On normal numbers and self-similar measures

Let $b$ be an integer greater or equal to $2$. Let $T_{b}$ be the times-$b$ map,

A number $x\in\mathbb{R}$ is called $b$-normal, or normal in base $b$, if its orbit $\{T_{b}^{k}(x)\}_{k\in\mathbb{N}}$ equidistributes for the Lebesgue measure on $[0,1]$. In 1909 Borel proved that Lebesgue almost every $x$ is absolutely normal, that is, normal in all bases. In the absence of obvious obstructions, it is believed that this phenomenon should continue to hold true for typical elements of well structured sets with respect to appropriate measures. The purpose of this paper is to make a contribution in this direction for a well studied class of fractal measures.

Before stating our main result, we recall that one may define digit expansions of numbers in non-integer bases as well: Following Rényi [25], for $\beta>1$ we define the $\beta$-expansion of $x\in(0,1)$ to be the lexicographically largest sequence $x_{n}\in\{0,...,[\beta]\}$ such that $x=\sum_{n=1}^{\infty}x_{n}\beta^{-n}$. This sequence is obtained from the orbit of $x$ under the map $T_{\beta}(x)=\beta\cdot x\mod 1$ similarly to the integer case. It is known that $T_{\beta}$ admits a unique absolutely continuous invariant measure, commonly referred to as the Parry measure [20]. When $\beta$ is an integer then the Parry measure is just the Lebesgue measure restricted to $[0,1)$. Thus, in analogy with the integer case, we say that $x$ is $\beta$-normal if it equidistributes under $T_{\beta}$ for the Parry measure. Our main result will allow us to conclude that for certain fractal measures, a typical element will be $\beta$-normal, where $\beta>1$ may be a non-integer.

In this paper we will work with self-similar measures on $\mathbb{R}$. These are defined as follows: Let $\Phi=\{f_{1},...,f_{n}\}$ be a finite set of real non-singular contracting similarity maps of a compact interval $J\subset\mathbb{R}$. That is, for every $i$ we can write $f_{i}(x)=s_{i}\cdot x+t_{i}$ where $s_{i}\in(-1,1)\setminus\{0\}$ and $t_{i}\in\mathbb{R}$, and $f_{i}(J)\subseteq J$. We will refer to $\Phi$ as a self similar IFS (Iterated Function System). It is well known that there exists a unique compact set $\emptyset\neq K=K_{\Phi}\subseteq J$ such that

The set $K$ is called a self-similar set, and the attractor of the IFS $\Phi$. We always assume that there exist $i\neq j$ such that the fixed point of $f_{i}$ is not equal to the fixed point of $f_{j}$. This ensures that $K$ is infinite.

Next, let $\textbf{p}=(p_{1},...,p_{n})$ be a strictly positive probability vector, that is, $p_{i}>0$ for all $i$ and $\sum_{i}p_{i}=1$. It is well known that there exists a unique Borel probability measure $\mu$ such that

The measure $\mu$ is called a self-similar measure, and is supported on $K$. Our assumptions that $K$ is infinite and that $p_{i}>0$ for every $i$ are known to imply that $\mu$ is non-atomic. In particular, all self-similar measures in this paper are non-atomic.

Recall that $\beta>1$ is called a Pisot number if it is an algebraic integer whose algebraic conjugates are all of modulus strictly less than $1$. Note that every integer larger than $1$ is a Pisot number. Also, we will write $a\sim b$ if the real numbers $a,b\neq 0$ satisfy that $\frac{\log|a|}{\log|b|}\in\mathbb{Q}$. Otherwise, we write $a\not\sim b$, in which case $a,b$ are said to be multiplicatively independent. Finally, given a measure $\mu$ on $\mathbb{R}$ and $\beta>1,$ we say that $\mu$ is pointwise $\beta$-normal if $\mu$ almost every $x$ is $\beta$-normal. We are now ready to state the main result of this paper:

We emphasize that no separation condition is imposed on the underlying IFS. Theorem 1.1 is sharp in the sense that there are many IFSs such that $s_{i}=n$ for all $i$ and some integer $n$, and no element in the attractor is $n$-normal - for example, no element in the middle-thirds Cantor set is $3$-normal. Nonetheless, at least for integer $\beta$, we will soon recall some recent results showing that even if all the contractions $s_{i}\sim\beta$ it is still possible that every self-similar measure is pointwise $\beta$-normal. We also note that there are other ways self-similar measures can lead to equidistribution - see e.g. [3]

Theorem 1.1 extends a long line of research on pointwise normality for dynamically defined measures, and provides a unified proof for many recent results regarding self similar measures. The first instances of a special case of Theorem 1.1 were obtained by Cassels [5] and Schmidt [28] around 1960. They considered the Cantor-Lebesgue measure on the middle thirds Cantor set, proving that almost every point is normal to base $n$ as long as $n\not\sim 3$. This was later generalized by Feldman and Smorodinsky [8] to all non-degenerate Cantor-Lebesgue measures with respect to any base $b$. We remark that the work of Host [12] and the subsequent works of Meiri [18] and Lindenstrauss [17] about pointwise normality for $T_{n}$-invariant measures are also related to this, though in the somewhat different context of Furstenberg’s $\times 2,\times 3$ Conjecture.

In 2015, Hochman and Shmerkin [11, Theorem 1.4] proved Theorem 1.1 under the additional assumptions that $s_{i}>0$ for all $i$, and the intervals $f_{i}(J)$, $i\in J$ are disjoint except potentially at their endpoints (a condition stronger than the open set condition). In fact, [11, Theorem 1.2] provides a general criterion for a uniformly scaling measure (that is, a measure such that at almost every point its scenery flow equidistributes for the same distribution) to be pointwise $\beta$-normal. We refer the reader to Section §1.2 for the relevant definitions about the scenery flow. The main reason they imposed this separation condition was to ensure that the corresponding self-similar measures are indeed uniformly scaling. This fact had previously been established by Hochman [10, Section 4.3]. Recently, Pyörälä [23] proved that it suffices to assume that the IFS satisfies the weak separation condition [15, 32] in order for self similar measures to be uniformly scaling. Utilizing this result and the methods from [11], Pyörälä obtained a version of Theorem 1.1 under the additional assumption that the IFS satisfies the weak separation condition [23, Corollary 5.4]. We note that there are many self-similar sets which satisfy neither the open set nor the weak separation condition; indeed, this is often the generic behaviour in parametrized families of self-similar sets: see [22].

Very recently, Algom, Rodriguez Hertz, and Wang, proved that if the Fourier transform of a self similar measure decays to $0$ at $\infty$ then almost every point is absolutely normal [2, Theorem 1.4]. Combining this with recent progress on the Fourier decay problem for self similar measures leads to many special cases of Theorem 1.1: For example, by a result of Li and Sahlsten [16], if there are $i,j$ such that $s_{i}\not\sim s_{j}$ then all self similar measures have this property, and consequently are supported on absolutely normal numbers. Furthermore, via [2, Theorem 1.4] and the recent work of Brémont [4] one obtains many instances of self similar measures that are supported on $n$-normal numbers even though the underlying IFS satisfies $s_{i}\sim n$ for all $i$. Examples of this form were also recently obtained by Dayan, Ganguly, and Weiss [7].

The results of [2] are related to a classical theorem of Davenport-Erdős-LeVeque [6]. This result states that if the Fourier transform of a Borel measure decays to zero sufficiently quickly, then a typical number with respect to this measure will be normal (for integer bases). Recall that by [2, Theorem 1.4], this property holds for self-similar measures with a decaying Fourier transform, regardless of the rate of decay. In [19] it was shown that for any homogeneous IFS with non-atomic self-similar measure $\mu$, if $g$ is a $C^{2}$ function satisfying $g^{\prime\prime}>0$ then the Fourier transform of $g\mu$ decays to zero sufficiently quickly so that the Davenport-Erdős-LeVeque theorem applies. For dynamically defined measures there are many recent results that establish a sufficiently fast rate of decay for this result to apply: [1, 13, 27, 16, 30, 31, 24] to name just a few examples (see e.g. [2] for many more references).

Let us compare Theorem 1.1 with the results of [2, 11, 23]: First, we fully relax the separation conditions on the IFS from [11, 23]. Second, while no separation conditions are needed for [2, Theorem 1.4], it does not cover two important cases where Theorem 1.1 does apply: The general case when $s_{i}\sim s_{j}$ for all $i,j$, and pointwise normality for non-integer Pisot numbers. Third, our work gives a simple and unified approach to this problem that is relatively self-contained. In particular, we do not need to invoke any results on the Fourier transform of self-similar measures as in [2]. On the other hand, we note that the methods and results of both [11] and [2] apply for more general self-conformal IFS’s (i.e. they allow for non-affine smooth maps in the IFS), and that most of the results of [11] work for a broader class of measures on the fractal.

We conclude this section with a brief informal discussion of our method. It consists of three steps: Let $\mu$ be a self similar measure as in Theorem 1.1, and let $\beta$ be a Pisot number such that $\beta\not\sim s_{i}$ for some $i$. We want to show that $\mu$ is pointwise $\beta$-normal. The first and most critical step is to express $\mu$ as an integral over a certain family of random measures. This is based upon a technique that first appeared in [9], and was subsequently applied in [14] and [26], to study the dimension of $\mu$. What was important for these authors was that these random measures could be expressed as an infinite convolution. The crucial new feature in our construction is to ensure that these random measures typically satisfy a certain dynamical self-similarity relation with strong separation. We are also able to preserve the arithmetic condition $\beta\not\sim s_{i}$, in some sense, into these random measures.

In the second step of the proof, we show that the scenery flow of typical random measures satisfying this dynamical self-similarity relation with strong separation equidistributes for the same non-trivial ergodic distribution, which we construct explicitly.

The third and final step of the proof is to use spectral analysis of this distribution together with independence from $\beta$ to conclude, via [11, Theorem 1.2], that almost every random measure is pointwise $\beta$-normal. Since in the first step we disintegrated $\mu$ according to these measures, Theorem 1.1 follows. The assertion made in Theorem 1.1 about $C^{1}$ pushforwards of $\mu$ also follows along these lines, by noting that $C^{1}$ images of typical random measures will still be pointwise $\beta$-normal by [11, Theorem 1.2].

In the next section, after recalling some definitions, we make this sketch precise and formally state the main steps in the proof.

We begin by recalling the notion of a model (as in e.g. [26]): Let $I$ be a finite set of iterated function systems of similarities $\Phi^{(i)}=(f_{1}^{(i)},...,f_{k_{i}}^{(i)})$, $i\in I$. We assume that each IFS is homogeneous and uniformly contracting. That is, for every $i\in I$ there is some $r_{i}\in(-1,0)\cup(0,1)$ such that for every $1\leq j\leq k_{i}$ we have

We allow $k_{i}$ to equal $1$ for some $i$, i.e. to have degenerate iterated function systems.

Let $\Omega=I^{\mathbb{N}}$. Given a sequence $\omega=(\omega_{n})_{n\in\mathbb{N}}\in\Omega$ we define the space of words of length $n$ (possibly with $n=\infty$) with respect to $\omega$ via

Next, for every $\omega$ we define subsets of $\mathbb{R}$ via

Thus, for every $\omega$ we have a surjective coding map $\Pi_{\omega}:X^{(\omega)}_{\infty}\rightarrow Y^{(\omega)}$ defined by

Let $\sigma$ denote the left-shift on $\Omega$ (or other shift spaces), and note that for every $\omega\in\Omega$ the following dynamical self-similarity relation holds:

We say that the model satisfies the Strong Separation Condition (SSC) if the union in (2) is disjoint for all $\omega\in\Omega$. Here we adopt the convention that the union of a single set is considered to be disjoint.

Next, for each $i\in I$, let $\mathbf{p}_{i}=(p_{1}^{(i)},...,p_{k_{i}}^{(i)})$ be a probability vector with strictly positive entries. On each $X_{\infty}^{(\omega)}$ we can then define the product measure $\bar{\eta}^{(\omega)}:=\prod_{n=1}^{\infty}\mathbf{p}_{\omega_{n}}$. The projection of $\bar{\eta}^{(\omega)}$ via the coding map is a Borel probability measure $\eta^{(\omega)}$ supported on $Y^{(\omega)}$. For every $\omega$ the measure $\eta^{(\omega)}$ also satisfies a dynamical self similarity relation, namely,

Finally, let $\mathbb{P}$ be a $\sigma$-invariant measure on $\Omega$. We refer to the triple $\Sigma=(\Phi^{(i)}_{i\in I},(\mathbf{p}_{i})_{i\in I},\mathbb{P})$ as the model under consideration. We say that the model is non-trivial if $\eta^{(\omega)}$ is non-atomic for $\mathbb{P}$-a.e. $\omega$. We say that the model is ergodic if the selection measure $\mathbb{P}$ is an ergodic measure on $\Omega$. We say that the model is Bernoulli if the selection measure $\mathbb{P}$ is a Bernoulli measure on $\Omega$.

As indicated earlier, the proof of Theorem 1.1 consists of three main steps. The first step is to prove that every self similar measure admits a disintegration over the typical measures of a model. Furthermore, this model is Bernoulli, and the arithmetic properties of the original IFS are preserved in an appropriate sense:

This theorem is proved in Section §2.

The second step in the proof of Theorem 1.1 is a general result about random model measures being uniformly scaling. Before stating this result, we recall the definition of the scaling scenery of a measure, and related notions. We follow the notations as in [11].

Let $\mathcal{P}(X)$ denote the family of Borel probability measures on a metrizable space $X$, and define

For $\mu\in\mathcal{P}(\mathbb{R})$ and $x\in\text{supp}(\mu)$, we define the translated and renormalized measure $\mu_{x}\in\mathcal{M}^{\square}$ by

Here $c=\mu([x-1,x+1])^{-1}$ is a normalizing constant. Note that in the definition of $\mu_{x}$, the measure $\mu$ does not need to be supported on $[-1,1]$ (but in any case $\mu_{x}\in\mathcal{M}^{\square}$).

For $\mu\in\mathcal{M}^{\square}$, we define the scaled measure $S_{t}\mu\in\mathcal{M}^{\square}$ by

where $c^{\prime}=\mu([-e^{-t},e^{-t}])^{-1}$ is again a normalizing constant.

The scaling flow is the Borel $\mathbb{R}^{+}$ flow $S=(S_{t})_{t\geq 0}$ acting on $\mathcal{M}^{\square}$. The scenery of $\mu$ at $x\in\text{supp}(\mu)$ is the orbit of $\mu_{x}$ under $S$, that is, the one parameter family of measures $\mu_{x,t}:=S_{t}(\mu_{x})$ for $t\geq 0$. Thus, the scenery of the measure at some point $x$ is what one sees as one “zooms into” the measure with focal point $x$.

Notice that $\mathcal{P}(\mathcal{M}^{\square})\subseteq\mathcal{P}(\mathcal{P}([-1,1]))$. As is standard in this context, we shall refer to elements of $\mathcal{P}(\mathcal{P}([-1,1]))$ as distributions (and denote them by capital letters such as $Q$), and to elements of $\mathcal{P}(\mathbb{R})$ as measures (and denote them by Greek letters such as $\mu$). A measure $\mu\in\mathcal{P}(\mathbb{R})$ generates a distribution $P\in\mathcal{P}(\mathcal{P}([-1,1]))$ at $x\in\text{supp}(\mu)$ if the scenery at $x$ equidistributes for $P$ in $\mathcal{P}(\mathcal{P}([-1,1]))$, i.e. if

(Any limits involving measures are understood to be in the weak topology.) We say that $\mu$ generates $P$, and call $\mu$ a uniformly scaling measure, if it generates $P$ at $\mu$ almost every $x$. If $\mu$ generates $P$, then $P$ is supported on $\mathcal{M}^{\square}$ and is $S$-invariant [10, Theorem 1.7]. Moreover, $P$ satisfies a sort of translation invariance known as the quasi-Palm condition (see [10]); we will not use this fact directly but it plays a key role in the proof of [11, Theorem 1.2]) which is one of the main tools in the proof of Theorem 1.1 (and is recalled as Theorem 4.1 below).

We say that $P$ is trivial if it is the distribution supported on the atom $\delta_{0}\in\mathcal{M}^{\square}$ - a fixed point of $S$. We can now state the second main step towards the proof of Theorem 1.1:

In fact, we will derive an explicit expression for $Q$ as a factor of a suspension flow over a Markov measure. This theorem is proved in Section §3.

The third and final step is to deduce Theorem 1.1 from Theorem 1.2 and Theorem 1.3. To do this, we study the pure point spectrum of the corresponding distribution $Q$, showing that it does not contain a non-zero integer multiple of $\frac{1}{\log\beta}$. Once this is established, we apply Theorem 4.1 to obtain the desired pointwise normality result. See Section §4 for more details.

Write $\mathcal{A}:=\{1,...,n\}$ and let $\mathbf{p}$ be a strictly positive probability vector on $\mathcal{A}$. For every $M\in\mathbb{N}$ and $I=(i_{1},...,i_{m})\in\mathcal{A}^{M}$ write $\varphi_{I}=\varphi_{i_{1}}\circ\cdots\circ\varphi_{i_{M}}$, and let

Recall that we are assuming that the attractor $K_{\Phi}$ is infinite. It is not hard to check (see [29, Proof of Lemma 4.2]) that there exist some $M\in\mathbb{N}$ and $I,J\in\mathcal{A}^{M}$ such that

where $\text{Conv}(X)$ denotes the convex hull of a set $X$. In addition, notice that $\mu$, the self similar measure corresponding to $\Phi$ and $\mathbf{p}$, is also a self similar measure with respect to the IFS $\Phi^{m}$, with weights

The contraction factors of $\Phi^{m}$ include $s_{i}^{m}$, so the hypothesis that some $s_{i}\not\sim\beta$ is preserved by this iteration. This shows that we may assume, without loss of generality, that there exists $i,j\in\mathcal{A}$ such that

After relabeling, we may also assume $i=1,j=2$. We will use (5) to show that the model satisfies the SSC.

We now define $I:=\{j:\,j=0\text{ or }j\in\mathcal{A}\setminus\{1,2\}\}$. For every index in $I$ we can associate an IFS as follows:

• •

  For $j=0$ we associate the IFS $\{\varphi_{1},\varphi_{2}\}$.

• •

  For every $j\in I\setminus\{0\},$ we associate the degenerate IFS $\{\varphi_{j}\}$.

Notice that the construction above gives us a dichotomy: Each IFS in $I$ is either degenerate, or is homogeneous with strong separation in the sense of (5).

Recall that $\mathbf{p}$ was our fixed probability vector on $\mathcal{A}$. We now define a probability vector $\mathbf{q}$ on $I$ as follows:

• •

  The mass $\mathbf{q}$ gives to $0$ is $p_{1}+p_{2}$, that is, $q_{0}=p_{1}+p_{2}$.

• •

  For every $j\in I\setminus\{0\}$, the singleton $j$ gets mass $q_{j}=p_{j}$.

Recalling our notation from Section §2.1, let $\mathbb{P}$ be the corresponding Bernoulli measure $\mathbf{q}^{\mathbb{N}}$ on $I^{\mathbb{N}}:=\Omega$. This $\mathbb{P}$ will be our (Bernoulli) selection measure.

Next, for $0\in I$ we define the probability vector

For every $j\in I\setminus\{0\}$ we define the (degenerate) probability vector

Recall from Section §1.2 that with these probability vectors, on each $X_{\infty}^{(\omega)}$ we can then define the product measure $\bar{\eta}^{(\omega)}$, and the projection of $\bar{\eta}^{(\omega)}$ via the coding map is a Borel probability measure $\eta^{(\omega)}$ supported on $Y^{(\omega)}$.

In this section we show that the model $\Sigma$ constructed in Section §2.1 satisfies the conclusion of Theorem 1.2. First, the model $\Sigma$ is Bernoulli by definition. Secondly, the union in (2) is disjoint for all $\omega$: if $\omega_{1}\neq 0$ there is nothing to do; otherwise, this follows from (5) and the fact that all sets $Y^{(\omega^{\prime})}$ are contained in the original attractor $K$. Since among the $|r_{i}|$ we find integer powers of all the original $|s_{j}|$, if there is $j$ such that $s_{j}\not\sim\beta$, then there is also some $i\in I$ such that $r_{i}\not\sim\beta$.

We proceed to prove part (1) of Theorem 1.2: Let $\mu$ be the self similar measure that corresponds to the weights $\mathbf{p}$. We will show that

Since $\mu$ is the unique Borel probability measure satisfying this self-similarity relation, it will then follow that

We have

Which is what we claimed.

So far we have established all the claims in Theorem 1.2, except for the non-degeneracy of the model: We claim that for $\mathbb{P}$-a.e. $\omega$, the measure $\eta^{(\omega)}$ is non-atomic. Indeed, by the SSC for any $\omega$ the coding map $\Pi_{\omega}$ is injective. So by our choice of weights it follows that for every $n\in\mathbb{N}$ and $I_{n}\in X_{n}^{(\omega)}$, we have

Since $\mathbb{P}$ is a Bernoulli measure on $I^{\mathbb{N}}$ with strictly positive weights, $\mathbb{P}$-a.s. the digit $0$ occurs in $\omega$ infinitely times. This shows that the right hand side of (6) tends to $0$ as $n\rightarrow\infty$ for $\mathbb{P}$-a.e. $\omega$. Since any point in $X_{\infty}^{(\omega)}$ is covered by sets $\Pi_{\omega}\left(I_{n}\right)$ for all $n$, the measures $\eta^{(\omega)}$ are $\mathbb{P}$-a.s. non-atomic, as claimed. This concludes the proof of Theorem 1.2.

Assume the conditions of Theorem 1.3 hold true. In particular, recall that we are assuming the SSC (2), and that the selection measure $\mathbb{P}$ is a non-degenerate Bernoulli measure. Without loss of generality, we may assume that the sets $(f_{u}^{(\omega_{1})}(Y^{(\sigma\omega)}))_{u\in X_{1}^{(\omega)}}$ are at distance $>2$ from each other for all $\omega\in\Omega$. See the discussion in §3.3 on how to treat the general case.

First, we define the random pair $(\omega,u)$, where $\omega$ is drawn according to $\mathbb{P}$ and $u$ is drawn according to $\overline{\eta}^{(\omega)}.$ Let $\mathbb{P}^{\prime}$ denote this distribution. In fact, $\mathbb{P}^{\prime}$ is the Bernoulli measure on the symbols $\{(i,u):i\in I,1\leq u\leq k_{i}\}$ with weights $q_{i,u}=\mathbb{P}([i])\cdot p_{u}^{(i)}$. We let $M$ be the shift map on $\Omega^{\prime}=\operatorname{supp}(\mathbb{P}^{\prime})$. In particular, $\mathbb{P}^{\prime}$ is $M$-invariant and ergodic. Write $\eta_{\omega,u}=\eta^{(\omega)}_{\Pi_{\omega}(u)}$ for simplicity; recall that this is the translation of $\eta^{(\omega)}$ that centers it at $\Pi_{\omega}(u)$.

We remark that our proof of Theorem 1.3 in the orientation preserving case also works under the weaker assumption that $\Sigma$ is a non-trivial ergodic model. In this case $\mathbb{P}^{\prime}$ is no longer Bernoulli but can still be seen to be ergodic. Our proof of this theorem in the orientation reversing case relies upon the ergodicity of certain group extensions. The ergodicity of these group extensions does not always hold if our model is just assumed to be ergodic. Fortunately however, ergodicity can be shown to hold if our model is Bernoulli.

Throughout this section we will assume that our model is such that there exists at least one $i$ satisfying $r_{i}<0$. This case is slightly more complicated because as we zoom in on $\eta_{\omega,x},$ we are not necessarily going to see $\eta_{M(\omega,x)}$. Because of the presence of orientation reversing maps, it is possible that as we zoom in on $\eta_{\omega,x},$ we may see the image of $\eta_{M(\omega,x)}$ reflected around the origin. Our analogue of the map $M$ has to take this behaviour into account, and our distribution $Q$ also has to see these reflected measures.

We again assume the conditions of Theorem 1.3 are satisfied and that the sets $(f_{u}^{(\omega_{1})}(Y^{(\sigma\omega)}))_{u\in X_{1}^{(\omega)}}$ are at distance $>2$ from each other. We let $\mathbb{P}^{\prime}$ be as in the orientation preserving case, and recall that its support is denoted by $\Omega^{\prime}$. We define a map $M^{\prime}:\Omega^{\prime}\times\mathbb{Z}_{2}\to\Omega^{\prime}\times\mathbb{Z}_{2}$ as follows:

We let $\mathfrak{m}$ denote the Haar measure on $\mathbb{Z}_{2}$ and let $\mathbb{P}^{\prime\prime}=\mathbb{P}^{\prime}\times\mathfrak{m}.$

To each element of $\Omega^{\prime}\times\mathbb{Z}_{2}$ we associate a measure as follows: Let $E\subset[-1,1]$ be a Borel set, then

What remains of the proof of Theorem 1.3 in the orientation reversing case is an almost identical argument to that presented in the orientation preserving case after Claim 3.1 is established. For the purpose of completion, we mention that in this case the distribution $Q$ is defined as the push-forward of $\overline{Q}$ under the factor map $(\omega,x,a,t)\mapsto S_{t}\eta_{\omega,x,a}$, where $\overline{Q}$ is the suspension measure over $\mathbb{P}^{\prime\prime}$ with roof function $-\log|r_{\omega_{1}}|$.

Finally, we discuss the general case when the smallest distance between the sets $(f_{u}^{(\omega_{1})}(Y^{(\sigma\omega)}))_{u\in X_{1}^{(\omega)}}$ is not larger than $2$ (note that by compactness, under the SSC there is indeed a smallest positive such distance). There are (at least) two ways to get around this. The first is to note that this can be achieved by rescaling all the translations of the IFS’s in the model by a common factor. The second is, assuming $2t_{0}>0$ is smaller than the smallest distance between the sets $(f_{u}^{(\omega_{1})}(Y^{(\sigma\omega)}))_{u\in X_{1}^{(\omega)}}$, to condition our measures not on $[-1,1]$ but rather on $[-t_{0},t_{0}]$ in the magnification process. By the discussion in [10, Section 1 and Section 3.1], the first way does not affect the uniformly scaling nature of the measures or the generated distributions, and the second has a corresponding very mild such effect.