_0\in \mathbb {N}$ such that $\nu ([\mathcal {U}_n]) \geq 1-\zeta $ for all n, and

(4.13) $$ \begin{align} \mu([\varphi_{i,j} \mathcal{U}_n]) \leq \zeta \end{align} $$

for all $i,j,n$ satisfying $n \geq 2(n_0 + \max (i,j))$ .

Since $\zeta <\frac 12$ , we have $\nu ([\mathcal {U}_n])> \frac 12$ for all n and since $\nu $ is an MME, Proposition 4.1 gives $i,j\in \{0,1,\ldots , M\}$ (depending on n) with

$$ \begin{align*} \#(\varphi_{i,j}(\mathcal{U}_n) \cap \mathcal{G}) \geq \theta e^{nh}; \end{align*} $$

thus, the Gibbs property (4.5) on $\mathcal {G}$ gives

$$ \begin{align*} \mu([\varphi_{i,j} (\mathcal{U}_n) \cap \mathcal{G}]) \geq c e^{-(n-(i+j)) h} \#(\varphi_{i,j}(\mathcal{U}_n) \cap \mathcal{G}) \geq c e^{(i+j) h} \theta \geq c\theta. \end{align*} $$

For all $n\geq 2(n_0 + M)$ and this choice of $i,j$ , we also have (4.13), which leads to

$$ \begin{align*} \zeta \geq \mu([\varphi_{i,j} \mathcal{U}_n]) \geq \mu([\varphi_{i,j} (\mathcal{U}_n) \cap \mathcal{G}]) \geq c\theta. \end{align*} $$

This contradicts our choice of $\zeta $ and completes the proof of Proposition 4.7.

The proof of ergodicity follows [Reference Pacifico, Yang and YangPYY22, Proposition 7.4].

Proof. Suppose for a contradiction that $P\subset X$ is invariant and $0 < \mu (P) < 1$ . Let $\nu $ and $\nu '$ be the normalizations of $\mu $ restricted to P and $P^c$ , respectively. Fix $\zeta < \min (\frac 12, \frac 12 c\theta ^2)$ and apply Lemma 4.6 twice—once with $\nu _1=\nu $ and $\nu _2=\nu '$ , and once with the roles reversed—to obtain $\mathcal {U},\mathcal {U}' \subset \mathcal {L}$ and $n_0 \in \mathbb {N}$ such that $\nu ([\mathcal {U}_n]) \geq 1-\zeta $ and $\nu '([\mathcal {U}^{\prime }_n]) \geq 1-\zeta $ for every n, and moreover,

(4.14) $$ \begin{align} \nu([\varphi_{i,j}\mathcal{U}_n']) \leq \zeta \quad \text{and} \quad \nu'([\varphi_{i,j}\mathcal{U}_n]) \leq \zeta \end{align} $$

for all $i,j,n$ satisfying $n \geq 2(n_0 + \max (i,j))$ .

Each of $\nu $ and $\nu '$ is an MME since entropy depends affinely on the measure, and the MME $\mu $ is a convex combination of $\nu $ and $\nu '$ . Thus, we can apply Proposition 4.1 to both $\mathcal {U}$ and $\mathcal {U}'$ , obtaining $i,j,i',j' \in \{0,1,\ldots , M\}$ (depending on n) such that

$$ \begin{align*} \mathcal{V}(n) := \varphi_{i,j}(\mathcal{U}_n) \cap \mathcal{G} \subset \mathcal{L}_{n-(i+j)} \quad \text{and} \quad \mathcal{V}'(n) := \varphi_{i',j'}(\mathcal{U}_n')\cap \mathcal{G} \subset \mathcal{L}_{n-(i'+j')} \end{align*} $$

both have cardinality at least $\theta e^{nh}$ .

Let $k = n-(i+j) + T$ , so that $|v| + T = k$ for every $v\in \mathcal {V}(n)$ . Given any such $v$ and any $v' \in \mathcal {V}'(n)$ , the ‘two-step’ Gibbs property in (4.6) gives

$$ \begin{align*} \mu([v] \cap \sigma^{-k}[v']) \geq c e^{-(|v| + |v'|)h} \geq c e^{-2nh}. \end{align*} $$

Summing over all such $v$ and $v'$ gives

(4.15) $$ \begin{align} \mu([\mathcal{V}(n)] \cap \sigma^{-k} [\mathcal{V}'(n)]) \geq c e^{-2nh} (\#\mathcal{V}(n)) (\#\mathcal{V}'(n)) \geq c \theta^2. \end{align} $$

We will bound this measure above, for a contradiction. Given $x\in [\mathcal {V}(n)] \cap \sigma ^{-k}[\mathcal {V}'(n)]$ , we either have $x\in P^c$ or $x\in P = \sigma ^{-k} P$ (since P is invariant) and, thus,

(4.16) $$ \begin{align} [\mathcal{V}(n)] \cap \sigma^{-k}[\mathcal{V}'(n)] \subset ([\mathcal{V}(n)] \cap P^c) \cup \sigma^{-k} ([\mathcal{V}'(n)] \cap P). \end{align} $$

Observe that whenever $n\geq 2(n_0 + M)$ , (4.14) gives

$$ \begin{align*} \mu([\mathcal{V}(n)] \cap P^c) = \nu'([\mathcal{V}(n)]) \mu(P^c) \leq \nu'([\varphi_{i,j}(\mathcal{U}_n)]) \leq \zeta \end{align*} $$

and similarly

$$ \begin{align*} \mu(\sigma^{-k}([\mathcal{V}'(n)] \cap P)) = \mu([\mathcal{V}'(n)] \cap P) \leq \zeta, \end{align*} $$

so that by (4.16), we have

$$ \begin{align*} \mu([\mathcal{V}(n)] \cap \sigma^{-k}[\mathcal{V}'(n)]) \leq 2\zeta. \end{align*} $$

Comparing this with (4.15) gives $2\zeta \geq c\theta ^2$ , which contradicts our choice of $\zeta $ and completing the proof of Proposition 4.8.