2.1 Subshift and factor map

Let $\mathcal {B}$ be a finite set. Let $ k \in \mathbb {N} $ and $ 0 \leq a < b \leq k $ . For $ x = (x_{i})_{i=1}^{\infty } \in \mathcal {B}^{\mathbb {N}}$ , write $ x|_{a}^{b} = x_{a+1} \cdots x_{b}$ and $ x|k = x_{1}\cdots x_{k} $ . By convention, we set $ x|0 = \varnothing $ and $\lvert \varnothing \rvert = 0 $ , where $\varnothing $ denotes the empty word. For $ I = i_{1}\cdots i_{k} \in \mathcal {B}^{k} $ , define $ I|_{a}^{b} = i_{a+1}\cdots i_{b}$ , $ \lvert I\rvert = k $ , and $ [I] = \{ x \in \mathcal {B}^{\mathbb {N}} \colon x|k = I \}$ . For $ I \in \mathcal {B}^{k_{1}}$ and $ J \in \mathcal {B}^{k_{2}}$ , let $ IJ $ denote the juxtaposition of $ I$ and J. Endow $\mathcal {B}^{\mathbb {N}}$ with the product topology. Let $ \sigma $ be the shift on $\mathcal {B}^{\mathbb {N}}$ defined by $ \sigma ( (x_{k})_{k=1}^{\infty } ) = (x_{k+1})_{k=1}^{\infty } $ . A closed subset $ X $ of $\mathcal {B}^{\mathbb {N}}$ is called a subshift of $ \mathcal {B}^{\mathbb {N}}$ if $\sigma (X) \subset X $ . Let $ \mathcal {M}_{\sigma }(X) $ denote the set of $\sigma $ -invariant probability measures on $ X $ . Define

$$ \begin{align*} \mathcal{L}_{k}(X_{}) = \{ I \in \mathcal{B}^{k} \colon [I] \operatorname*{\mathrm{\cap}} X \neq \emptyset \} ,\,\, k\in\mathbb{N}\quad\text{and}\quad {\mathcal{L}(X)} = \operatorname*{\mathrm{\bigcup}}_{k=1}^{\infty} \mathcal{L}_{k}(X_{}) \cup \{\varnothing\}. \end{align*} $$

Recall $\mathcal {A} = \prod _{j=1}^{d}\{ 0, \ldots , m_{j}-1 \}$ and $ s = \# \{ m_{i} \colon 1 \leq i \leq d \} $ . Throughout this paper, a subshift refers by default to a subshift of $ \mathcal {A}^{\mathbb {N}} $ . Since $ m_{1} \geq \cdots \geq m_{d} \geq 2 $ , there are integers $ n_{1}> \cdots > n_{s} $ and $ 0 = d_{0} < d_{1} < \cdots < d_{s} = d $ such that for $ 1 \leq i \leq s $ ,

(2.1) $$ \begin{align} m_{j} = n_{i}, \quad d_{i-1} < j \leq d_{i}. \end{align} $$

By convention, we set $n_{0} = \infty $ and $\mathcal {A}_{0} = \mathcal {A} $ . For $ 1 \leq i \leq s $ , define

$$ \begin{align*} \mathcal{A}_{i} = \prod_{j>d_{i-1}} \{0, \ldots, m_{j}-1\}. \end{align*} $$

Let $ \pi _{1} \colon \mathcal {A} \to \mathcal {A}_{1}$ be the identity map. For $ 2 \leq i \leq s $ , let $ \pi _{i} \colon \mathcal {A}_{i-1} \to \mathcal {A}_{i} $ be the projection by removing first $ (d_{i-1} - d_{i-2}) $ coordinates, that is,

$$ \begin{align*} \pi_{i}( (x_{d_{i-2}+1}, \ldots, x_{d_{i-1}}, x_{d_{i-1}+1}, \ldots, x_{d})) = (x_{d_{i-1}+1}, \ldots, x_{d}). \end{align*} $$

Naturally extend $ \pi _{i} $ from $ \mathcal {A}_{i-1}$ to $\mathcal {A}_{i-1}^{\mathbb {N}}$ by

$$ \begin{align*} \pi_{i}(x) = (\pi_{i}(x_{k})) \quad\text{for } x = (x_{k}) \in \mathcal{A}_{i-1}^{\mathbb{N}}. \end{align*} $$

For $ 1 \leq i \leq s $ , define $ \tau _{i} \colon \mathcal {A}^{\mathbb {N}} \to \mathcal {A}_{i}^{\mathbb {N}}$ by $ \tau _{i} = \pi _{i} \circ \cdots \circ \pi _{1} $ . By abuse of notation, we keep using $\sigma $ to denote the shift on $\mathcal {A}_{i}^{\mathbb {N}}$ for $1 \leq i \leq s $ , which shall not cause confusion since the domain of $ \sigma $ is always clear from the context. Then, $ \pi _{i} \circ \sigma = \sigma \circ \pi _{i} $ and $ \tau _{i} \circ \sigma = \sigma \circ \tau _{i} $ , which means $ \pi _{i}$ and $\tau _{i}$ are factor maps. For a subshift $ X $ and $ 1 \leq i \leq s$ , define a subshift of $ \mathcal {A}_{i}^{\mathbb {N}}$ by $ X_{i} = \tau _{i}(X) $ . Hence, the following diagram commutes.

For $1 \leq i \leq s $ , by Definition 2.1 and the surjectivity of $ \tau _{i}$ , if $ X $ satisfies weak specification, then so does $ X_{i} $ .

2.2 Measure of maximal entropy

Let $ \mathcal {B} $ be a finite set and $ X $ be a subshift of $ \mathcal {B}^{\mathbb {N}}$ . The topological entropy $ h(X)$ of $ X $ is

(2.2) $$ \begin{align} h(X) = \lim_{k\to\infty} \frac{1}{k} \log \# \mathcal{L}_{k}(X), \end{align} $$

where the limit exists by a subadditivity argument. For $ \mu \in \mathcal {M}_{\sigma }(X) $ , the measure-theoretic entropy $h(\mu )$ of $\mu $ is

(2.3) $$ \begin{align} h(\mu) = \lim_{k\to\infty} \frac{1}{k} \sum_{I \in \mathcal{L}_{k}(X)} - \mu(I) \log \mu(I), \end{align} $$

where the limit exists by the invariance of $ \mu $ and a subadditivity argument. Note that $ h(\mu ) \leq h(X)$ since $ \sum _{I \in \mathcal {L}_{k}(X)} - \mu (I) \log \mu (I) \leq \log \# \mathcal {L}_{k}(X)$ by the convexity. We call $\mu $ a measure of maximal entropy if $h(\mu ) = h(X)$ .

Suppose X satisfies weak specification. Applying [Reference Feng16, Theorem 5.5] with $ \phi \equiv 1 $ on $ X $ shows that there is a unique measure of maximal entropy with the Gibbs property, which is an extension of the classical result of Parry [Reference Parry35].

Proposition 2.2. Let $ X $ be a subshift satisfying weak specification. Then, there is a unique measure of maximal entropy $ \mu $ on $ X $ . Moreover, $ \mu $ is the unique ergodic measure satisfying the following Gibbs property:

$$ \begin{align*} \mu(I) \approx \frac{1}{\#\mathcal{L}_{\lvert I\rvert}(X)} \approx \exp (- \lvert I\rvert\, h(X) ) \quad \text{for } I \in \mathcal{L}(X), \end{align*} $$

where $ h(X) $ is as in (2.2).

2.3 Measure of full dimension

Roughly speaking, a measure of full dimension is an invariant measure having the same Hausdorff dimension as its support. It is an important topic in dynamical systems to determine the existence and uniqueness of the measure of full dimension on invariant sets [Reference Barral and Feng2, Reference Feng16, Reference Gatzouras and Peres22, Reference Kenyon and Peres26, Reference Luzia30, Reference Olivier33, Reference Rams39]; see [Reference Gatzouras and Peres21] for a survey. In their celebrated work, Das and Simmons [Reference Das and Simmons9] constructed a self-affine sponge whose Hausdorff dimension is strictly greater than the supremum of the Hausdorff dimensions of its invariant measures; thus showing that the measure of full dimension may not exist in an expanding non-conformal system. For other dimensional properties of invariant measures in non-conformal settings, very recently, Kolossváry [Reference Kolossváry28] computed the $L^{q}$ dimensions of self-affine measures on self-affine sponges assuming a suitable separation condition, which generalizes the results in [Reference King27, Reference Olsen34].

For a subshift $X $ and $K = R(X)$ , we call an invariant measure $ \mu $ on $ X $ a measure of full dimension if $ \dim _{\mathrm {H}} R\mu = \dim _{\mathrm {H}} K $ . Kenyon and Peres [Reference Kenyon and Peres26] established a Ledrappier–Young formula for the Hausdorff dimensions of ergodic measures and proved the existence of the measure of full dimension. For $ \mu \in \mathcal {M}_{\sigma }(\mathcal {A}^{\mathbb {N}}) $ , following [Reference Kenyon and Peres26], we define

(2.4) $$ \begin{align} \mathrm{d}_{\mathrm{LY}}(\mu) = \sum_{i=1}^{s} \bigg ( \frac{1}{\log n_{i}} - \frac{1}{\log n_{i-1}}\bigg ) h(\tau_{i}\mu). \end{align} $$

Proposition 2.3. [Reference Kenyon and Peres26, Theorem 1.1 and Lemma 4.3]

Let $ X $ be any subshift and $ {K = R(X)}$ . Then, for each ergodic measure $ \mu \in \mathcal {M}_{\sigma }(X) $ ,

(2.5) $$ \begin{align} \dim_{\mathrm{H}} R\mu = \mathrm{d}_{\mathrm{LY}}(\mu) \end{align} $$

and

(2.6) $$ \begin{align} \dim_{\mathrm{H}} K = \max_{\mu\in\mathcal{M}_{\sigma}(X)} \mathrm{d}_{\mathrm{LY}}(\mu). \end{align} $$

By studying the optimization problem in (2.6), a part of the so-called weighted theormodynamic formalism, Feng [Reference Feng16] proved that when $ X $ satisfies weak specification, there is a unique measure of full dimension with certain lower Gibbs property. Before stating this result in Proposition 2.4, we need some notation. For $ 1\leq i \leq s $ , write

(2.7) $$ \begin{align} \alpha_{i} = \frac{\log n_{i}}{\log n_{i-1}}. \end{align} $$

Let X be a subshift. Define $ \phi ^{(1)} \equiv 1 $ on $ \mathcal {L}(X) $ . For $ 2 \leq i \leq s $ , recursively define $ \phi ^{(i)} \colon \mathcal {L}(X_{i}) \to (0,\infty )$ by

(2.8) $$ \begin{align} \phi^{(i)}(J) = \sum_{I \in \mathcal{L}(X_{i-1}) \colon \pi_{i}(I) = J } \phi^{(i-1)}(I)^{\alpha_{i-1}} \quad\text{for } J \in \mathcal{L}(X_{i}). \end{align} $$

Then, define $ Z \colon \mathbb {N} \to (0, \infty ) $ as

$$ \begin{align*} Z(k) = \sum_{I \in \mathcal{L}_{k}(X_{s})} \phi^{(s)}(I)^{\alpha_{s}} \end{align*} $$

and

(2.9) $$ \begin{align} P = \lim_{k\to\infty} \frac{\log Z(k)}{k}, \end{align} $$

where the limit exists by a subadditivity argument.

Proposition 2.4. [Reference Feng16, Theorems 5.5 and 7.3]

Let $ X $ be a subshift satisfying weak specification and $ K = R(X)$ . Then, there is a unique measure of full dimension $ \eta $ on X, that is,

$$ \begin{align*} \dim_{\mathrm{H}} R\eta = \mathrm{d}_{\mathrm{LY}}(\eta) = \max_{\mu\in\mathcal{M}_{\sigma}(X)} \mathrm{d}_{\mathrm{LY}}(\mu) = \dim_{\mathrm{H}} K. \end{align*} $$

Moreover, $ \eta $ is ergodic and has the following lower Gibbs property:

(2.10) $$ \begin{align} \eta(I) \gtrsim \psi(I) \quad \text{for } I \in \mathcal{L}(X), \end{align} $$

where

$$ \begin{align*} \psi(I) = \frac{1}{Z(\lvert I\rvert)} \prod_{i = 1}^{s} \phi^{(i)}(\tau_{i}(I))^{\alpha_{i}-1}. \end{align*} $$

Additionally, $ \tau _{s}\eta $ has the following Gibbs property:

(2.11) $$ \begin{align} \tau_{s}\eta(J) \approx \frac{\phi^{(s)}(J)^{\alpha_{s}}}{\exp(\lvert J\rvert P)} \quad \text{for } J \in \mathcal{L}(X_{s}), \end{align} $$

where P is as in (2.9).

Combining Proposition 2.3 and [Reference Feng16, Proposition 3.7] (a relative variational principle for subadditive potentials) gives a formula for the Hausdorff dimension of all compact T-invariant sets, an analog of which for certain sofic affine-invariant sets was previously established in [Reference Kenyon and Peres25, Theorem 1.1].

Lemma 2.5. Let $ X $ be a subshift and $ K = R(X) $ . Then,

(2.12) $$ \begin{align} \dim_{\mathrm{H}} K = \frac{P}{\log n_{s}}, \end{align} $$

where P is as in (2.9).

2.4 Approximate cube

In this subsection, we introduce the approximate cubes with side lengths $ \approx n_{s}^{-k} $ ( $k\in \mathbb {N}$ ) by ‘slowing down’ the iterations along the stronger unstable directions. For $ 1 \leq i \leq s $ , write

(2.13) $$ \begin{align} \theta_{i} = \frac{\log n_{s}}{\log n_{i}}. \end{align} $$

For $ k \in \mathbb {N} $ , define $R_{k} \colon \operatorname *{\mathrm {\bigcup }}_{n \geq k}\mathcal {A}^{n} \to \mathbb {T}^{d} $ by

(2.14) $$ \begin{align} R_{k}(I) := \sum_{\ell=1}^{k} \Lambda^{-\ell} i_{\ell} \quad \text{for } I = i_{1} \cdots i_{k}\cdots i_{n}. \end{align} $$

Combining (1.2) and (2.14) gives

(2.15) $$ \begin{align} R(x) = R_{k}(x|k) + \Lambda^{-k} R(\sigma^{k}x) \quad\text{for } x \in \mathcal{A}^{\mathbb{N}}. \end{align} $$

Let $ \lfloor x \rfloor $ denote the integer part of $ x \in \mathbb {R} $ . For $ I \in \mathcal {A}^{k} $ , the $ k $ -level approximate cube is

(2.16) $$ \begin{align} Q_{k}(I) : = \prod_{i = 1}^{s} \prod_{j = d_{i-1} + 1}^{d_{i}} [R_{\lfloor \theta_{i} k \rfloor }(I)_{j}, R_{\lfloor \theta_{i} k \rfloor }(I)_{j} + n_{i}^{-\lfloor \theta_{i} k \rfloor } ), \end{align} $$

where $ v_{j} $ denotes the $ j $ th coordinate of a vector $ v \in \mathbb {R}^{d} $ . For $ x \in \mathcal {A}^{\mathbb {N}} $ , we define $ Q_{k}(x) = Q_{k}(x|k) $ . Combining (2.1), (2.13), and (2.16) shows that $ \lvert Q_{k}(x) \rvert \approx n_{s}^{-k} $ .

We begin with a lemma relating the measure of approximate cubes to that of a collection of cylinders in the symbolic space. For $ x \in X $ and $ k \in \mathbb {N} $ , define

(2.17) $$ \begin{align} \Gamma_{k}(x) = \{ I \in \mathcal{L}_{k}(X) \colon \tau_{i}(I|\lfloor \theta_{i} k \rfloor ) = \tau_{i}(x|\lfloor \theta_{i} k \rfloor ) \text{ for } 1 \leq i \leq s \}. \end{align} $$

Lemma 2.6. Let X be a subshift and $ \mu $ be a Borel measure on $ X $ . Then, for $ x \in X $ ,

(2.18) $$ \begin{align} R\mu(Q_{k}(x)) = \sum_{I \in \Gamma_{k}(x)} \mu(I). \end{align} $$

Proof. Let $ K = R(X)$ . It follows from (2.16) and (2.17) that

$$ \begin{align*} R^{-1}(Q_{k}(x) \operatorname*{\mathrm{\cap}} K) \subset \operatorname*{\mathrm{\bigcup}}_{I \in \Gamma_{k}(x)} [I] \end{align*} $$

and

$$ \begin{align*} R^{-1}(Q_{k}(x) \operatorname*{\mathrm{\cap}} K ) \operatorname*{\mathrm{\bigcap}} \, [I] \neq \emptyset \quad \text{for } I \in \Gamma_{k}(x). \end{align*} $$

Since $ R\mu $ is supported on $ K $ ,

$$ \begin{align*} R\mu(Q_{k}(x)) = R\mu(Q_{k}(x) \operatorname*{\mathrm{\cap}} K) = \sum_{I\in \Gamma_{k}(x)} \mu(I), \end{align*} $$

which finishes the proof.

Next, we give a lemma for counting the number of all $ k $ -level approximate cubes. Denote

(2.19) $$ \begin{align} \mathcal{Q}_{k}(X) = \{ Q_{k}(x) \colon x \in X \}. \end{align} $$

Then, $ \mathcal {Q}_{k}(X) $ covers $ K $ by the $ k $ -level approximate cubes. To estimate $ \#\mathcal {Q}_{k}(X) $ , following [Reference McMullen32], we introduce a collection of vectors

(2.20) $$ \begin{align} \mathcal{D}_{k}(X) := \bigg \{ (J_{1}, \ldots, J_{s}) \in \prod_{i=1}^{s} & \mathcal{L}_{\lfloor \theta_{i} k \rfloor -\lfloor \theta_{i-1} k \rfloor } (X_{i}) \colon \text{there exists } I \in \mathcal{L}_{k}(X) \notag\\ & \text{such that }\tau_{i}(I|_{\lfloor \theta_{i-1} k \rfloor }^{\lfloor \theta_{i} k \rfloor }) = J_{i} \text{ for } 1 \leq i \leq s \bigg\}. \end{align} $$

Proof. It suffices to construct a bijective map $ f \colon \mathcal {Q}_{k}(X) \to \mathcal {D}_{k}(X) $ . For $ Q \in \mathcal {Q}_{k}(X) $ , there exists $ x \in X $ such that $ Q = Q_{k}(x) $ . Define

$$ \begin{align*} f(Q) := (\tau_{1}(x|_{\lfloor \theta_{-1} k \rfloor}^{\lfloor \theta_{1} k \rfloor}), \ldots, \tau_{s}(x|_{\lfloor \theta_{s-1} k \rfloor }^{\lfloor \theta_{s} k \rfloor })). \end{align*} $$

Then, $ f $ is injective since for $ x, y \in X $ , $ Q_{k}(x) = Q_{k}(y) $ if and only if $ \tau _{i}(x|_{\lfloor \theta _{i-1} k \rfloor }^{\lfloor \theta _{i} k \rfloor }) = \tau _{i}(y|_{\lfloor \theta _{i-1} k \rfloor }^{\lfloor \theta _{i} k \rfloor }) $ for $ 1 \leq i \leq s $ . Next, we show that $ f $ is surjective. For $ \mathbf {J} = (J_{1}, \ldots , J_{s}) \in \mathcal {D}_{k}(X) $ , by (2.20), there is $ I \in \mathcal {L}_{k}(X) $ such that $ \tau _{i}(I|_{\lfloor \theta _{i-1} k \rfloor }^{\lfloor \theta _{i} k \rfloor }) = J_{i} $ for $ 1 \leq i \leq s $ . Take some $ x \in [I] \operatorname *{\mathrm {\cap }} X $ , then $ f(Q(x)) = \mathbf {J} $ .

An upper bound on the upper box dimension of $ R(X) $ follows immediately.

Lemma 2.8. Let $ X $ be a subshift and $ K = R(X) $ . Then,

$$ \begin{align*} \overline{\dim}_{\mathrm{B}} K \leq \sum_{i=1}^{s}\bigg (\frac{1}{\log n_{i}}-\frac{1}{\log n_{i-1}}\bigg) h(X_{i}). \end{align*} $$

Proof. By definition of $ \mathcal {D}_{k}(X) $ ,

(2.21) $$ \begin{align} \log \# \mathcal{D}_{k}(X) \leq \log \# \bigg (\prod_{i=1}^{s} \mathcal{L}_{\lfloor \theta_{i} k \rfloor -\lfloor \theta_{i-1} k \rfloor } (X_{i}) \bigg ) \leq \sum_{i=1}^{s} \log \# \mathcal{L}_{\lfloor \theta_{i} k \rfloor -\lfloor \theta_{i-1} k \rfloor } (X_{i}). \end{align} $$

For $ 1\leq i \leq s $ , since $ h(X_{i}) = \lim _{k\to \infty } (1/k) \log \# \mathcal {L}_{k}(X_{i}) $ and $ \theta _{i} - \theta _{i-1}> 0 $ ,

(2.22) $$ \begin{align} &\lim_{k\to\infty} \frac{\log \# \mathcal{L}_{\lfloor \theta_{i} k \rfloor -\lfloor \theta_{i-1} k \rfloor } (X_{i})}{k}\notag\\ &\quad = \lim_{k\to\infty} \frac{\lfloor \theta_{i} k \rfloor - \lfloor \theta_{i-1} k \rfloor }{k} \cdot \frac{\log \# \mathcal{L}_{\lfloor \theta_{i} k \rfloor -\lfloor \theta_{i-1} k \rfloor } (X_{i})}{ \lfloor \theta_{i} k \rfloor - \lfloor \theta_{i-1} k \rfloor } \notag\\ &\quad = (\theta_{i}-\theta_{i-1}) h(X_{i}). \end{align} $$

Then,

$$ \begin{align*} \overline{\dim}_{\mathrm{B}} K & \leq \limsup_{k\to\infty} \frac{\log \# \mathcal{Q}_{k}(X) }{ \log n_{s}^{k} } \\ & = \limsup_{k\to\infty} \frac{\log \# \mathcal{D}_{k}(X) }{ \log n_{s}^{k} } & \text{(by Lemma}~{2.7})\\ & \leq \frac{1}{\log n_{s}} \sum_{i=1}^{s} \limsup_{k\to\infty} \frac{\log \# \mathcal{L}_{\lfloor \theta_{i} k \rfloor -\lfloor \theta_{i-1} k \rfloor } (X_{i})}{k} & \text{(by}~({2.21}))\\ & = \frac{1}{\log n_{s}} \bigg (\sum_{i=1}^{s}(\theta_{i}-\theta_{i-1}) h( X_{i} )\bigg ) & \text{(by}~({2.22})), \end{align*} $$

which finishes the proof.

We end this section with the following variant of the Rogers–Taylor theorem [Reference Rogers and Taylor40], which is a useful tool to estimate the Hausdorff measure.

3.1 The equivalence of Theorem 3.1(a) and (b)

We first compute the box dimension, which relies on the next lemma about the number of approximate cubes.

Lemma 3.2. Let $ X $ be a subshift satisfying weak specification. Then, for $ k \in \mathbb {N} $ ,

$$ \begin{align*} \# \mathcal{Q}_{k}(X) \approx \exp\bigg (k \sum_{i=1}^{s} (\theta_{i} - \theta_{i-1}) h(X_{i})\bigg ). \end{align*} $$

Proof. For $ 1\leq i \leq s $ , applying Proposition 2.2 to $ X_{i} $ gives

(3.1) $$ \begin{align} \#\mathcal{L}_{k}(X_{i}) \approx \exp(k h(X_{i})). \end{align} $$

By Lemma 2.7 and (2.20),

$$ \begin{align*} \# \mathcal{Q}_{k}(X) = \# \mathcal{D}_{k}(X) \leq \prod_{i=1}^{s} \# \mathcal{L}_{\lfloor \theta_{i} k \rfloor -\lfloor \theta_{i-1} k \rfloor } (X_{i}) \approx \exp\bigg (k \sum_{i=1}^{s} (\theta_{i} - \theta_{i-1}) h(X_{i})\bigg ). \end{align*} $$

Next, we control $ \# \mathcal {Q}_{k}(X) $ from below. Let $ p $ be as in Definition 2.1. For $ k \in \mathbb {N} $ large, define

(3.2) $$ \begin{align} \mathcal{J}_{k} = \prod_{i=1}^{s} \mathcal{L}_{\lfloor \theta_{i} k \rfloor -\lfloor \theta_{i-1} k \rfloor -p}(X_{i}) \end{align} $$

and

$$ \begin{align*} \mathcal{W} = \bigg \{\prod_{i=1}^{s} (W_{i}^{l}, W_{i}^{r}) \in \prod_{i=1}^{s}\mathcal{L}(X) \times \mathcal{L}(X) \colon \lvert W_{i}^{l}\rvert + \lvert W_{i}^{r}\rvert = p \bigg \}. \end{align*} $$

We proceed to construct a map $ g $ from a subset of $ {\mathcal {J}_{k} \times \mathcal {W}} $ to $ \mathcal {D}_{k}(X) $ . For $ {\mathbf {J} = (J_{1}, \ldots , J_{s}) \in \mathcal {J}_{k}} $ , by the surjectivity of $\tau _{i}$ , there exists $ \mathbf {I} = (I_{1}, \ldots , I_{s}) \in \mathcal {L}(X)^{s}$ such that $ \tau _{i}(I_{i}) = J_{i} $ for $ 1 \leq i \leq s $ . Then, we define $ W_{i}^{l}$ and $ W_{i}^{r} $ recursively for $ 1 \leq i \leq s $ . Take $ W_{1}^{l} = \varnothing $ and $ W_{1}^{r} \in \mathcal {L}_{p}(X) $ such that $ I_{1} W_{1}^{r} \in \mathcal {L}(X) $ . Suppose for some $ 1 \leq i \leq s-1 $ , we have found $ W_{1}^{l}, W_{1}^{r}, \ldots , W_{i}^{l}, W_{i}^{r}$ such that

$$ \begin{align*} W_{1}^{l}I_{1}W_{1}^{r} \cdots W_{i}^{l}I_{i}W_{i}^{r} \in \mathcal{L}(X) \quad\text{and}\quad \lvert W_{j}^{l}\rvert + \lvert W_{j}^{r}\rvert = p \quad\text{for } 1\leq j \leq i. \end{align*} $$

By Definition 2.1, there exists $ W_{i+1}^{l} \in \mathcal {L}(X) $ with $ \lvert W_{i+1}^{l}\rvert \leq p $ such that $ W_{i}^{r} W_{i+1}^{l} I_{i+1} \in \mathcal {L}(X) $ . Take $ W_{i+1}^{r} \in \mathcal {L}_{p - \lvert W_{i+1}^{l}\rvert }(X_{}) $ such that $ I_{i+1}W_{i+1}^{r} \in \mathcal {L}(X) $ . Hence, we find $ {\mathbf {W} = ( W_{1}^{l}, W_{1}^{r}, \ldots , W_{s}^{l}, W_{s}^{r})} $ such that

(3.3) $$ \begin{align} W_{1}^{l} I_{1} W_{1}^{r} \cdots W_{s}^{l} I_{s} W_{s}^{r} \in \mathcal{L}(X) \quad\text{and}\quad \lvert W_{i}^{l}\rvert + \lvert W_{i}^{r}\rvert = p \quad\text{for } 1 \leq i \leq s. \end{align} $$

Thus, for $ \mathbf {J} \in \mathcal {J}_{k} $ , we find some $ \mathbf {W} \in \mathcal {W} $ as above and define

$$ \begin{align*} g(\mathbf{J}, \mathbf{W}) = \prod_{i = 1}^{s} \tau_{i}(W_{i}^{l} I_{i} W_{i}^{r} ). \end{align*} $$

Denote the domain and range of g by $\mathrm {Dom}(g)$ and $\mathrm {Ran}(g)$ , respectively. Since $ g $ is defined for all $ \mathbf {J} \in \mathcal {J}_{k} $ , we have $ \# \mathrm {Dom}(g) \geq \# \mathcal {J}_{k} $ . By (3.3) and (2.20), ${\mathrm {Dom}(g) \subset \mathcal {J}_{k}\times \mathcal {W} }$ and $\mathrm {Ran}(g) \subset \mathcal {D}_{k}(X) $ . Note that $ \# g^{-1}( \mathbf { U}) \leq \# \mathcal {W} $ for $ \mathbf {U} \in \mathrm {Ran}(g) $ by the definition of $ g $ . Hence,

$$ \begin{align*} \# \mathcal{Q}_{k}(X) & = \# \mathcal{D}_{k}(X) && \text{(by Lemma}~{2.7}) \\ & \geq \# \mathrm{Ran}(g) && \text{(by } \mathrm{Ran}(g) \subset \mathcal{D}_{k}(X) )\\ & \geq \frac{\# \mathrm{Dom}(g)}{\# \mathcal{W} } && \text{(by } \# g^{-1}( \mathbf{U}) \leq \# \mathcal{W},\, \mathbf{U} \in \mathrm{Ran}(g) )\\ & \geq \frac{\# \mathcal{J}_{k}}{\# \mathcal{W} } && \text{(by } \# \mathrm{Dom}(g) \geq \# \mathcal{J}_{k} ) \\ & \gtrsim \prod_{i=1}^{s} \# \mathcal{L}_{\lfloor \theta_{i} k \rfloor -\lfloor \theta_{i-1} k \rfloor -p}(X_{i}) && \text{(by}~({3.2}) \text{ and } \# \mathcal{W} \leq (\# \mathcal{A})^{2p})\\ & \approx \exp\bigg (k \sum_{i=1}^{s} (\theta_{i} - \theta_{i-1}) h(X_{i})\bigg ) && \text{(by}~({3.1})), \end{align*} $$

for $ k $ large enough depending on p.

Since $ \mathcal {Q}_{k}(X) $ covers $ K $ by approximate cubes with side length $ \approx n_{s}^{-k} $ , Lemma 3.2 implies the following formula for the box dimension of K.

Proposition 3.3. Let $ X $ be a subshift satisfying weak specification and $ K = R(X)$ . Then,

(3.4) $$ \begin{align} \dim_{\mathrm{B}} K = \sum_{i=1}^{s} \bigg ( \frac{1}{\log n_{i}} - \frac{1}{\log n_{i-1}}\bigg ) h(X_{i}), \end{align} $$

where $ h(X_{i}) $ is as in (2.2).

It is worth pointing out that, recently, Jurga [Reference Jurga23] showed that the box dimension of $ K = R(X) $ may not exist if $ X $ does not satisfy weak specification.

The equivalence of Theorem 3.1(a) and (b) follows immediately.

Proof of the equivalence of Theorem 3.1(a) and (b)

By Proposition 2.3, let $ \eta $ be an ergodic measure on $ X $ such that $ \mathrm {d}_{\mathrm {LY}}(\eta ) = \dim _{\mathrm {H}} K $ . By (2.4) and Proposition 3.3, Theorem 3.1(a) is equivalent to

(3.5) $$ \begin{align} \sum_{i=1}^{s} \bigg (\frac{1}{n_{i}} - \frac{1}{n_{i-1}}\bigg ) ( h(X_{i}) - h(\tau_{i} \eta) ) = 0. \end{align} $$

Since $ h(\tau _{i}\eta ) \leq h(X_{i}) $ and $ n_{i} < n_{i-1} $ for $ 1\leq i \leq s $ , (3.5) holds if and only if

$$ \begin{align*} h(X_{i}) = h(\tau_{i}\eta) \quad\text{for } 1 \leq i \leq s. \end{align*} $$

The proof is finished by Proposition 2.2.

3.2 The equivalence of Theorem 3.1(b), (c), and (d)

Proof of the equivalence of Theorem 3.1(b), (c), and (d)

By Proposition 2.2, for $ 1\leq i \leq s $ and $ I \in \mathcal {L}_{k}(X_{i}) $ ,

(3.6) $$ \begin{align} \mu_{i}(I) \approx \frac{1}{\# \mathcal{L}_{k}(X_{i})} \approx \exp(-k h(X_{i})). \end{align} $$

Fix any $ i \in \{1, \ldots , s\} $ and $ k \in \mathbb {N} $ .

If (b) holds, then for $ I \in \mathcal {L}_{k}(X_{i})$ ,

$$ \begin{align*} \exp(-k h(X_{i})) \approx \mu_{i}(I) = \tau_{i}\mu_{1}(I) = \sum_{J \in \tau_{i}^{-1}(I) } \mu(J) \approx \# \tau_{i}^{-1}(I) \cdot \exp(-k h(X_{1})). \end{align*} $$

This implies statement (c).

It is immediate that statement (d) follows from statement (c) by taking $ \unicode{x3bb} _{i} = h(X_{1}) - h(X_{i})$ .

Next, we show that statement (d) implies statement (b). By (3.6) and (d), for $ I \in \mathcal {L}_{k}(X_{i})$ ,

(3.7)

Then,

Since $\# \mathcal {L}_{k}(X_{i}) \approx \exp (kh(X_{i})) $ by (3.6), we have $ \unicode{x3bb} _{i} = h(X_{1}) - h(X_{i}) $ . Combining (3.7) and (3.6) gives

$$ \begin{align*} \tau_{i}\mu_{1}(I) \approx \exp(-kh(X_{i})) \approx \mu_{i}(I). \end{align*} $$

This shows (b) by Proposition 2.2.

5.1 Proof of Theorem 1.1(ii)

We first show that the Hausdorff measures of R-images of cylinders with the same length are comparable to each other.

Lemma 5.1. Let $ X $ be a subshift satisfying weak specification and $ K = R(X)$ . Let $ \varphi $ be a gauge function. Then, for $ k \in \mathbb {N} $ and $ I \in \mathcal {L}_{k}(X)$ ,

$$ \begin{align*} \mathcal{H}^{\varphi}(R([I]) \operatorname*{\mathrm{\cap}} K ) \approx \mathcal{H}^{\varphi}(\Lambda^{-k} K). \end{align*} $$

Proof. For $ J \in \mathcal {L}(X)$ , write $ \mathcal {D}_{J} = \{ j \in \mathcal {A} \colon Jj \in \mathcal {L}(X)\}$ . By the translation-invariance of $ \mathcal {H}^{\varphi }$ and (2.15),

(5.1) $$ \begin{align} \mathcal{H}^{\varphi}(R([J]\operatorname*{\mathrm{\cap}} X)) = \sum_{j\in\mathcal{D}_{J}} \mathcal{H}^{\varphi}(R([Jj]\operatorname*{\mathrm{\cap}} X)) = \sum_{j\in\mathcal{D}_{J}} \mathcal{H}^{\varphi}(\Lambda^{-\lvert J\rvert}R([j]\operatorname*{\mathrm{\cap}} X)). \end{align} $$

Then, since $ \mathcal {D}_{I} \subset \mathcal {L}_{1}(X)$ , $ X = \operatorname *{\mathrm {\bigcup }}_{j\in \mathcal {L}_{1}(X)} [j]\operatorname *{\mathrm {\cap }} X$ , and $ K = R(X)$ ,

(5.2) $$ \begin{align} \mathcal{H}^{\varphi}(R([I]\operatorname*{\mathrm{\cap}} X)) \leq \sum_{j\in\mathcal{L}_{1}(X)} \mathcal{H}^{\varphi}(\Lambda^{-k}R([j]\operatorname*{\mathrm{\cap}} X)) = \mathcal{H}^{\varphi}(\Lambda^{-k}R(X)) = \mathcal{H}^{\varphi}(\Lambda^{-k}K). \end{align} $$

Next, we give a lower bound on $\mathcal {H}^{\varphi }(R([I]\operatorname *{\mathrm {\cap }} X))$ . Let $ p $ be as in Definition 2.1. For $ j \in \mathcal {L}_{1}(X) $ , there exists $ W \in \mathcal {L}(X)$ with $ \lvert W\rvert \leq p $ such that $ I W j \in \mathcal {L}(X)$ . Then,

(5.3) $$ \begin{align} \begin{aligned} \mathcal{H}^{\varphi}(R([I]\operatorname*{\mathrm{\cap}} X)) & \geq \mathcal{H}^{\varphi}(R( [IWj] \operatorname*{\mathrm{\cap}} X ) ) \\ & \geq \mathcal{H}^{\varphi}(\Lambda^{-k-\lvert W\rvert} R([j]\operatorname*{\mathrm{\cap}} X) ) & \text{(by}~({5.1})) \\ & \geq \mathcal{H}^{\varphi}(\Lambda^{-k-p} R([j]\operatorname*{\mathrm{\cap}} X) ), \end{aligned} \end{align} $$

where the last inequality is by $ \lvert W\rvert \leq p $ and $ \Lambda ^{-1}$ is contracting. Summing (5.3) over $ j \in \mathcal {L}_{1}(X)$ gives

(5.4) $$ \begin{align} \mathcal{H}^{\varphi}(R([I]\operatorname*{\mathrm{\cap}} X)) \geq \frac{1}{\#\mathcal{A}}\mathcal{H}^{\varphi}(\Lambda^{-k-p} R(X)). \end{align} $$

Notice that $ \Lambda ^{-k} T^{-p}(E) = \Lambda ^{-k-p}E + \Lambda ^{-k} \sum _{J\in \mathcal {A}^{p}} R_{p}(J)$ for $ E \subset \mathbb {T}^{d}$ . By the translation-invariance of $\mathcal {H}^{\varphi }$ ,

$$ \begin{align*} \mathcal{H}^{\varphi}(\Lambda^{-k-p} E) = \frac{1}{(\# \mathcal{A})^{p}} \mathcal{H}^{\varphi}(\Lambda^{-k}T^{-p}(E)). \end{align*} $$

Then, by (5.4), taking $ E = R(X) $ in the above equation implies that

(5.5) $$ \begin{align} \mathcal{H}^{\varphi}(R([I]\operatorname*{\mathrm{\cap}} X)) \geq \frac{1}{(\#\mathcal{A})^{p+1}}\mathcal{H}^{\varphi}(\Lambda^{-k}T^{-p}(R(X))) \geq \frac{1}{(\#\mathcal{A})^{p+1}}\mathcal{H}^{\varphi}(\Lambda^{-k}K), \end{align} $$

where the last inequality is by $ K = R(X)$ and $ K \subset T^{-p}(K) $ . Combining (5.2) and (5.5) finishes the proof.

Now, we are ready to prove Theorem 1.1(ii), which is contained in the following proposition.

Proposition 5.2. Let X be a subshift satisfying weak specification and $ K = R(X)$ . Let $ \mu $ be the measure of maximal entropy on $ X $ . If $ 0 < \mathcal {H}^{\varphi }(K) < \infty $ for some gauge function $ \varphi $ , then

$$ \begin{align*} R\mu \approx \mathcal{H}^{\varphi}|_{K}, \end{align*} $$

where $ \mathcal {H}^{\varphi }|_{K} $ denotes the restriction of $ \mathcal {H}^{\varphi } $ to $ K $ . In particular, $ \dim _{\mathrm {H}} R\mu = \dim _{\mathrm {H}} K $ .

Proof. Fix any $ k \in \mathbb {N} $ . By Lemma 5.1,

(5.6) $$ \begin{align} \mathcal{H}^{\varphi}(R([I])\operatorname*{\mathrm{\cap}} K ) \approx \mathcal{H}^{\varphi}(R([J]) \operatorname*{\mathrm{\cap}} K) \quad\text{for } I, J \in \mathcal{L}_{k}(X). \end{align} $$

Since $ 0 < \mathcal {H}^{\varphi }(K) < \infty $ and $ \{ R([I])\}_{I\in \mathcal {L}_{k}(X)}$ covers $ K $ ,

(5.7) $$ \begin{align} 1 \approx \mathcal{H}^{\varphi}|_{K}(K) = \sum_{I \in \mathcal{L}_{k}(X)} \mathcal{H}^{\varphi}|_{K}(R([I])). \end{align} $$

Combining (5.6) and (5.7) gives

$$ \begin{align*} \mathcal{H}^{\varphi}|_{K}(R([I])) \approx \frac{1}{\# \mathcal{L}_{k}(X)} \quad \text{for } I \in \mathcal{L}_{k}(X). \end{align*} $$

Then, Proposition 2.2 implies that

$$ \begin{align*} R\mu(R([I])) = \mu(I) \approx \frac{1}{\# \mathcal{L}_{k}(X)} \approx \mathcal{H}^{\varphi}|_{K}(R([I])) \quad \text{for } I \in \mathcal{L}_{k}(X). \end{align*} $$

Note that $ R\mu (R([I])) = 0 = \mathcal {H}^{\varphi }|_{K}(R([I])) $ for $ I \in \mathcal {A}^{k}\setminus \mathcal {L}_{k}(X) $ since $\mu $ is supported on X. Hence,

$$ \begin{align*} R\mu(R([I])) \approx \mathcal{H}^{\varphi}|_{K}(R([I])) \quad \text{for all } I \in \mathcal{A}^{k}. \end{align*} $$

This finishes the proof because the collection of sets $ \{ R([I]) \colon I \in \operatorname *{\mathrm {\bigcup }}_{k=1}^{\infty } \mathcal {A}^{k} \}$ generates the Borel $ \sigma $ -algebra of $ \mathbb {T}^{d} $ .

5.2 Proof of Theorem 1.1(iii)

The proof of Theorem 1.1(iii) follows immediately by combining the following proposition with Theorem 1.1(i) and (ii).

Proof. By the definition of $ \mu $ and the assumption on $ \mu $ ,

$$ \begin{align*} h(\mu) = h(X) \quad\text{and}\quad \dim_{\mathrm{H}} R\mu = \dim_{\mathrm{H}} K. \end{align*} $$

When $ s = 1 $ . By Propositions 2.4 and 3.3,

$$ \begin{align*} \dim_{\mathrm{H}} K = \dim_{\mathrm{H}} R\mu = \mathrm{d}_{\mathrm{LY}}(\mu) = \frac{h(\mu)}{\log n_{1}} = \frac{h(X)}{\log n_{1}} = \dim_{\mathrm{B}} K. \end{align*} $$

Alternatively, this follows from the result of Furstenberg [Reference Furstenberg20] since $ T $ is conformal.

When $ s = 2 $ . Recall $ \theta _{1} = \alpha _{2} \theta _{2} = \alpha _{2} $ from (2.7) and (2.13). By Lemma 2.5,

(5.8) $$ \begin{align} \dim_{\mathrm{H}} K = \frac{P}{\log n_{2} }, \end{align} $$

where

(5.9) $$ \begin{align} P = \lim_{k\to\infty} \frac{1}{k} \log \sum_{J \in \mathcal{L}_{k}(X_{2})} (\# \pi_{2}^{-1}(J))^{\theta_{1}}. \end{align} $$

Let $k \in \mathbb {N}$ and $ J \in \mathcal {L}_{k}(X_{2})$ . Since $ \tau _{2} = \pi _{2} $ , Proposition 2.4 implies that

(5.10) $$ \begin{align} \pi_{2}\mu(J) \approx \frac{(\# \pi_{2}^{-1}(J))^{\theta_{1}}}{\exp(kP)}. \end{align} $$

Since $ \mu (I) \approx \exp (-k h(X_{1})) $ for $ I \in \mathcal {L}_{k}(X_{1})$ by Proposition 2.2,

(5.11) $$ \begin{align} \pi_{2}\mu(J) = \sum_{I \in \pi_{2}^{-1} (J)} \mu(I) \approx \frac{\# \pi_{2}^{-1}(J)}{ \exp(kh(X_{1}))}. \end{align} $$

Combining (5.10) and (5.11) gives

(5.12) $$ \begin{align} (\# \pi_{2}^{-1}(J))^{\theta_{1}} \approx \exp\bigg (k \frac{\theta_{1}}{1 - \theta_{1} } (h(X_{1}) - P)\bigg ). \end{align} $$

By Proposition 2.2,

(5.13) $$ \begin{align} \#\mathcal{L}_{k}(X_{2}) \approx \exp(kh(X_{2})). \end{align} $$

Applying (5.12) and (5.13) to (5.9), we solve that

$$ \begin{align*} P = \theta_{1} h(X_{1}) + (1-\theta_{1}) h(X_{2}). \end{align*} $$

This shows $ \dim _{\mathrm {H}} K = \dim _{\mathrm {B}} K $ by Proposition 3.3 and (5.8).

In the above proof, the assumption of $ s \leq 2 $ allows the potential $ \phi ^{(2)} $ as in (2.8) to be estimated by combining the Gibbs property of $ \mu $ and $\tau _{2} \mu $ . However, this argument does not extend to $ s \geq 3 $ due to the lack of the information about the regularity of $ \tau _{i} \mu $ for $ 2 \leq i \leq s - 1 $ . In fact, the examples in §7 show that Lemma 5.3 does not hold when $ s \geq 3 $ .

6.1 Dimensional results

In this subsection, we include some results of Kenyon and Peres about the dimensions of Bedford–McMullen sponges. Let $\mathcal {D}$ be a non-empty subset of $ \mathcal {A}$ . Define $ \mathcal {D}_{i} = \tau _{i} (\mathcal {D}) $ for $ 1\leq i \leq s $ . Define $ Z^{(1)} \equiv 1 $ on $\mathcal {D}_{1} $ . For $ 2 \leq i \leq s $ , recursively define

(6.1) $$ \begin{align} Z^{(i)}(x) = \sum_{y\in \mathcal{D}_{i-1} \colon \pi_{i}(y) = x } Z^{(i-1)}(y)^{\alpha_{i-1}} \quad\text{for } x \in \mathcal{D}_{i}. \end{align} $$

Define

(6.2) $$ \begin{align} Z = \sum_{x \in \mathcal{D}_{s}} Z^{(s)}(x)^{\alpha_{s}}. \end{align} $$

Proposition 6.1. [Reference Kenyon and Peres26, Theorem 1.2]

Let $ K = R(\mathcal {D}^{\mathbb {N}}) $ for some $ \emptyset \neq \mathcal {D} \subset \mathcal {A} $ . Then,

$$ \begin{align*} \dim_{\mathrm{H}} K = \frac{\log Z}{\log n_{s}}, \end{align*} $$

where $ Z $ is as in (6.2). There is a unique measure of full dimension $\eta _{p} $ which is the Bernoulli measure with marginal $ p = p(x)_{x\in \mathcal {D}} $ , where

(6.3) $$ \begin{align} p(x) = \frac{1}{Z} \prod_{i = 1}^{s} Z^{(i)}(\tau_{i}(x))^{\alpha_{i}-1} \quad\text{for } x \in \mathcal{D}. \end{align} $$

There is a simple criterion for Bedford–McMullen sponges to have equal Hausdorff and box dimensions. For $ 1 \leq i \leq s $ , define

(6.4) $$ \begin{align} f_{i}(x) = \#\pi_{i}^{-1}(\tau_{i}(x)) \quad\text{for } x \in \mathcal{D}. \end{align} $$

Proposition 6.2. [Reference Kenyon and Peres26, Proposition 1.3]

Let $ K = R(\mathcal {D}^{\mathbb {N}}) $ for some $\emptyset \neq \mathcal {D} \subset \mathcal {A} $ . Then, $\dim _{\mathrm {H}} K = \dim _{\mathrm {B}} K$ if and only if

(6.5) $$ \begin{align} f_{i}(x) = f_{i}(y) \quad \text{ for all } 1 \leq i \leq s \text{ and } x, y\in \mathcal{D}. \end{align} $$

As a direct corollary of Proposition 6.1, the following condition characterizes when the measures of maximal entropy and full dimension coincide.

Corollary 6.3. Let $ K = R(\mathcal {D}^{\mathbb {N}}) $ for some $ \emptyset \neq \mathcal {D} \subset \mathcal {A} $ . Then, the measures of maximal entropy and full dimension coincide if and only if

(6.6) $$ \begin{align} \prod_{i = 1}^{s} Z^{(i)}(\tau_{i}(x))^{\alpha_{i}-1} = \prod_{i = 1}^{s} Z^{(i)}(\tau_{i}(y))^{\alpha_{i}-1} \quad\text{for } x, y \in \mathcal{D}. \end{align} $$

When $ s \leq 3 $ , (6.6) is equivalent to

(6.7) $$ \begin{align} \prod_{i=1}^{s} f_{i}(x)^{\theta_{i-1}-1} = \prod_{i=1}^{s} f_{i}(y)^{\theta_{i-1}-1} \quad\text{for } x, y \in \mathcal{D}. \end{align} $$

Proof. Since the measure of maximal entropy on $ \mathcal {D}^{\mathbb {N}}$ is the Bernoulli measure with marginal $(1/\#\mathcal {D}, \ldots , 1/\#\mathcal {D})$ , the first equivalence follows directly from Proposition 6.1. Next, we show the second equivalence. Since $ f_{1}(x) = Z^{(1)}(x) = 1 $ and $ f_{2}(x) = Z^{(2)}(\tau _{2}(x)) $ for $ x\in \mathcal {D}$ , (6.6) and (6.7) are equivalent when $ s \leq 2 $ . Suppose $ s = 3 $ . Let $ x \in \mathcal {D}$ . For $ z \in \pi _{3}^{-1}(\tau _{3}(x)) $ , there exists $ y \in \mathcal {D} $ such that $ \tau _{2}(y) = z $ , and so $ \tau _{3}(y) = \pi _{3}(\tau _{2}(y)) = \pi _{3}(z) = \tau _{3}(x)$ . Then, either (6.6) or (6.7) implies $ f_{2}(x) = f_{2}(y) $ , and thus $ Z^{(2)}(z) = Z^{(2)}(\tau _{2}(y)) = f_{2}(y) = f_{2}(x)$ . Hence, by (6.1),

$$ \begin{align*} Z^{(3)}(\tau_{3}(x)) = \sum_{z \in \pi_{3}^{-1}(\tau_{3}(x))} Z^{(2)}(z)^{\alpha_{2}} = f_{3}(x) f_{2}(x)^{\alpha_{2}}. \end{align*} $$

Thus,

$$ \begin{align*} \prod_{i = 1}^{3} Z^{(i)}(\tau_{i}(x))^{\alpha_{i}-1} = f_{2}(x)^{\alpha_{2}-1} (f_{3}(x) f_{2}(x)^{\alpha_{2}} )^{\alpha_{3}-1} = f_{2}(x)^{\theta_{1}-1}f_{3}(x)^{\theta_{2}-1}, \end{align*} $$

which finishes the proof.

It is easy to see that (6.5) is equivalent to (6.7) if $ s \leq 2 $ , but stronger than (6.7) if $ s = 3 $ . This motivates us to give the examples in §7.

6.2 Proof of Theorem 1.2

Let $ X $ be a subshift and $\mu $ be a measure on X. For a gauge function $\varphi $ , define

$$ \begin{align*} \Theta_{k}^{\varphi}(\mu, x) = \frac{R\mu(Q_{k}(x))}{\varphi(n_{s}^{-k})} \quad\text{for } x \in X \text{ and } k\in \mathbb{N}. \end{align*} $$

Write $ \Theta _{k}^{\varphi } $ as $ \Theta _{k}^{\gamma } $ when $\varphi (r) = r^{\gamma }$ for some $ \gamma \geq 0 $ . Let $ E[X] $ and $ \operatorname {\mathrm {Var}}(X)$ respectively denote the expectation and variance of a random variable $ X $ .

Proof of Theorem 1.2

By Theorem 1.1(i), it remains to show that if $ 0 < \mathcal {H}^{\varphi }(K) < \infty $ for some gauge function $ \varphi $ , then $ \dim _{\mathrm {H}} K = \dim _{\mathrm {B}} K $ .

Suppose $ 0 < \mathcal {H}^{\varphi }(K) < \infty $ . Write $ K = R(\mathcal {D}^{\mathbb {N}})$ for some $ \mathcal {D} \subset \mathcal {A} $ . Let $ \mu $ be the measure of maximal entropy on $ \mathcal {D}^{\mathbb {N}} $ , that is, $ \mu $ is the Bernoulli measure with marginal $ ( 1/ \# \mathcal {D}, \ldots , 1/\# \mathcal {D})$ . Let $ x = (x_{j})_{j=1}^{\infty } \in \mathcal {D}^{\mathbb {N}}$ be a random sequence with law $\mu $ . For $ k \in \mathbb {N} $ and $ 1 \leq j \leq k $ , define the random variables

(6.8) $$ \begin{align} X_{j}^{(k)} = \log \# \tau_{i}^{-1}(\tau_{i}(x_{j})) \quad\text{for } 1 \leq i \leq s \text{ and } \lfloor \theta_{i-1} k \rfloor + 1 \leq j \leq \lfloor \theta_{i} k \rfloor. \end{align} $$

Note that $ x_{1}, \ldots , x_{k}$ are independent and identically distributed random variables with common law $ \sum _{y\in \mathcal {D}} 1/(\#\mathcal {D}) \delta _{y} $ . Then, $ X_{1}^{(k)}, \ldots , X_{k}^{(k)} $ are independent and uniformly bounded. Write $ E_{k} = \sum _{j=1}^{k} E[ X_{j}^{(k)}]$ and $ V_{k}^{2} = \sum _{j=1}^{k} \operatorname {\mathrm {Var}}( X_{j}^{(k)}) $ . For $ k \in \mathbb {N} $ , define

(6.9) $$ \begin{align} L(k) = \log ( (\#\mathcal{D})^{k} \varphi(n_{s}^{-k}) ). \end{align} $$

For $ x = (x_{j})_{j=1}^{\infty } \in \mathcal {D}^{\mathbb {N}} $ and $ k \in \mathbb {N} $ , (2.17) implies

(6.10) $$ \begin{align} \begin{aligned} \# \Gamma_{k}(x) & = \# \big\{ I \in \mathcal{L}_{k}(\mathcal{D}^{\mathbb{N}}) \colon \tau_{i}(I|_{\lfloor \theta_{i-1} k \rfloor }^{\lfloor \theta_{i} k \rfloor } ) = \tau_{i}(x|_{\lfloor \theta_{i-1} k \rfloor }^{\lfloor \theta_{i} k \rfloor }) \text{ for } 1 \leq i \leq s \big\} \\ & = \prod_{i=1}^{s}\prod_{j=\lfloor \theta_{i-1} k \rfloor +1}^{\lfloor \theta_{i} k \rfloor } \# \tau_{i}^{-1}(\tau_{i}(x_{j})). \end{aligned} \end{align} $$

Since $ \mu (I) = 1/(\#\mathcal {D})^{k}$ for $ I \in \mathcal {D}^{k}$ , Lemma 2.6 shows

(6.11) $$ \begin{align} \begin{aligned} R\mu(Q_{k}(x)) & = \sum_{I \in \Gamma_{k}(x)} \mu(I) = \frac{ \#\Gamma_{k}(x)}{(\# \mathcal{D})^{k}}. \end{aligned} \end{align} $$

Combining (6.8), (6.9), (6.10), and (6.11) gives

(6.12) $$ \begin{align} \Theta_{k}^{\varphi}(\mu, x) = \exp \bigg( \kern-2pt\sum_{j=1}^{k} X_{j}^{(k)} - L(k) \bigg) \quad\text{for } x \in \mathcal{D}^{\mathbb{N}}. \end{align} $$

By Proposition 5.2 and the upper density theorem for the Hausdorff measures (see Proposition B.1 or [Reference Davies and Samuels10]), there exists $ C> 0 $ such that

$$ \begin{align*} \mu \bigg \{ x \colon \frac{1}{C} \leq \limsup_{k\to\infty} \Theta_{k}^{\varphi}(\mu, x) \leq C \bigg\} = 1. \end{align*} $$

From this and (6.12), there exists $ M> 0 $ such that

(6.13) $$ \begin{align} \mathbb{P} \bigg \{ \limsup_{k\to\infty} \bigg\lvert\!\sum_{j=1}^{k} X_{j}^{(k)} - L(k) \bigg\rvert < M \bigg \} = 1, \end{align} $$

where $ \mathbb {P} $ denotes the underlying probability (that is, a probabilistic notation for $ \mu $ ).

Suppose on the contrary that $ \dim _{\mathrm {H}} K < \dim _{\mathrm {B}} K $ . It follows from Proposition 6.2 that $ \# \tau _{i_{0}}^{-1}(\tau _{i_{0}}(x)) \neq \# \tau _{i_{0}}^{-1}(\tau _{i_{0}}(y))$ for some $ 1 \leq i_{0} \leq s $ and $ x, y \in \mathcal {D}$ . Let $ Y $ be the random variable with law $ \sum _{x\in \mathcal {D}} (1/\#\mathcal {D}) \delta _{\# \tau _{i_{0}}^{-1}(\tau _{i_{0}}(x))}$ , then $ \operatorname {\mathrm {Var}}(Y)> 0 $ . By (6.8), $Y $ has the same law as $ X_{j}^{(k)}$ for $ \lfloor \theta _{i-1} k \rfloor +1 \leq j \leq \lfloor \theta _{i} k \rfloor $ . Then, by the independence of $ X_{j}^{(k)}$ ,

$$ \begin{align*} V_{k}^{2} \geq \sum_{j=\lfloor \theta_{i-1} k \rfloor +1}^{\lfloor \theta_{i} k \rfloor } \operatorname{\mathrm{Var}}(X_{j}^{(k)}) \gtrsim \operatorname{\mathrm{Var}}(Y) (\theta_{i} - \theta_{i-1}) k \, \to \infty, \end{align*} $$

as $ k \to \infty $ . By Lyapunov’s central limit theorem (see e.g. [Reference Chung7, Theorem 7.1.2]), the normalized random sum $ Y_{k} = (\sum _{j=1}^{k} X_{j}^{(k)} - E_{k})/ V_{k}$ converges to the standard normal random variable $ N(0,1) $ in distribution. Since the distribution of $ N(0,1)$ is absolutely continuous and $ \lim _{k\to \infty } M/V_{k} = 0 $ ,

(6.14) $$ \begin{align} \begin{aligned} \mathbb{P} \bigg\{ \bigg\lvert\kern-2pt\sum_{j=1}^{k} X_{j}^{(k)} - L(k) \bigg\rvert < M \bigg\} = \mathbb{P} \bigg\{ \bigg\lvert Y_{k} - \frac{L(k)-E_{k}}{V_{k}} \bigg\rvert < \frac{M}{V_{k}} \bigg\} \to 0, \end{aligned} \end{align} $$

as $ k \to \infty $ . This implies

$$ \begin{align*} \mathbb{P} \bigg \{\kern-3pt \limsup_{k\to\infty} \bigg\lvert\kern-3pt\sum_{j=1}^{k} X_{j}^{(k)} - L(k) \bigg\rvert < M \bigg \} \leq \limsup_{k\to\infty} \mathbb{P} \bigg \{ \bigg\lvert\kern-3pt\sum_{j=1}^{k} X_{j}^{(k)} - L(k) \bigg\rvert < M \bigg \} = 0, \end{align*} $$

which contradicts (6.13).

Remark 6.4. The proof of Theorem 1.2 works for some other subshift X satisfying weak specification if we can write $\log R\mu (Q_{k}(x))$ as a sequence of random variables such that a certain limit theorem can be applied to conclude an analog of (6.14). Next, we give an example to illustrate it. Let $m_{1}, m_{2}, m_{3}$ be integers such that $ m_{1}> m_{2} > m_{3} \geq 2, m_{1}, m_{2} \geq 3$ . Let $ \mathcal {A}, \theta _{i}, \pi _{i}, \tau _{i}, 1 \leq i \leq s = 3 $ be defined as in §2.1. Let

$$ \begin{align*} \mathcal{D} = \left \{ \begin{pmatrix}0 \\ 0 \\ 0\end{pmatrix}, \begin{pmatrix}0 \\ 1 \\ 0\end{pmatrix}, \begin{pmatrix}1 \\ 1 \\ 0\end{pmatrix}, \begin{pmatrix}0 \\ 2 \\ 1\end{pmatrix}, \begin{pmatrix}1 \\ 2 \\ 1\end{pmatrix}, \begin{pmatrix}2 \\ 2 \\ 1\end{pmatrix} \right \}. \end{align*} $$

Enumerate $\mathcal {D}$ as $ \{ z_{1}, \ldots , z_{6}\}$ according to the above order. Let $ A = (A_{i,j}) $ be a $ 6 \times 6 $ matrix such that $ A_{i,i} = 0 $ and $ A_{i,j} = 1$ if $ i \neq j$ for $ 1 \leq i, j \leq 6$ . Consider the following subshift of finite type:

$$ \begin{align*} \Sigma_{A} &= \{ (x_{k})_{k=1}^{\infty} \in \mathcal{D}^{\mathbb{N}} \colon \text{ for all }\, k \in \mathbb{N}, \\ &\qquad\text{there exists }\, 1\leq i, j\leq 6, \, x_{k} = z_{i}, x_{k+1} = z_{j}, \, A_{i,j} = 1 \}. \end{align*} $$

Let $ K = R(\Sigma _{A})$ . By [Reference Parry35] (see also [Reference Walters41]), the unique measure of maximal entropy $\mu $ on $ \Sigma _{A} $ is the Markov measure generated by the probability vector $ (1/6, \ldots , 1/6)$ and the random matrix $ A / 5 $ . By computation, $ \tau _{2}\mu $ is the Markov measure on $ \tau _{2}(\Sigma _{A})$ generated by $ (1/6, 1/3, 1/2)$ and $ \bigg (\!\begin {smallmatrix} 0 & 2/5 & 3/5 \\ 1/5& 1/5 & 3/5 \\ 1/5& 2/5 & 2/5 \end {smallmatrix}\!\bigg ) $ . Since the computation shows that $ \tau _{2}\mu $ is not the measure of maximal entropy on $ \tau _{2}(\Sigma _{A})$ , it follows from Theorem 3.1(b) that $ \dim _{\mathrm {H}} K < \dim _{\mathrm {B}} K $ . By the definition of $ \mathcal {D} $ , a similar estimate as in the proof of Theorem 1.2 gives

$$ \begin{align*} \log R\mu(Q_{k}(x)) \approx \theta_{1} k + 3 (1-\theta_{2}) k + \sum_{j = \lfloor \theta_{1} k \rfloor + 1}^{\lfloor \theta_{2} k \rfloor} Y((\tau_{2}x)_{j}), \end{align*} $$

where $ Y \colon \tau _{2}\mathcal {D} \to \{1,2,3\} $ is a function defined as $ Y((0, 0)^{t}) = 1, Y((1, 0)^{t}) = 2$ and $ Y((2, 1)^{t}) = 3 $ . Since the law of $\tau _{2}x$ is the Markov measure $ \tau _{2}\mu $ if $ x $ is a random sequence with law $ \mu $ , we can apply the central limit theorem for Markov chains (see e.g. [Reference Durrett11, Example 8.3.4]) and adapt the proof of Theorem 1.2 to conclude that $ \mathcal {H}^{\varphi }(K) = 0 $ or $\infty $ for every gauge function $ \varphi $ .

6.3 Infinite Hausdorff measure

In this subsection, we provide a sufficient condition (see Proposition 6.6) for Bedford–McMullen sponges to have infinite Hausdorff measures at their respective Hausdorff dimensions. This is based on the method of Peres [Reference Peres37] and a relation (see Lemma 6.8) satisfied by the measure of full dimension. In [Reference Peres37, §5], Peres mentioned that [Reference Peres37, Theorem 1 and Proposition 2] can extend to Bedford–McMullen sponges. However, he did not give a detailed justification. In fact, the proof of [Reference Peres37, Proposition 2] does not work for Bedford–McMullen sponges, as indicated by the examples in §7. For the extension of [Reference Peres37, Theorem 1], it is not clear how to construct a measure with zero density like that in the proof of [Reference Peres37, Theorem 1] since the measure of full dimension becomes more complicated when $ s \geq 3 $ (see Proposition 6.1).

Throughout this subsection, we fix a non-empty subset $ \mathcal {D} $ of $ \mathcal {A} $ and let $ K = R(\mathcal {D}^{\mathbb {N}})$ . Let $p = (p(x))_{x\in \mathcal {D}} $ be the probability vector given in (6.3). Write $ \gamma = \dim _{\mathrm {H}} K $ . For a finite set $ \mathcal {B} $ and probability vectors $ q = (q(x))_{x \in \mathcal {B}}$ and $ q' = (q'(x))_{x\in \mathcal {B}}$ , following [Reference Peres37], we define

(6.15) $$ \begin{align} \Delta(q\| q') = \sum_{x \in \mathcal{B}} (q(x)- q'(x)) \log q(x), \end{align} $$

where we require $ q'(x) = 0 $ if $ q(x) = 0 $ .

The following proposition is essentially contained in [Reference Peres37]. For completeness, we include a proof of Proposition 6.5 in Appendix A.

Proposition 6.5. (Peres [Reference Peres37])

If there exist a probability vector $ q $ on $\mathcal {D} $ and $ c> 0 $ such that

(6.16) $$ \begin{align} \sum_{i=1}^{s} \Delta(\tau_{i}p\| \tau_{i} q) \int_{\theta_{i-1}k}^{\theta_{i}k} \frac{1}{\log t} \,d t \geq c \frac{k}{(\log k)^{2}} \quad\text{for large } k \in \mathbb{N}, \end{align} $$

then $ \mathcal {H}^{\varphi }(K) = \infty $ , where

(6.17) $$ \begin{align} \varphi(r) = r^{\gamma} \exp\bigg(\kern-2pt-\widetilde{c} \frac{\lvert\log r\rvert}{(\log \lvert\log r\rvert)^{2}} \bigg), \quad r> 0, \end{align} $$

for some $ \widetilde {c}> 0 $ depending on $ c $ explicitly. In particular, $ \mathcal {H}^{\gamma }(K) = \infty $ .

In the remaining part of this subsection, we prove the following proposition which is a sufficient condition such that the Hausdorff measures of Bedford–McMullen sponges are infinite at their Hausdorff dimensions. We do not expect condition (6.18) to be necessary, but it is sufficient for determining the Hausdorff measure of the examples in §7 at their respective Hausdorff dimensions. For a probability vector $ q = (q(x))_{x\in \mathcal {B}}$ on a finite set $\mathcal {B} $ , we say $q $ is uniform if $q = (1/\#\mathcal {B}, \ldots , 1/\#\mathcal {B})$ .

Proposition 6.6. If

(6.18) $$ \begin{align} 1 \leq \# \{ 1 \leq i \leq s \colon \tau_{i} p \text{ is not uniform}\} \leq 2, \end{align} $$

then $ \mathcal {H}^{\dim _{\mathrm {H}} K}(K) = \infty $ .

Proposition 6.6 extends some results in [Reference Peres37] since for Bedford–McMullen carpets, the upper bound in (6.18) trivially holds when $ d = 2 $ , and the lower bound in (6.18) follows from $ \dim _{\mathrm {H}} K < \dim _{\mathrm {B}} K $ and Theorem 3.1(b). Next, we give two lemmas for the proof of Proposition 6.6.

Proof. The first statement is by a direct computation. The second statement follows from

$$ \begin{align*} \Delta(q\| q') = \frac{1}{2\#\mathcal{B}} \sum_{x\in\mathcal{B}} \sum_{y\in\mathcal{B}} (q(x)-q'(y))(\log q(x)- \log q'(y))> 0, \end{align*} $$

when $ q' $ is uniform.

The following lemma is the key ingredient for the verification of condition (6.16) when $ s \geq 3 $ .

Lemma 6.8. For every probability vector $ q $ on $ \mathcal {D} $ ,

(6.19) $$ \begin{align} \sum_{i=1}^{s} (\theta_{i}- \theta_{i-1})\Delta( \tau_{i}p\| \tau_{i} q) = 0. \end{align} $$

Proof. For $ 1\leq i \leq s $ , write $ p_{i} = \tau _{i}p $ , $ q_{i} = \tau _{i} q $ , and $\Delta _{i} = \Delta (p_{i}\| q_{i})$ . By (6.15),

$$ \begin{align*} \begin{aligned} \Delta_{i} & = \sum_{x \in \mathcal{D}_{i}} (p_{i}(x)-q_{i}(x)) \log p_{i}(x) &&\\ & = \sum_{x\in\mathcal{D}_{i}} \log p_{i}(x) \sum_{y\in\tau_{i}^{-1}(x)} (p_{1}(y) - q_{1}(y)) && \text{(by }p_{i}=\tau_{i}p_{1}, q_{i}=\tau_{i}q_{1})\\ & = \sum_{x\in\mathcal{D}_{i}} \sum_{y\in\tau_{i}^{-1}(x)} (p_{1}(y) - q_{1}(y))\log p_{i}(\tau_{i}(y)) && \text{(by } \tau_{i}(y) = x \text{ for } y \in \tau_{i}^{-1}(x))\\ & = \sum_{y\in\mathcal{D}_{1}} (p_{1}(y) - q_{1}(y))\log p_{i}(\tau_{i}(y)).&& \end{aligned} \end{align*} $$

Then,

$$ \begin{align*} \begin{aligned} \sum_{i=1}^{s} (\theta_{i}- \theta_{i-1})\Delta_{i} & = \sum_{x\in\mathcal{D}_{1}} (p_{1}(x) - q_{1}(x)) \sum_{i=1}^{s} (\theta_{i}-\theta_{i-1}) \log p_{i}(\tau_{i}(x)). \end{aligned} \end{align*} $$

Since $ \sum _{x\in \mathcal {D}_{1}}p_{1}(x) = \sum _{x\in \mathcal {D}_{1}} q_{1}(x) = 1$ , it suffices to show

(6.20) $$ \begin{align} \sum_{i=1}^{s} (\theta_{i}-\theta_{i-1}) \log p_{i}(\tau_{i}(x)) = - \log Z \quad\text{for } x \in \mathcal{D}_{1}. \end{align} $$

By (6.3), for $ x \in \mathcal {D}_{1}$ and $ 2 \leq i \leq s$ ,

$$ \begin{align*} p_{1}(x) = p_{i}(\tau_{i}(x)) \frac{1}{Z^{(i)}(\tau_{i}(x))} \prod_{j=1}^{i-1} Z^{(j)}(\tau_{j}(x))^{\alpha_{j}-1}. \end{align*} $$

Thus,

$$ \begin{align*} \begin{aligned} \sum_{i=1}^{s} & (\theta_{i}-\theta_{i-1}) \log p_{i}(\tau_{i}(x)) \\ & = \log p_{1}(x) - \sum_{i=2}^{s} (\theta_{i-1}-\theta_{i}) Z^{(i)}(\tau_{i}(x)) - \sum_{i=2}^{s}\sum_{j=1}^{i-1} (\theta_{i}-\theta_{i-1}) (\alpha_{j}-1)\log Z^{(j)}(\tau_{j}(x)) \\ & = \log p_{1}(x) - \sum_{i=2}^{s} (\theta_{i-1}-\theta_{i}) Z^{(i)}(\tau_{i}(x)) - \!\sum_{j=1}^{s-1}\sum_{i=j+1}^{s} (\theta_{i}-\theta_{i-1}) (\alpha_{j}-1)\log Z^{(j)}(\tau_{j}(x)) \\ & = \log p_{1}(x) - \sum_{i=2}^{s} (\theta_{i-1}-\theta_{i}) Z^{(i)}(\tau_{i}(x)) - \sum_{j=1}^{s-1}(1-\theta_{j}) (\alpha_{j}-1)\log Z^{(j)}(\tau_{j}(x)) \\ & = \log p_{1}(x) - \sum_{i=1}^{s} (\alpha_{i}-1) Z^{(i)}(\tau_{i}(x)) \\ & = - \log Z, \end{aligned} \end{align*} $$

where the second equality is by changing the index of $i, j$ ; the second last equality is by $ Z^{(1)} \equiv 1 $ and $ \theta _{i}\alpha _{i} = \theta _{i-1}$ for $ 1 \leq i \leq s$ ; the last equality is by (6.3).

Now, we are ready to prove Proposition 6.6.

Proof of Proposition 6.6

For a probability vector $ q $ on $ \mathcal {D} $ and $ 1 \leq i \leq s $ , write $ p_{i} = \tau _{i}p $ , $ q_{i} = \tau _{i} q $ , and $\Delta _{i} = \Delta (p_{i}\| q_{i})$ . Define $ I = \{ 1 \leq i \leq s \colon p_{i} \text { is not uniform} \}$ . It follows from Lemma 6.7 that $ \Delta _{i} = 0 $ for $ i \notin I $ . By Lemma 6.8,

(6.21) $$ \begin{align} \sum_{i \in I} (\theta_{i}-\theta_{i-1}) \Delta_{i} = \sum_{i=1}^{s} (\theta_{i}-\theta_{i-1}) \Delta_{i} = 0. \end{align} $$

We first show $ \# I \neq 1 $ . Suppose otherwise that $ I = \{i_{1}\}$ . Take a probability vector $ q $ on $\mathcal {D}$ such that $ \tau _{i_{1}}q $ is uniform. Then, $ \Delta _{i_{1}}> 0 $ by Lemma 6.7, which contradicts (6.21). Then, $ \# I = 2 $ by assumption (6.18).

Write $ I = \{ i_{1}, i_{2}\}$ for some $ 1 \leq i_{1} < i_{2} \leq s $ . Take a probability vector $ q $ on $\mathcal {D}$ such that $ \tau _{i_{1}}q $ is uniform. Then, $ \Delta _{i_{1}}> 0 $ by Lemma 6.7. By (6.21),

$$ \begin{align*} \Delta_{i_{2}} = - \frac{\theta_{i_{1}}-\theta_{i_{1}-1}}{\theta_{i_{2}}-\theta_{i_{2}-1}}\Delta_{i_{1}} < 0. \end{align*} $$

Since $ \Delta _{i} = 0 $ for $i \neq I$ ,

(6.22) $$ \begin{align} \begin{aligned} \sum_{i=1}^{s} & \Delta_{i} \int_{\theta_{i-1}k}^{\theta_{i}k} \frac{1}{\log t} \,d t \\ & = \Delta_{i_{1}}(\theta_{i_{1}}-\theta_{i_{1}-1})\bigg( \frac{1}{\theta_{i_{1}}-\theta_{i_{1}-1}} \int_{\theta_{i_{1}-1}k}^{\theta_{i_{1}}k}\frac{1}{\log t} \, dt - \frac{1}{\theta_{i_{2}}-\theta_{i_{2}-1}} \int_{\theta_{i_{2}-1}k}^{\theta_{i_{2}}k}\frac{1}{\log t} \, dt \bigg). \end{aligned} \end{align} $$

By a change of variable, for $ 0 < a < b \leq c < d$ , there exists $ \delta> 0 $ such that

(6.23) $$ \begin{align} \frac{1}{b-a} \int_{ak}^{bk} \frac{1}{\log t} \, dt - \frac{1}{d-c} \int_{ck}^{dk} \frac{1}{\log t} \, dt \geq \delta \frac{k}{(\log k)^{2}} \quad\text{for } k \in \mathbb{N} \operatorname*{\mathrm{\cap}} \bigg [\frac{2}{a},\infty\bigg). \end{align} $$

Combining (6.22) and (6.23) verifies condition (6.16). Finally, the proof is completed by Proposition 6.5.

Remark 6.9. From the proof of Proposition 6.6, we see that

(6.24) $$ \begin{align} \# \{ 1 \leq i \leq s \colon \tau_{i}p \text{ is not uniform}\} \neq 1. \end{align} $$

Based on (6.24), there is an alternative way to prove that for a Bedford–McMullen carpet K, $ 0 < \mathcal {H}^{\varphi }(K) < \infty $ implies $ \dim _{\mathrm {H}} K = \dim _{\mathrm {B}} K $ . Suppose otherwise that $ \dim _{\mathrm {H}} K < \dim _{\mathrm {B}} K $ . Combining Proposition 5.2 and Theorem 3.1(b) gives $ \# \{ 1 \leq i \leq s \,{\colon} \tau _{i}p \text { is not uniform}\} = 1 $ , which contradicts (6.24).