2.1 Assouad dimension of sets and connection to weak tangents

We are mainly focused on dimensions of measures in this paper, but to place the results in a wider context, we recall some results concerning the Assouad dimensions of sets. The Assouad dimension of a set $F\subset X$ is defined by

\begin{align*} \dim_{\mathrm{A}} F& =\inf\left\{ s>0\colon \exists C>0,\textrm{ s.t. for all }x\in F,\; 0< r< R,\vphantom{\frac{R}{r}}\right.\\ & \quad \left.N_r(B(x,R)\cap F)\leq \left(\frac{R}{r}\right)^s\right\}, \end{align*}

where $N_r(E)$ denotes the smallest number of open balls of diameter $r$ needed to cover the set $E\subset X$. A convenient way to bound the Assouad dimension of a set from below is given by the weak tangent approach. Recall that a map $T\colon X\to X$ is a similarity if there exists $c>0$, such that $d(T(x),T(y))=c d(x,y)$, for all $x,y\in X$. The constant $c$ is called the similarity ratio (of $T$).

For simplicity we give the definition of weak tangents when $X\subset \mathbb {R}^d$ and make a brief remark that they can be defined in complete metric spaces using pointed convergence in the Gromov–Hausdorff distance [Reference Keith and Laakso15, Reference Mackay and Tyson20]. A closed set $F\subset B(0,1)$ is said to be a weak tangent of a compact set $X\subset \mathbb {R}^d$ if there is a sequence of similarities $T_n\colon \mathbb {R}^d\to \mathbb {R}^d$, such that

\begin{equation*} T_n(X)\cap B(0,1)\to F, \end{equation*}

in the Hausdorff distance. The collection of weak tangents of $X$ is denoted by $\mathrm {Tan}(X)$. The following proposition gives an easy way to bound the Assouad dimension from below. For the proof in the general setting see e.g. [Reference Mackay and Tyson20, proposition 6.1.5].

2.2 Assouad dimension of measures and the doubling property

Let us now turn our attention to the dimensions of measures. When referring to (1.1) we sometimes use the term global Assouad dimension to avoid ambiguity with the pointwise variant. Originally, the global Assouad dimension of a measure was called the upper regularity dimension in [Reference Käenmäki, Lehrbäck and Vuorinen16], but due to the intimate connections between this notion of dimension and the Assouad dimension for sets, the term Assouad dimension of a measure is now widely used. A simple volume argument implies that for a measure $\mu$ fully supported on a metric space $X$, we have the inequality $\dim _{\mathrm {A}} X\leq \dim _{\mathrm {A}}\mu$. Moreover, in [Reference Luukkainen and Saksman19, Reference Vol'berg and Konyagin24] it was shown that

\[ \dim_{\mathrm{A}} X=\inf\{\dim_{\mathrm{A}}\mu\colon \mu\textrm{ is a measure fully supported on }X\}, \]

which further supports the current terminology.

The Assouad dimension of a measure has two important properties. First of all, it characterizes the doubling property, which the measure $\mu$ is said to satisfy if there is a constant $C\geq 1$, such that for any $x\in X$, $r>0$, we have

(2.1)\begin{equation} \mu(B(x,2r))\leq C\mu(B(x,r)). \end{equation}

Measures that satisfy (2.1) are called doubling measures and it is a simple exercise to show that a measure has finite Assouad dimension if and only if it is doubling [Reference Fraser9, lemma 4.1.1].

Secondly, the Assouad dimension is ‘the greatest of all dimensions’ [Reference Fraser9], that is

\[ \dim_{\mathrm{H}}\mu\leq \dim_{\mathrm{P}}\mu\leq\dim_{\mathrm{A}}\mu, \]

where $\dim _{\mathrm {H}}\mu := {\mathrm{ess}\,\inf} _{x\in X}\underline {\dim }_{\mathrm {loc}}(\mu,x)$ and $\dim _{\mathrm {P}}\mu := {\mathrm{ess}\,\inf} _{x\in X}\overline {\dim }_{\mathrm {loc}}(\mu,x)$ are the Hausdorff dimension and the packing dimension of the measure $\mu$ respectively. As we will see in the next section, the pointwise Assouad dimension enjoys properties which are analogous to the two properties mentioned.

3.1 Relationships to other dimensions

Next we investigate the relationships between the pointwise Assouad dimension and other common notions of dimension in fractal geometry. We will focus on the global Assouad dimension, the packing dimension, and the upper Minkowski dimension of measures and the Assouad dimension of the support of the measure.

As proposition 3.1 shows, the global Assouad dimension provides an upper bound for the pointwise Assouad dimension at every point. The natural question that arises is if the converse holds at some point, i.e. is it true that $\sup _{x\in X}\dim _{\mathrm {A}}(\mu,x)=\dim _{\mathrm {A}}\mu$. It turns out that generally speaking this is not the case, even in compact spaces. The following is an example of a non-doubling measure, which is pointwise doubling at all points of its support. By proposition 3.1(1) and the analogous fact for the global Assouad dimension, we see that this measure has finite pointwise Assouad dimension at all points, but infinite global Assouad dimension.

Example 3.3 Let $x_n=2^{-n}$, and let $\mu =\sum _{n=0}^{\infty }(3^{-n}\delta _{-x_n}+2^{-n}\delta _{x_n})$, where $\delta _x$ denotes the point mass centred at $x$. Clearly the measure is a finite Borel measure fully supported on the set $X=\{0\}\cup \bigcup _{n=0}^{\infty }\{x_n,-x_n\}$. By considering $y_k=-x_k$, and $r_k=2^{-k}$, it follows by a simple calculation that

\[ \frac{\mu(B(y_k,2r_k))}{\mu(B(y_k,r_k))}\geq\frac{\sum_{n=k-1}^{\infty}3^{{-}n}+\sum_{n=k}^{\infty}2^{{-}n}}{\sum_{n=k-1}^{\infty}3^{{-}n}}=1+\left(\frac{3}{2}\right)^{k-2} \to \infty, \]

as $k\to \infty$, which shows that $\mu$ is not doubling.

The fact that $\mu$ is pointwise doubling at $X\setminus \{0\}$ follows from properties (1) and (3) of proposition 3.1, and a standard calculation shows that for any $2^{-k}\leq r < 2^{-k+1}$, we have

\[ \frac{\mu(B(0,2r))}{\mu(B(0,r))}\leq \frac{2\sum_{n=k-2}^{\infty}2^{{-}n}}{\sum_{n=k}^{\infty}2^{{-}n}}\leq\frac{2^{4-k}}{2^{1-k}}=8, \]

which shows that $\mu$ is pointwise doubling at $0$.

The upper Minkowski dimension of a measure was introduced in [Reference Falconer, Fraser and Käenmäki7] and it is defined by

\begin{align*} \overline{\dim}_{\mathrm{M}}\mu& =\inf\{s>0\colon \textrm{there exists a constant }c>0\textrm{ such that}\\ & \quad\mu(B(x,r))\geq cr^s, \textrm{ for all }x\in \text{supp}(\mu)\textrm{ and }0< r<1\}. \end{align*}

In [Reference Falconer, Fraser and Käenmäki7, proposition 4.1] it was shown that

(3.1)\begin{equation} \dim_{\mathrm{P}}\mu\leq \overline{\dim}_{\mathrm{M}}\mu\leq \dim_{\mathrm{A}}\mu. \end{equation}

By definition of $\dim _{\mathrm {P}}\mu$ and proposition 3.1(2) we have the general relationship $\dim _{\mathrm {P}}\mu \leq \dim _{\mathrm {A}}(\mu,x)$, for $\mu$-almost every $x$. Recalling (3.1), it is natural to ask if a similar inequality holds for the upper Minkowski dimension in some direction. This turns out not to be the case. For some self-affine measures on Bedford–McMullen carpets (see § 6 for the definitions), we have $\overline {\dim }_{\mathrm {M}}\mu \leq \dim _{\mathrm {A}}(\mu,x)$, for $\mu$-almost all $x$. Example 6.6 provides an explicit example satisfying this property. However, it is also possible to have $\overline {\dim }_{\mathrm {M}}\mu >\dim _{\mathrm {A}}(\mu,x)$ for all $x$, as the following example shows.

Example 3.4 Let $\mu =\sum _{n=0}^{\infty }(3^{-n}\delta _{-2^{-n}}+2^{-n}\delta _{2^{-n}})$ be the measure of example 3.3. Let us first show that $\overline {\dim }_{\mathrm {M}}\mu \geq \frac {\log 3}{\log 2}$. Let $s<\frac {\log 3}{\log 2}$, $x_n=-2^{-n}$, and $r_n=2^{-(n+2)}$. Then

\[ \mu(B(x_n,r_n))=3^{{-}n}=r_n^{\frac{n\log 3}{(n+2)\log 2}}< r_n^{s}, \]

for all large enough $n$. Since $r_n\to 0$ with $n$, this implies that $\overline {\dim }_{\mathrm {M}}\mu \geq s$, and taking $s\to \frac {\log 3}{\log 2}$ gives the claim.

Since $\mu$ has an atom at every $x\in X\setminus \{0\}$, by proposition 3.1(3), $\dim _{\mathrm {A}}(\mu,x)=0<\overline {\dim }_{\mathrm {M}}\mu$. At the origin, a simple calculation shows that for any $2^{-L}< r \leq 2^{-L+1}$ and $2^{-N-1}\leq R < 2^{-N}$, we have

\[ \frac{\mu(B(0,R))}{\mu(B(0,r))}\leq \frac{2\sum_{n=N}^{\infty}2^{{-}n}}{\sum_{n=L}^{\infty}2^{{-}n}}=\frac{2^{2-N}}{2^{1-L}}\leq 8\left(\frac{R}{r}\right), \]

and therefore $\dim _{\mathrm {A}}(\mu,0)\leq 1<\frac {\log 3}{\log 2}\leq \overline {\dim }_{\mathrm {M}}\mu$.

Finally, a natural question to ask is if the Assouad dimension of the support of a measure is a lower bound for the pointwise Assouad dimension of the measure, as it is for the global one. Our next example shows that this is also not generally the case, in fact, there are measures supported on sets of full Assouad dimension, which have $0$ pointwise Assouad dimension at all points. The example is original, but builds on a construction by Le Donne and Rajala [Reference Le Donne and Rajala18, example 2.20].

Example 3.5 Let $x_{n,k}=2^{-2^n}+k4^{-2^n}$ and let $X=\{0\}\cup \bigcup _{n=1}^{\infty }\bigcup _{k=0}^{n-1}\{x_{n,k}\}$. Define the measure $\mu$ as

\[ \mu=\sum_{n=1}^{\infty}\sum_{k=0}^{n-1}\frac{2^{{-}n}}{n}\delta_{x_{n,k}}. \]

It is straightforward to show that $\mu$ is a finite doubling measure fully supported on $X$. We show that $\dim _{\mathrm {A}}X=1$, and $\dim _{\mathrm {A}}(\mu,x)=0$, for every $x\in X$.

To show that $\dim _{\mathrm {A}}X= 1$, by proposition 2.1 it is enough to show that $[0,1]$ is a weak tangent for the set $X$. For each $n\in \mathbb {N}$ define a similarity $T_n:\mathbb {R}\to \mathbb {R}$ by

\[ T_n(x)=n^{{-}1}4^{2^n}(x-2^{{-}2^n}), \]

and note that

\[ T_n(X)\cap[0,1]=\bigcup_{k=0}^{n-1}\left\{\frac{k}{n}\right\}\to[0,1], \]

in the Hausdorff distance as $n\to \infty$, that is $[0,1]$ is a weak tangent to $X$.

Next we show that $\dim _{\mathrm {A}}(\mu,x)=0$, for every $x\in X$. Note that every point $x\in X\setminus \{0\}$ is an atom so by proposition 3.1(3), $\dim _{\mathrm {A}}(\mu,x)=0$, so we only need to consider the case $x=0$. Fix $0< r< R<1$, and choose numbers $L,N\in \mathbb {N}$, such that $2^{-2^L}< r \leq 2^{-2^{L-1}}$ and $2^{-2^{N+1}}\leq R < 2^{-2^{N}}$. Clearly $[0,x_{L+1,L}]\subset B(0,r)$ and $B(x,R)\subset [0,x_{N,N-1}]$, so we have

\[ \frac{\mu(B(0,R))}{\mu(B(0,r))}\leq\frac{\mu([0,x_{N,N-1}])}{\mu([0,x_{L+1,L}])}\leq 2^{L-N+1}\leq 2\frac{\log r}{\log R}. \]

Note that for any $s>0$, the function $\phi (t)=t^s\log t$ is decreasing for $0< t< e^{-\frac {1}{s}}$, so for all $0< r< R< e^{-\frac {1}{s}}$ we have

\[ \frac{\mu(B(0,R))}{\mu(B(0,r))}\leq 2\left(\frac{R}{r}\right)^s. \]

Since this holds for arbitrary $s>0$, we have $\dim _{\mathrm {A}}(\mu,0)=0$.

4.1 Self-conformal sets

Next we define self-conformal sets which act as a support for the measures we study in this section. Recall that an IFS $\{\varphi _i\}_{i\in \Lambda }$ on $\mathbb {R}^d$ is called self-conformal if it satisfies the following assumptions:

1. (C1) There is a set $\Omega \subset \mathbb {R}^d$, which is open, bounded and connected, and a compact set $X\subset \Omega$ with non-empty interior, such that

   \[ \varphi_{i}(X)\subset X, \]
   for all $i\in \Lambda$.

2. (C2) For each $i\in \Lambda$, the map $\varphi _i$ is a $C^{1+\varepsilon }$-diffeomorphism, and $\varphi _i\colon \Omega \to \Omega$ is conformal. That is, for all $x\in \Omega$, the linear map $\varphi _i'(x)$ is a similarity. In particular, for every $y\in \Omega$, we have

   \[ |\varphi_i'(x)y|= |\varphi_i'(x)||y|, \]
   where $|\varphi _i'(x)|$ denotes the operator norm of the linear map $\varphi _i'(x)$.

The use of the bounded open set $\Omega$ here is essential since contractive conformal maps defined on whole $\mathbb {R}^d$ are in fact similarities. The limit set $F$ of an IFS satisfying (C1) and (C2) is called a self-conformal set. In the following we let $||\varphi '_{\mathbf {i}}||=\sup _{x\in \Omega }|\varphi '_{\mathbf {i}}(x)|$. It follows from the fact that each $\varphi _i$ is a contraction, that $||\varphi '_{\mathbf {i}}||<1$, for all $\mathbf {i}\in \Sigma _*$, and that for a fixed $\mathbf {i}\in \Sigma$, $||\varphi '_{\mathbf {i}|_n}||$ is strictly decreasing in $n$. Let us recall some key lemmas for the proof of theorem 4.1.

The first property in lemma 4.3 is commonly called the Bounded Distortion Property (BDP) and it originates in [Reference Mauldin and Urbański21]. Property (ii) is a special case of [Reference Fan and Lau8, lemma 2.3] and property (iii) is proved in [Reference Mauldin and Urbański21].

Lemma 4.4 For all $\mathbf {i},\mathbf {j}\in \Sigma _*$, we have

\[ C^{{-}1}||\varphi'_{\mathbf{i}}||\cdot||\varphi_{\mathbf{j}}'||\leq||\varphi_{\mathbf{i}\mathbf{j}}'||\leq ||\varphi_{\mathbf{i}}'||\cdot||\varphi_{\mathbf{j}}'||, \]

where $C>1$ is the constant of lemma 4.3.

Proof. Using the chain rule, and the conformality of the IFS, it is easy to see that for all $x\in F$ we have

\[ |\varphi_{\mathbf{i}\mathbf{j}}'(x)|= |(\varphi_{\mathbf{i}}\circ\varphi_{\mathbf{j}})'(x)| =|\varphi_{\mathbf{i}}'(\varphi_{\mathbf{j}}(x))\cdot\varphi_{\mathbf{j}}'(x)|=|\varphi_{\mathbf{i}}'(\varphi_{\mathbf{j}}(x))|\cdot|\varphi_{\mathbf{j}}'(x)|. \]

Applying lemma 4.3 we get that for all $y\in F$

\[ C^{{-}1}|\varphi_{\mathbf{i}}'(y)|\cdot|\varphi_{\mathbf{j}}'(x)|\leq |\varphi_{\mathbf{i}\mathbf{j}}'(x)| \leq |\varphi_{\mathbf{i}}'(\varphi_{\mathbf{j}}(x))|\cdot|\varphi_{\mathbf{j}}'(x)|\leq ||\varphi_{\mathbf{i}}'||\cdot||\varphi_{\mathbf{j}}'||. \]

The result follows by taking suprema.

Remark 4.5 Let $\mathbf {i}\in \Sigma$ be $n$-periodic. Notice that, by applying the previous lemma iteratively, we have

\[ C^{{-}k}||\varphi'_{\mathbf{i}|_n}||^k\leq||\varphi'_{\mathbf{i}|_{kn}}||\leq ||\varphi'_{\mathbf{i}|_n}||^k. \]

The exponential growth of the distortion in the lower bound is a problem in § 5 when we want to establish a lower bound for the Assouad dimension of the measure we investigate. The following lemma provides a precise estimate for this purpose.

Lemma 4.6 If $x= \pi (\mathbf {i})$, where $\mathbf {i}\in \Sigma$ is $n$-periodic for some $n\in \mathbb {N}$, then for any $k\in \mathbb {N}$ we have

\[ |\varphi_{\mathbf{i}|_{kn}}'(x)|=|\varphi_{\mathbf{i}|_{n}}'(x)|^k. \]

Proof. Let $\mathbf {i}\in \Sigma$ be $n$-periodic, and let $x=\pi (\mathbf {i})$. By the definition of $\pi$, this implies that

(4.2)\begin{equation} \varphi_{\mathbf{i}|_n}(x)=x. \end{equation}

Let $k\in \mathbb {N}$. Using the chain rule, (4.2) and the conformality of the IFS we find that

\begin{align*} |\varphi_{\mathbf{i}|_{kn}}'(x)|& =|((\varphi_{i_1}\circ\ldots\circ\varphi_{i_n})\circ\underset{k\text{ times}}{\ldots}\circ(\varphi_{i_1}\circ\ldots\circ\varphi_{i_n}))'(x)|\\ & =|(\varphi_{i_1}\circ\ldots\circ \varphi_{i_n})'(x)\cdot\underset{k\text{ times}}{\ldots}\cdot(\varphi_{i_1}\circ\ldots\circ \varphi_{i_n})'(x)|\\ & =|\varphi_{\mathbf{i}|_n}'(x)^k|=|\varphi_{\mathbf{i}|_n}'(x)|^k. \end{align*}

4.2 Quasi–Bernoulli measures

A probability measure $\nu$ on $\Sigma$ is called quasi-Bernoulli if there exists a uniform constant $C\geq 1$, such that for all $\mathbf {i},\mathbf {j}\in \Sigma _*$, we have

\[ C^{{-}1}\nu([\mathbf{i}])\nu([\mathbf{j}])\leq \nu([\mathbf{i}\mathbf{j}])\leq C\nu([\mathbf{i}])\nu([\mathbf{j}]), \]

where here and hereafter $\mathbf {i}\mathbf {j}$ denotes the concatenation of the finite words $\mathbf {i}$ and $\mathbf {j}$. Note the similarity to lemma 4.4. If $C$ can be taken to equal $1$, then the measure is called a Bernoulli measure.

To simplify notation, from hereafter we write $A\lesssim B$ to mean that $A$ is bounded from above by $B$ multiplied by a uniform constant. Similarly, we say that $A\gtrsim B$, if $B\lesssim A$ and $A\approx B$ if $B\lesssim A\lesssim B$.

Recall that two measures $\mu$ and $\nu$ are said to be equivalent if $\mu (A)=0$ if and only if $\nu (A)=0$. In the following $\sigma \colon \Sigma \to \Sigma$ denotes the left-shift defined by $\sigma (\mathbf {i})=i_2i_2\ldots$. Also recall that a measure $\nu$ on $\Sigma$ is ergodic if either $\nu (A)=0$ or $\nu (A)=1$, for all $\sigma$-invariant sets $A\subset \Sigma$.

For the rest of this section, let $\mathcal {N}\subset \Sigma$ denote the set of infinite words, which contain all finite words as a substring, that is

\[ \mathcal{N} = \{\mathbf{i}\in\Sigma\colon \mathbf{j}\sqsubset\mathbf{i},\text{ for all }\mathbf{j}\in\Sigma_*\}. \]

The following lemma is simple, but crucial to the proof of theorem 4.1.

Proof. It is well known that if $\nu$ is a quasi-Bernoulli measure, then the measure $\tilde {\nu }$ obtained as a weak-$*$ accumulation point of the sequence

\[ \tilde{\nu}_n:=\frac{1}{n}\sum_{j=0}^{n-1}\nu\circ\sigma^{{-}j}, \]

is a $\sigma$-invariant and ergodic quasi-Bernoulli measure, which is equivalent with $\nu$ [Reference Heurteaux11]. Therefore, we may assume without loss of generality that $\nu$ is $\sigma$-invariant and ergodic. Now for every $\mathbf {j}\in \Sigma _*$, we see by applying Birkhoff's ergodic theorem, that

\[ \lim_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n}\chi_{[\mathbf{j}]}(\sigma^n\mathbf{i})=\int_{\Sigma}\chi_{[\mathbf{j}]}\,{\rm d}\nu = \nu([\mathbf{j}])>0, \]

for $\nu$-almost every $\mathbf {i}\in \Sigma$, where $\chi _{[\mathbf {j}]}$ denotes the indicator function of the set $[\mathbf {j}]$. In particular, this implies that for almost every $\mathbf {i}$, there is $n\in \mathbb {N}$, such that $\chi _{[\mathbf {j}]}(\sigma ^n\mathbf {i})=1$, that is $\mathbf {j}\sqsubset \mathbf {i}$. Let $\Sigma _{\mathbf {j}}=\{\mathbf {i}\in \Sigma \colon \mathbf {j}\sqsubset \mathbf {i}\}$, so by the previous $\nu (\Sigma \setminus \Sigma _{\mathbf {j}})=0$. By definition of $\mathcal {N}$, we have

\[ \nu(\Sigma\setminus\mathcal{N})=\nu\left(\Sigma\setminus\bigcap_{\mathbf{j}\in\Sigma_*}\Sigma_{\mathbf{j}}\right)=\nu\left(\bigcup_{\mathbf{j}\in\Sigma_*}\Sigma\setminus\Sigma_{\mathbf{j}}\right)\leq \sum_{\mathbf{j}\in\Sigma_*}\nu(\Sigma\setminus\Sigma_{\mathbf{j}})=0. \]

We say that a measure $\mu$ supported on a self-conformal set $F$ is quasi-Bernoulli if it is the projection of a quasi-Bernoulli measure $\nu$ supported on $\Sigma$ under the natural projection $\pi \colon \Sigma \to F$ defined as in (4.1). Next we show that the quasi-Bernoulli measures supported on self-conformal sets satisfying the SSC are doubling, which in particular implies that the Assouad dimensions of these measures are finite. After that, we prove the main theorem of this section, theorem 4.1.

Proof. Let $\delta = \min _{i\ne j}d(\varphi _i(F),\varphi _j(F))$, which is positive since $F$ is strongly separated. Fix an integer $k$ satisfying $\max _{\mathbf {i}\in \Sigma _k}\left |\left |\varphi _{\mathbf {i}}'\right |\right |< \frac {\delta }{2C^4}$, where $C$ is the maximum of the constant given by lemma 4.3. Finally, let $c=\min _{\mathbf {i}\in \Sigma _{k+1}}\nu ([\mathbf {i}])$.

Since $F$ is compact and $\mu$ is fully supported on $F$, it is easy to see that it suffices to consider only uniformly small values of $r>0$. Therefore let

\[ 0< r<\min\{||\varphi_{\mathbf{i}|_k}'||\colon \mathbf{i}\in\Sigma_k\}. \]

Note that the right-hand side is positive since $X$ is compact so this is possible. Also fix $x\in F$, let $\mathbf {i}\in \Sigma$ be such that $\pi (\mathbf {i})=x$, and choose $n\in \mathbb {N}$ as the unique integer satisfying $C||\varphi _{\mathbf {i}|_n}'||< r \leq C||\varphi _{\mathbf {i}|_{n-1}}'||$. This immediately implies that $\varphi _{\mathbf {i}|_n}(F)\subset B(x,r)$. Note that by the assumption on $r$ we have

\[ ||\varphi_{\mathbf{i}|_n}'|| <||\varphi_{\mathbf{i}|_k}'||. \]

so in particular $n>k$. For any $l\in \mathbb {N}$ and $i\in \Lambda$ let $\mathbf {i}|_li$ denote the word $(i_1,i_2,\ldots, i_l,i)$ and notice that by the strong separation condition and lemma 4.3(1)–(3), we have for all large enough $l\in \mathbb {N}$ and $i\ne j$ that

(4.3)\begin{equation} d(\varphi_{\mathbf{i}_l i}(F),\varphi_{\mathbf{i}_l j}(F)):=\inf_{\substack{x\in \varphi_{\mathbf{i}_l i}(F)\\ y\in \varphi_{\mathbf{i}_l j}(F)}}|x-y|\geq \frac{\delta}{C^2}\cdot \mathrm{diam}(\varphi_{\mathbf{i}|_l}(F))\geq \frac{\delta}{C^3}||\varphi_{\mathbf{i}|_l}'||. \end{equation}

Now using lemma 4.4, we have for every $y\in B(x,2r)$

\begin{align*} {\rm d}(y,\varphi_{\mathbf{i}|_{n-k-1}}(F))& \leq 2r< \frac{\delta}{C^4\max_{\mathbf{i}\in\Sigma_k}||\varphi_{\mathbf{i}}'||}r\leq \frac{\delta}{C^3\max_{\mathbf{i}\in\Sigma_k}\left|\left|\varphi_{\mathbf{i}}'\right|\right|}||\varphi_{\mathbf{i}|_{n-1}}'||\\ & \leq\frac{\delta}{C^3}\frac{||\varphi_{\mathbf{i}|_{n-1}}'||}{||\varphi_{\sigma^{n-k-1}\mathbf{i}|_{k}}'||}\leq \frac{\delta}{C^3} ||\varphi_{\mathbf{i}|_{n-k-1}}'||, \end{align*}

in particular, combining this with estimate (4.3), we have $B(x,2r)\cap F\subset \varphi _{\mathbf {i}|_{n-k-1}}(F)$. Therefore, using the quasi-Bernoulli property, we have

\[ \frac{\mu(B(x,2r))}{\mu(B(x,r))}\leq \frac{\mu(\varphi_{\mathbf{i}|_{n-k-1}}(F))}{\mu(\varphi_{\mathbf{i}|_{n}}(F)))}=\frac{\nu([\mathbf{i}|_{n-k-1}])}{\nu([\mathbf{i}|_{n}])}\lesssim \frac{1}{\nu([\sigma^{n-k-1}\mathbf{i}|_{k+1}])}\leq \frac{1}{c}. \]

Since the upper bound is independent of $x$ and $r$, the claim follows.

Proof of theorem 4.1 It follows from proposition 4.8 that $\dim _{\mathrm {A}}\mu$ is finite. Let $s<\dim _{\mathrm {A}}\mu$ and $c>0$. Now there is a point $y\in F$ and radii $0< r< R$, satisfying

\[ \frac{\mu(B(y,R))}{\mu(B(y,r))}> c\left(\frac{R}{r}\right)^s. \]

Let $\mathbf {i}\in \mathcal {N}$, $x=\pi (\mathbf {i})$ and let $\mathbf {j}\in \Sigma$, such that $\pi (\mathbf {j})=y$. Now choose $k,n\in \mathbb {N}$ as the unique integers which satisfy

\[ ||\varphi'_{\mathbf{j}|_{n+1}}||\leq R < ||\varphi'_{\mathbf{j}|_{n}}||, \text{and}\ ||\varphi'_{\mathbf{j}|_{k}}||< r \leq ||\varphi'_{\mathbf{j}|_{k-1}}||. \]

Then $\varphi _{\mathbf {j}|_k}(F)\subset B(y,Cr)$ and $B\left (y,\frac {\delta }{2C}R\right )\cap F\subset \varphi _{\mathbf {j}|_n}(F)$, where $C$ is the constant of lemma 4.3. Using the quasi-Bernoulli property and the fact that $\mu$ is doubling, we get that

\[ c\left(\frac{R}{r}\right)^s<\frac{\mu(B(y,R))}{\mu(B(y,r))}\lesssim \frac{\mu(\varphi_{\mathbf{j}|_n}(F))}{\mu(\varphi_{\mathbf{j}|_k}(F))}=\frac{\nu([\mathbf{j}|_n])}{\nu([\mathbf{j}|_k])}\lesssim \nu([\sigma^{n}\mathbf{j}|_{k-n}])^{{-}1}. \]

Now since $\mathbf {i}\in \mathcal {N}$, there is an index $l\in \mathbb {N}$, such that $\sigma ^l\mathbf {i}|_{k-n}=\sigma ^{n}\mathbf {j}|_{k-n}$. Let $R'=||\varphi '_{\mathbf {i}|_{l}}||$ and $r'= ||\varphi '_{\mathbf {i}|_{l+n}}||$, and observe that by lemma 4.4,

\[ \frac{R'}{r'}=\frac{||\varphi'_{\mathbf{i}|_{l}}||}{||\varphi'_{\mathbf{i}|_{l+n}}||}\lesssim \frac{1}{||\varphi'_{\sigma^l\mathbf{i}|_{n}}||}=\frac{1}{||\varphi'_{\sigma^{n}\mathbf{j}|_{k-n}}||}\lesssim\frac{ ||\varphi'_{\mathbf{j}|_{n}}||}{||\varphi'_{\mathbf{j}|_{k}}||}=\frac{R}{r}. \]

Again, it is easy to see that $\varphi _{\mathbf {i}|_{l}}(F)\subset B(x,CR')$ and $B\left (x,\frac {\delta }{2C}r'\right )\cap F\subset \varphi _{\mathbf {i}|_{l+n}}(F)$, so using the doubling and quasi-Bernoulli properties of $\mu$, we see that

\begin{align*} \frac{\mu(B(x,R'))}{\mu(B(x,r'))}& \gtrsim\frac{\mu(\varphi_{\mathbf{i}|_{l}}(F))}{\mu(\varphi_{\mathbf{i}|_{l+k-n}}(F))}=\frac{\nu([\mathbf{i}|_{l}])}{\nu([\mathbf{i}|_{l+k-n}])}\gtrsim \nu([\sigma^l\mathbf{i}|_{k-n}])^{{-}1}\\ & =\nu([\sigma^{n}\mathbf{j}|_{k-n}])^{{-}1} \gtrsim c\left(\frac{R}{r}\right)^s\gtrsim c\left(\frac{R'}{r'}\right)^s. \end{align*}

This shows that $\dim _{\mathrm {A}}(\mu,x)\geq s$, and taking $s\to \dim _{\mathrm {A}}\mu$ gives $\dim _{\mathrm {A}}(\mu,x)\geq \dim _{\mathrm {A}}\mu$. Since this holds for all $\mathbf {i}\in \mathcal {N}$, the claim follows from lemma 4.7.

5.1 Place-dependent invariant measures

We assume that our IFS $\{\varphi _i\}_{i\in \Lambda }$ is self-conformal, that is it satisfies (C1) and (C2). In contrast to the case of self-conformal measures where we assign a uniform measure $p_i$ on the set $\varphi _i(F)$, now we allow the mass concentration to depend continuously on the point, that is we choose for each $i\in \Lambda$ a Hölder continuous function $p_i\colon X\to (0,1)$, which satisfy $\sum _{i\in \Lambda }p_i(x)\equiv 1$ and consider the probability measures satisfying the equation

\[ \int f(x)\,{\rm d}\mu(x)=\sum_{i\in\Lambda}\int p_i(x)f\circ \varphi_i(x)\,{\rm d}\mu(x), \]

for $f\in C(X)$ where here and hereafter $C(X)$ denotes the set of continuous real valued functions on $X$. We define the Ruelle operator $T\colon C(X)\to C(X)$ by

\[ (Tf)(x)=\sum_{i\in\Lambda}p_i(x)f\circ \varphi_i(x), \]

and let $T^*\colon M(X)\to M(X)$ denote the adjoint operator, where $M(X)$ is the set of Borel probability measures on $X$. Recall that for $\nu \in M(X)$, $T^*\nu$ is given by

\[ T^*\nu(B)=\sum_{i\in\Lambda}\int_{\varphi_i^{{-}1}(B)}p_i(x)\,{\rm d}\nu(x), \]

for all Borel subsets $B\subset X$. Barnsley et al. [Reference Barnsley, Demko, Elton and Geronimo2] as well as Fan and Lau [Reference Fan and Lau8] have studied the measures which are invariant under $T$ in a setting which is more general than ours. The next proposition, which is vital to this section, is a special case of [Reference Fan and Lau8, theorem 1.1] or [Reference Barnsley, Demko, Elton and Geronimo2, theorem 2.1] and we refer to the mentioned papers for the proof.

Proposition 5.1 Let $F$ be a self-conformal set satisfying the SSC and $p_i\colon X\to (0,1)$ be Hölder continuous for every $i\in \Lambda$. Then there is a unique Borel probability measure $\mu$ satisfying

\[ T^*\mu=\mu. \]

Furthermore, for every $f\in C(X)$, $T^nf$ converges uniformly to the constant $\int f(x)\,{\rm d}\mu (x)$.

We call the measure $\mu$ an invariant measure with place-dependent probabilities, which we shorten to just invariant measure for the remainder of this section. As was the case with self-similar measures and Bernoulli measures on the corresponding code space, there is also a natural correspondence between the invariant measure $\mu$ and a Gibbs measure on the code space. Let us define some useful notation for this section. For $\mathbf {i}\in \Sigma$ we slightly abuse notation by writing $p_i(\mathbf {i}):= p_i(\pi (\mathbf {i}))$ and $\varphi _i(\mathbf {i}):= \varphi _i(\pi (\mathbf {i}))$, where $\pi \colon \Sigma \to F$ is the natural projection given by (4.1). For $\mathbf {i}\in \Sigma$ and $n\in \mathbb {N}$ we let

\[ p_{\mathbf{i}|_n}(\mathbf{i}) =\prod_{k=1}^{n}p_{i_k}(\sigma^{k-1}\mathbf{i}). \]

Denote by $P(\Sigma )\subset \Sigma$ the set of periodic points of $\Sigma$. For $\mathbf {i}\in P(\Sigma )$ with period of length $n$, we let

\[ \overline{p}_{\mathbf{i}}=p_{\mathbf{i}|_{n}}(\mathbf{i}), \]

and

\[ |\varphi'_{\mathbf{i}}|=|\varphi'_{\mathbf{i}|_{n}}(\mathbf{i})|. \]

The following lemma is a well-known consequence of the Ruelle–Perron–Frobenius theorem [Reference Bowen4], and it follows from proposition 1.3 and theorem 1.6 of [Reference Fan and Lau8].

Lemma 5.2 There exists a unique $\sigma$-invariant probability measure $\nu$ on $\Sigma$, and a constant $C>1$ such that for any $x\in F$, $\mathbf {i}\in \Sigma$ and $n\in \mathbb {N}$, we have

\[ C^{{-}1}p_{\mathbf{i}|_n}(\mathbf{i})\leq \nu([\mathbf{i}|_n])\leq Cp_{\mathbf{i}|_n}(\mathbf{i}). \]

Furthermore, $\nu$ is quasi-Bernoulli and we have $\mu =\pi _*\nu$, where $\mu$ is the measure of proposition 5.1.

The previous lemma together with proposition 4.8 immediately imply that the measure $\mu$ is doubling. We are now ready to state the main result of this section. Recall that $P(\Sigma )$ denotes the set of periodic words in $\Sigma$.

Theorem 5.3 Let $\mu$ be a place-dependent invariant measure fully supported on a self-conformal set $F$, which satisfies the SSC. Then

\[ \dim_{\mathrm{A}}\mu=\sup_{\mathbf{i}\in P(\Sigma)}\frac{\log \overline{p}_{\mathbf{i}}}{\log |\varphi_{\mathbf{i}}'|}. \]

Proof. Let us start with the upper bound. For the rest of the proof let $s=\sup _{\mathbf {i}\in P(\Sigma )}\frac {\log \overline {p}_{\mathbf {i}}}{\log |\varphi _{\mathbf {i}}'|}$. Let $x\in F$ and $\mathbf {i}\in \Sigma$, such that $\pi (\mathbf {i})=x$. Let $0< r< R$ and choose integers $k$ and $n$ which satisfy

\[ |\varphi'_{\mathbf{i}|_{n+1}}(x)|\leq R < |\varphi'_{\mathbf{i}|_{n}}(x)|,\text{ and }|\varphi'_{\mathbf{i}|_{k}}(x)|< r \leq |\varphi'_{\mathbf{i}|_{k-1}}(x)|. \]

This immediately implies that $\varphi _{\mathbf {i}|_k}(F)\subset B(x,Cr)$, where $C$ is the constant of lemma 4.3. As before, let $\delta = \min _{i\ne j}\{{\rm d}(\varphi _i(F),\varphi _j(F))\colon i\ne j\}$. Then it is also easy to see that $B(x,\frac {\delta }{2C}R)\cap F\subset \varphi _{\mathbf {i}|_n}(F)$. Let us set $\mathbf {j}=(i_{k-n+1},i_{k-n+2},\ldots,i_k,i_{k-n+1},i_{k-n+2},\ldots,i_k,\ldots )\in P(\Sigma )$. Note that lemma 5.2 shows that

\[ \prod_{j=k-n+1}^kp_{i_j}(\sigma^{j-1}\mathbf{i})\gtrsim \prod_{l=1}^np_{j_l}(\sigma^{l-1}\mathbf{j}). \]

Using proposition 4.8 to conclude that $\mu$ is doubling and lemmas 4.3(1) and 4.4, we get

\begin{align*} \frac{\mu(B(x,R))}{\mu(B(x,r))}& \lesssim \frac{\mu(B(x,\frac{\delta}{2C}R))}{\mu(B(x,Cr))}\leq \frac{\mu(\varphi_{\mathbf{i}|_n}(F))}{\mu(\varphi_{\mathbf{i}|_k}(F))}\lesssim \frac{p_{\mathbf{i}|_n}(\mathbf{i})}{p_{\mathbf{i}|_k}(\mathbf{i})}\\ & =\frac{\prod_{j=1}^np_{i_j}(\sigma^{j-1}\mathbf{i})}{\prod_{j=1}^kp_{i_j}(\sigma^{j-1}\mathbf{i})} =\left(\prod_{j=k-n+1}^kp_{i_j}(\sigma^{j-1}\mathbf{i})\right)^{{-}1}\\ & \lesssim\left(\prod_{l=1}^np_{j_l}(\sigma^{l-1}\mathbf{j})\right)^{{-}1}= |\varphi'_{\mathbf{j}|_n}(\mathbf{j})|^{-\frac{\log \prod_{l=1}^np_{j_l}(\sigma^{l-1}\mathbf{j})}{\log |\varphi'_{\mathbf{j}|_n}(\mathbf{j})|}}\\ & \leq|\varphi'_{\mathbf{j}|_n}(\mathbf{j})|^{{-}s}\lesssim |\varphi'_{\mathbf{i}|_{k-n-1}}(x)|^{{-}s}\lesssim\left(\frac{|\varphi'_{\mathbf{i}|_{n}}(x)|}{|\varphi'_{\mathbf{i}|_{k}}(x)|}\right)^s\lesssim\left(\frac{R}{r}\right)^s. \end{align*}

This shows that $\dim _{\mathrm {A}}\mu \leq s$, for any $x\in F$.

For the lower bound, let $t< s$, and choose $\mathbf {i}\in P(\Sigma )$, such that

\[ \frac{\log \overline{p}_{\mathbf{i}}}{\log |\varphi'_{\mathbf{i}|_{n}}(x)|}\geq t, \]

where $x=\pi (\mathbf {i})$ and $n$ is the period of $\mathbf {i}$. For every $k\in \mathbb {N}$ let $r_k=|\varphi _{\mathbf {i}_{kn}}'(x)|=|\varphi _{\mathbf {i}_{n}}'(x)|^k$, where the second equality follows from lemma 4.6. Using the SSC and proposition 4.8 we get

\begin{align*} \mu(B(x,r_k))& \lesssim \mu(\varphi_{\mathbf{i}|_{kn}}(F))=\prod_{j=1}^{kn}p_{i_j}(\sigma^{j-1}\mathbf{i})\\ & =\overline{p}_{\mathbf{i}}^k=r_k^\frac{k\log \overline{p}_{\mathbf{i}}}{k\log |\varphi_{\mathbf{i}|_{n}}'(x)|}\leq r_k^t. \end{align*}

Taking logarithms and limits shows us that

\[ \overline{\dim}_{\text{loc}}(\mu,x)\geq t, \]

so in particular by proposition 3.1(2), $\dim _{\mathrm {A}}\mu \geq t$. Taking $t\to s$ finishes the proof.

Using the fact that the measure $\mu$ is quasi-Bernoulli, theorem 4.1 gives the following immediate corollary.

Corollary 5.4 If $\mu$ is a place-dependent invariant measure fully supported on a self-conformal set $F$ satisfying the SSC, then

\[ \dim_{\mathrm{A}}(\mu,x)=\sup_{\mathbf{i}\in P(\Sigma)}\frac{\log \overline{p}_{\mathbf{i}}}{\log|\varphi'_{\mathbf{i}}|}, \]

for $\mu$-almost every $x\in F$

Remark 5.5 By [Reference Fraser and Howroyd10, theorem 2.4], the Assouad dimension of a self-similar measure $\mu$ under the SSC is given by the formula

(5.1)\begin{equation} \dim_{\mathrm{A}}\mu=\max_{i\in\Lambda}\frac{\log p_i}{\log r_i}, \end{equation}

see § 6 for definitions. Our theorem 5.3 can be viewed as a generalization of this result. Indeed, an IFS consisting of similarities $\varphi _i$ with similarity ratios $r_i$ is a self-conformal IFS, with $|\varphi '_i(x)|=r_i$, for all $x\in F$. Moreover, when each $p_i(x)\equiv p_i$, the assumptions of theorem 5.3, and we have

\[ \dim_{\mathrm{A}}\mu=\sup_{\mathbf{i}\in P(\Sigma)}\frac{\log \overline{p}_{\mathbf{i}}}{\log |\varphi'_{\mathbf{i}}|}=\sup_{\mathbf{i}\in \Sigma_*}\frac{\log p_{\mathbf{i}}}{\log r_{\mathbf{i}}}=\max_{i\in\Lambda}\frac{\log p_i}{\log r_i}. \]

This together with corollary 5.4 gives the following slightly stronger version of [Reference Fraser and Howroyd10, theorem 2.4]: If $\mu$ is a self-similar measure satisfying the SSC, then

\[ \dim_{\mathrm{A}}(\mu,x)=\max_{i\in\Lambda}\frac{\log p_i}{\log r_i}=\dim_{\mathrm{A}}\mu, \]

for $\mu$-almost every $x\in F$.

6.1 Self-similar measures and the open set condition

The doubling properties of self-similar measures are quite well studied. The fact that self-similar measures satisfying the SSC are doubling follows, for example, from proposition 4.8, and as mentioned in remark 5.5, their Assouad dimension was explicitly computed by Fraser and Howroyd [Reference Fraser and Howroyd10], and it is given by the formula (5.1). Slightly surprisingly, relaxing the SSC to the OSC changes the situation dramatically. Yung [Reference Yung25] provides examples of self-similar sets satisfying the OSC for which (1) only the canonical self-similar measure is doubling, (2) all self-similar measures are doubling, (3) the measures are doubling for some (non-canonical) but not all choices of the weights $p_i$. In particular this shows that the Assouad dimension of self-similar measures satisfying the OSC can in many cases be infinite. Still, it is an interesting question to study the Assouad dimension of doubling self-similar measures which do not satisfy the SSC. In the main theorem of this section, we show that if a self-similar measure satisfying the OSC is doubling, then the Assouad dimension is given by the natural formula (5.1). Furthermore, we show that the pointwise Assouad dimension agrees with the global Assouad dimension almost everywhere, obtaining a stronger version of remark 5.5.

In this section, we use the notation

\[ c_{\mathbf{i}|_n}=\prod_{k=1}^nc_{i_k}, \]

for any parameters $c_i$, with $i\in \Lambda$. Recall that we may construct a Bernoulli measure $\nu$ on $\Sigma$ by setting for all $\mathbf {i}\in \Sigma _n$,

\[ \nu([\mathbf{i}])=p_{\mathbf{i}}, \]

and extending this to the whole space $\Sigma$ in the usual way. There is a natural correspondence between the Bernoulli measure $\nu$ and the self-similar measure on $F$, namely

(6.1)\begin{equation} \mu=\pi_*\nu, \end{equation}

where $\pi \colon \Sigma \to F$ is the coding map given by (4.1). The proof of the next theorem, which is our main result of this section, builds on ideas of [Reference Fraser and Howroyd10, Reference Yung25].

Theorem 6.1 Let $\mu$ be a self-similar measure satisfying the OSC. If $\mu$ is doubling, then

\[ \dim_{\mathrm{A}}\mu=\max_{i\in\Lambda}\frac{\log p_i}{\log r_i}, \]

and for $\mu$-almost every $x\in F$, we have

\[ \dim_{\mathrm{A}}(\mu,x)=\dim_{\mathrm{A}}\mu. \]

Proof. Let $s=\max _{i\in \Lambda }\frac {\log p_i}{\log r_i}$. We start by showing that $\dim _{\mathrm {A}}\mu \leq s$. Let $x\in F$, $0< r< R<1$ and let $\mathbf {i}\in \Sigma$, be a (not necessarily unique) word satisfying $\pi (\mathbf {i})=x$. Choose integers $k$ and $l$, such that $r_{\mathbf {i}|_k}\leq R < r_{\mathbf {i}|_{k-1}}$ and $\quad r_{\mathbf {i}|_{l+1}}< r \leq r_{\mathbf {i}|_{l}}$. We may assume from this point on that $l>k$, since otherwise $\frac {R}{r}$ would be bounded from above by a uniform constant, which does not bother us. Now $\varphi _{\mathtt {i}_{l+1}}(F)\subset B(x,r)$, so in particular $\mu (B(x,r))\geq p_{\mathbf {i}|_{l+1}}$. Define $\Lambda _{x,R}=\{\mathbf {j}\in \Sigma _*\colon r_{\mathbf {j}}\leq R < r_{\mathbf {j}^{-}},\,{\rm d}(x,\varphi _{\mathbf {j}}(F))\leq R\}$. Note that the OSC implies that there is a constant $M\geq 1$ independent of $x$ and $R$, such that

\[ \#\Lambda_{x,R}\leq M. \]

For the simple proof of this see [Reference Kigami17, proposition 1.5.8]. By definition, for every $\mathbf {j}\in \Lambda _{x,R}$ we have $\mathrm {diam}(\varphi _{\mathbf {j}}(F))=r_{\mathbf {j}}\leq R < r_{\mathbf {i}|_{k-1}}$, and since $x\in \varphi _{\mathbf {i}|_{k-1}}(F)$, this implies that $d(\varphi _{\mathbf {i}|_{k-1}}(F),\varphi _{\mathbf {j}}(F))\leq R < r_{\mathbf {i}|_{k-1}}$. Combining these estimates we see that $\varphi _{\mathbf {j}}(F)\subset B(\varphi _{\mathbf {i}|_{k-1}}(F),2r_{\mathbf {i}|_{k-1}})$, where $B(\varphi _{\mathbf {i}|_{k-1}}(F),2r_{\mathbf {i}|_{k-1}})$ denotes the open $2r_{\mathbf {i}|_{k-1}}$-neighbourhood of the set $\varphi _{\mathbf {i}|_{k-1}}(F)$. Therefore, since $\mu$ is doubling, we may apply theorem 1.1 of [Reference Yung25] to see that there is a constant $C>0$, such that

\[ p_{\mathbf{j}}\leq Cp_{\mathbf{i}|_{k-1}} \]

holds independently from $\mathbf {j}$ and $\mathbf {i}$. Furthermore, it is clear that

\[ B(x,R)\cap F\subset\bigcup_{\mathbf{j}\in\Lambda_{x,R}}\varphi_{\mathbf{j}}(F), \]

so we may estimate

\begin{align*} \frac{\mu(B(x,R))}{\mu(B(x,r))}& \leq \frac{\sum_{\mathbf{j}\in\Lambda_{x,R}}p_{\mathbf{j}}}{p_{\mathbf{i}|_{l+1}}}\leq MC\frac{p_{\mathbf{i}|_{k-1}}}{p_{\mathbf{i}|_{l+1}}}=\frac{MC}{p_{i_k}p_{i_l}}\frac{p_{\mathbf{i}|_k}}{p_{\mathbf{i}|_l}}\\ & \leq \frac{MC}{p_{\min}^2}\left(p_{i_{l-k+1}}p_{i_{l-k+2}}\cdots p_{i_l}\right)^{{-}1}\\ & \leq \frac{MC}{p_{\min}^2}\left(r_{i_{l-k+1}}^{\frac{\log p_{i_{k-l+1}}}{\log r_{i_{k-l+1}}}}r_{i_{l-k+2}}^{\frac{\log p_{i_{k-l+2}}}{\log r_{i_{k-l+2}}}}\cdots r_{i_l}^{\frac{\log p_{i_{l}}}{\log r_{i_{l}}}}\right)^{{-}1}\\ & \leq\frac{MC}{p_{\min}^2}\left(\frac{r_{\mathbf{i}|_{k}}}{r_{\mathbf{i}|_l}}\right)^{s} \leq \frac{MC}{p_{\min}^2}\left(\frac{R}{r}\right)^{s}, \end{align*}

which is enough to show that $\dim _{\mathrm {A}}\mu \leq s$.

To finish the proof, it is enough to show that the lower bound holds for the pointwise Assouad dimension at almost every point. For this, let $i\in \Lambda$ be the index maximizing $\frac {\log p_i}{\log r_i}$ and define $\mathcal {N}_n=\{\mathbf {i}\in \Sigma \colon (i,\ldots,i)\sqsubset \mathbf {i}\}$ and subsequently $\mathcal {N}=\bigcap _{n\in \mathbb {N}}\mathcal {N}_n$. Pick $x\in \pi (\mathcal {N})$ and note that as a special case of lemma 4.7, we have that $\pi (\mathcal {N})$ is a set of full measure. Let $\mathbf {i}\in \mathcal {N}$ be a (not necessarily unique) sequence such that $\pi (\mathbf {i})=x$. Now for any $n\in \mathbb {N}$ there is an integer $k$ such that

\[ \mathbf{i}=(i_1,\ldots,i_k,\underbrace{i,i,\ldots,i}_n,i_{k+n+1},\ldots). \]

Choose $R_n = r_{\mathbf {i}|_{k}}$ and $r_n=r_{\mathbf {i}|_{k+n}}$, so $\varphi _{\mathtt {i}_{k}}(F)\subset B(x,R_n)$, and thus

\[ \mu(B(x,R_n))\geq\mu(\varphi_{\mathtt{i}_{k}}(F))=p_{\mathbf{i}|_{k-1}}, \]

and by calculations similar to above,

\[ \mu(B(x,r_n))\leq MC p_{\mathbf{i}|_{k+n}}. \]

Therefore

\[ \frac{\mu(B(x,R_n))}{\mu(B(x,r_n))}\geq \frac{1}{MC}p_i^{{-}n}=\frac{1}{MC}(r_i^{{-}n})^{s}= \frac{1}{MC}\left(\frac{R_n}{r_n}\right)^{s}. \]

Since $\frac {R_n}{r_n}\to \infty$ as $n\to \infty$, this shows that $\dim _{\mathrm {A}}(\mu,x)\geq s$. This finishes the proof, since now at $\mu$-almost every $x$, we have $s\leq \dim _{\mathrm {A}}(\mu,x)\leq \dim _{\mathrm {A}}\mu \leq s$.

6.2 Self-affine measures on Bedford–McMullen sponges

A result similar to theorem 4.1 also holds for self-affine measures on very strongly separated Bedford–McMullen sponges, which we define as follows. We work in $\mathbb {R}^d$, with $d\geq 2$. Start by choosing integers $n_1< n_2<\ldots < n_d$, and after that choose a subset $\Lambda \subset \prod _{q=1}^d\{0,\ldots,n_q - 1\}$. The set $\Lambda$ is the code space associated with the Bedford–McMullen sponge. For all $\bar {\imath }=(i_1,i_2,\ldots,i_d)\in \Lambda$, we define an affine transform $\varphi _{\bar {\imath }}:[0,1]^d\to [0,1]^d$ by

\[ \varphi_{\bar{\imath}}(x_1,\ldots,x_d)=\left(\frac{x_1+i_1}{n_1},\ldots,\frac{x_d+i_d}{n_d}\right). \]

The limit set of this IFS is called a Bedford–McMullen carpet if $d=2$ or a Bedford–McMullen sponge if $d>2$. With this construction, we associate a probability vector $(p_{\bar {\imath }})_{\bar {\imath }\in \Lambda }$, and define the self-affine measure $\mu$ on $F$ as usual. Recall that $\mu$ is related to a Bernoulli measure $\nu$ on the code space $\Sigma$ by (6.1). To establish bounds for the measures of balls, we need a separation condition which is strictly stronger than the strong separation condition. Following Olsen [Reference Olsen22], we say that a Bedford–McMullen sponge $F$ satisfies the very strong separation condition (VSSC), if for words $(i_1,\ldots,i_d),(j_1,\ldots,j_d)\in \Lambda$ satisfying $i_k=j_k$, for all $k=1,\ldots, q-1$, and $i_q\ne j_q$, for some $q=1,\ldots,d$, we have $|i_q-j_q|>1$. We also need the following quantity. For $q=1,\ldots,d$ and $\bar {\imath }=(i_1,\ldots,i_d)$, define

(6.2)\begin{equation} p_q(\bar{\imath})=p(i_q|i_1,\ldots,i_{q-1})=\dfrac{\sum\limits_{\substack{\bar{\jmath}\in \Lambda\\ j_k=i_k,\,k=1,\ldots,q}}p_{\bar{\jmath}}}{\sum\limits_{\substack{\bar{\jmath}\in \Lambda\\ j_k=i_k,\,k=1,\ldots,q-1}}p_{\bar{\jmath}}}, \end{equation}

if $(i_1,\ldots,i_q,i_{q+1},\ldots,i_d)\in \Lambda$ for some $i_{q+1},\ldots,i_d$, and 0 otherwise. These numbers can be interpreted as the conditional probabilities that the $q$th digit of a randomly chosen member of $\Lambda$ equals the $q$th digit of $\bar {\imath }$, given that the first $q-1$ coordinates did. The following theorem was proved by Fraser and Howroyd [Reference Fraser and Howroyd10, theorem 2.6].

Theorem 6.3 Let $\mu$ be a self-affine measure on a Bedford–McMullen sponge satisfying the VSSC. Then

\[ \dim_{\mathrm{A}}\mu=\sum_{q=1}^d\max_{\bar{\imath}\in \Lambda}\frac{-\log p_q(\bar{\imath})}{\log n_q}. \]

Again, we extend this result and prove that the pointwise Assouad dimension coincides with this value at almost every point.

Theorem 6.4 Let $\mu$ be a self-affine measure on a Bedford–McMullen sponge $F$ satisfying the VSSC. Then

\[ \dim_{\mathrm{A}}(\mu,x)=\dim_{\mathrm{A}}\mu, \]

for $\mu$-almost every $x\in F$.

For the proof we need the concept of approximate cubes introduced by Olsen [Reference Olsen22]. For clarity, we use $\omega$ to represent members of the set $\Sigma$ instead of $\mathbf {i}$ which we used in the self-similar case. We denote the approximate cube of level $k\in \mathbb {N}$ centred at $\omega =(\bar {\imath }_1,\ldots )=((i_{1,1},\ldots,i_{1,d}),\ldots )\in \Sigma$ by $Q_k(\omega )$, and define it by

\[ Q_k(\omega)=\{\omega'=(\bar{\jmath}_1,\ldots)\in\Sigma\colon j_{t,q}=i_{t,q},\,\forall q=1,\ldots,d\text{ and }\forall t=1,\ldots L_q(k)\}, \]

where $L_q(k)$ is the unique number that satisfies

(6.3)\begin{equation} n_q^{{-}L_q(k)-1} < n_1^{{-}k}\leq n_q^{{-}L_q(k)}. \end{equation}

The geometric equivalent of the approximate cube $Q_k(\omega )$ is its image under the projection map $\pi :\Sigma \to \mathbb {R}^d$. The image $\pi (Q_k(\omega ))$ is contained in

\[ \prod_{q=1}^d\left[\frac{i_{1,q}}{n_q}+\ldots+\frac{i_{L_q(k),q}}{n_q^{L_q(k)}},\frac{i_{1,q}}{n_q}+\ldots+\frac{i_{L_q(k),q}}{n_q^{L_q(k)}}+\frac{1}{n_q^{L_q(k)}}\right], \]

which is a hypercuboid in $\mathbb {R}^d$ with all side lengths comparable to $n_1^{-k}$.

Olsen [Reference Olsen22] observed that the measure of an approximate cube is given by

(6.4)\begin{equation} \mu(\pi(Q_k(\omega)))=\prod_{q=1}^d\prod_{j=0}^{L_q(k)-1}p_q(\sigma^j\omega), \end{equation}

where $\sigma :\Sigma \to \Sigma$ is the left-shift and $p_q(\omega )=p(i_{1,q}|i_{1,1},\ldots,i_{1,q-1})$, where the right-hand side is as in equation (6.2). Recall also that a Bernoulli measure on the code space $\Sigma$ is shift invariant.

The following proposition by Olsen shows that we can approximate the balls centred at the Bedford–McMullen sponges by approximate cubes of comparable size.

The proof of the proposition can be found in [Reference Olsen22, proposition 6.2.1]. Let us now prove theorem 6.4. The proof follows ideas of Fraser and Howroyd [Reference Fraser and Howroyd10, theorem 2.6], where they carefully construct sequences of points and scales, which give the desired exponent for the lower bound. The difficulty we face when compared to the approach in [Reference Fraser and Howroyd10], is that where they have the freedom to choose the point they consider for each pair of scales independently, we have to find a single point where we see the desired behaviour at arbitrarily small scales. Due to the non-conformality of the sponge, this essentially means that we not only need to find long enough sequences of convenient symbols in the symbolic space, but we also have to control their location within the word.

Proof of theorem 6.4 First we note that $L_q(n)$ increases with $n$ and, since $n_q$ are strictly increasing, decreases with $q$. It is an elementary exercise to show that for every $k\in \mathbb {N}$, there is an integer $n_k$, such that for all $n\geq n_k$, we have

\[ L_d(n)< L_d(n+k)< L_{d-1}(n)< L_{d-1}(n+k)<\ldots< L_1(n)< L_1(n+k). \]

For $q=1,\ldots,d$, let $p_q^{\min }=\min _{\bar {\imath }\in \Lambda }p_q(\bar {\imath })$, and let $\bar {\imath }_q^{\min }$ be some element of $\Lambda$ which achieves this minimum. Define for every $k\in \mathbb {N}$ the set

\[ I_k=\bigcup_{n\geq n_k}\bigcap_{q=1}^{d}\sigma^{{-}L_q(n)}[\underbrace{\bar{\imath}_q^{\min},\ldots,\bar{\imath}_q^{\min}}_{L_q(n+k)-L_q(n)\text{ times}}]. \]

Note that an element $\omega \in I_k$ has the form

(6.5)\begin{align} \omega& =(\bar{\imath}_1,\ldots \bar{\imath}_{L_d(n)},\bar{\imath}_d^{\min},\ldots,\bar{\imath}_d^{\min},\bar{\imath}_{L_d(n+k)+1},\ldots,\bar{\imath}_{L_{2}(n)},\nonumber\\ & \quad\bar{\imath}_2^{\min},\ldots,\bar{\imath}_2^{\min}, \bar{\imath}_{L_2(n+k)+1},\ldots,\bar{\imath}_{L_1(n)},\bar{\imath}_1^{\min},\ldots,\bar{\imath}_1^{\min},\bar{\imath}_{L_1(n+k)+1},\ldots). \end{align}

It is also a simple exercise to show that if $\mathbf {i},\mathbf {j}\in \Sigma _*$, and $q,\ell \in \mathbb {N}$, such that $\ell > q+|\mathbf {i}|$, and $A,B\subset \Sigma$, with $A\subset \Lambda ^q\times [\mathbf {i}]$ and $B\subset \Lambda ^{\ell }\times [\mathbf {j}]$, then

(6.6)\begin{equation} \nu(A\cap B)=\nu(A)\nu(B). \end{equation}

Now we choose $m_1=n_k$ and then inductively $m_i=L_1(m_{i-1}+k)+1$, for every $i>1$, and define $A_i:= (\bigcap _{q=1}^{d}\sigma ^{-L_q(m_i)}[\underbrace {\bar {\imath }_q^{\min },\ldots,\bar {\imath }_q^{\min }}_{L_q(m_i+k)-L_q(m_i)\text { times}}])^c$. Noting that $I_k^c\subset \bigcap _{i\in \mathbb {N}}A_i$ and applying (6.6) inductively first to the sets $A_i$ and then to the sets $\sigma ^{-L_q(m_i)}[\underbrace {\bar {\imath }_q^{\min },\ldots,\bar {\imath }_q^{\min }}_{L_q(m_i+k)-L_q(m_i)\text { times}}]$, we obtain

\begin{align*} \nu(I_k^c)& \leq\nu\left(\bigcap_{i\in\mathbb{N}}A_i\right)= \prod_{i\in\mathbb{N}}\nu(A_i)\\ & =\prod_{i\in\mathbb{N}}\left(1-\nu\left(\bigcap_{q=1}^{d}\sigma^{{-}L_q(m_i)}\left[\underbrace{\bar{\imath}_q^{\min},\ldots,\bar{\imath}_q^{\min}}_{L_q(m_i+k)-L_q(m_i)\text{ times}}\right]\right)\right)\\ & =\prod_{i\in\mathbb{N}}\left(1-\prod_{q=1}^{d}(p_q^{\min})^{L_q(m_i+k)-L_q(m_i)}\right)\leq \prod_{i\in\mathbb{N}}\left(\underbrace{1-(p_q^{\min})^d}_{{<}1}\right)=0. \end{align*}

Thus $\nu (I_k)=1$, and moreover $\nu (I)=1$, where $I=\bigcap _{k\in \mathbb {N}}I_k$.

Now let $s=\dim _{\mathrm {A}}\mu$ given by theorem 6.3, $x=\pi (\omega )$, where $\omega \in I$, and let $R_k=(n_1+\ldots +n_d)n_1^{-n-1}$, and $r_k=2^{-1}n_1^{-(n+k)-1}$, where $k$ and $n$ are chosen, such that $\omega$ is given by equation (6.5). Observe that by proposition 6.5 and equations (6.3) and (6.4), we have

\begin{align*} \frac{\mu(B(x,R_k))}{\mu(B(x,r_k))}& =\frac{\prod_{q=1}^d\prod_{j=0}^{L_q(n)}p_q(\sigma^j\omega)}{\prod_{q=1}^d\prod_{j=0}^{L_q(n+k)}p_q(\sigma^j\omega)}=\frac{1}{\prod_{q=1}^d\prod_{j=L_q(n)-1}^{L_q(n+k)}p_q(\sigma^j\omega)}\\ & =\prod_{q=1}^d\left(\frac{1}{p_q^{\min}}\right)^{L_q(n+k)-L_q(n)+2}\geq \prod_{q=1}^d\left(\frac{1}{p_q^{\min}}\right)^{(n+k)\frac{\log n_1}{\log n_q}-n\frac{\log n_1}{\log n_q}+1}\\ & \geq (p_q^{\min})^{{-}d}\prod_{q=1}^d\left(\frac{1}{p_q^{\min}}\right)^{k\frac{\log n_1}{\log n_q}}=(p_q^{\min})^{{-}d}\prod_{q=1}^d\left(n_1^k\right)^{\frac{-\log p_q^{\min}}{\log n_q}}\\ & \geq (\min_{q}p_q^{\min})^{{-}d}\left(n_1^k\right)^{s}=C\left(\frac{R_k}{r_k}\right)^{s}, \end{align*}

where $C=(\min _{q}p_q^{\min })^{-d}\cdot (2(n_1+\ldots +n_d))^s>0$ is a constant. Taking $k\to \infty$, we see that $\frac {R_k}{r_k}\to \infty$, which is enough to prove that $\dim _{\mathrm {A}}(\mu,x)\geq s$. This holds for all $x=\pi (\omega )$, such that $\omega \in I$, where $I$ has full measure, proving the claim.

Example 6.6 Here we give an example of a measure $\mu$, with $\overline {\dim }_{\mathrm {M}}\mu <\dim _{\mathrm {A}}(\mu,x)$, for $\mu$-almost every $x$. Let $\mu$ be a self-affine measure on a Bedford–McMullen carpet. By [Reference Fraser9, theorem 8.6.2], the upper Minkowski dimension of $\mu$ is given, in the notation of § 6, by the formula

\[ \overline{\dim}_{\mathrm{M}}\mu=\max_{\bar{\imath}\in\Lambda}\left(\frac{-\log p_{\bar{\imath}}}{\log n_2}\right)+\max_{\bar{\imath}\in\Lambda}\left(\frac{\log p_1(\bar{\imath})}{\log n_2}+\frac{-\log p_1(\bar{\imath})}{\log n_1}\right), \]

and from theorems 6.3 and 6.4, it follows that the pointwise Assouad dimension is given by

\[ \dim_{\mathrm{A}}(\mu,x)=\max_{\bar{\imath}\in\Lambda}\left(\frac{-\log p_{\bar{\imath}}}{\log n_2}+\frac{\log p_1(\bar{\imath})}{\log n_2}\right)+\max_{\bar{\imath}\in\Lambda}\left(\frac{-\log p_1(\bar{\imath})}{\log n_1}\right), \]

at $\mu$-almost every $x$. By choosing the $p_{\bar {\imath }}$, for example in a way that $p_{\bar {\imath }}$ and $\frac {p_{\bar {\imath }}}{p_1(\bar {\imath })}$ are minimized in the same column, and $p_1(\bar {\imath })$ is minimized in a different column, we have

\[ \dim_{\mathrm{A}}(\mu,x)>\overline{\dim}_{\mathrm{M}}\mu, \]

for $\mu$-almost every $x$. For example, we may choose $n_1=3$ and $n_2=4$, and $\Lambda =\{(0,0),(0,3),(2,0)\}$, with $p_{(0,0)}=\frac {1}{8},p_{(0,3)}=\frac {5}{8}$ and $p_{(2,0)}=\frac {1}{4}$. Then we have

\[ \dim_{\mathrm{A}}(\mu,x)=\frac{\log 6}{\log 4}+\frac{\log 4}{\log 3}> \frac{\log2}{\log 4}+\frac{\log 4}{\log 3}=\overline{\dim}_{\mathrm{M}}\mu., \]

for $\mu$-almost every $x$.