Geometric and combinatorial properties of self-similar multifractal measures

2.1 Iterated function systems of similarities in ${\mathbb {R}}$

Let $\mathcal {I}$ be a non-empty finite index set. By an iterated function system of similarities (IFS) $\{S_{i}\}_{i\in \mathcal {I}}$ , we mean a finite set of similarities

(2.1) $$ \begin{align} S_{i}(x)=r_{i}x+d_{i}:\mathbb{R}\rightarrow \mathbb{R}\quad\text{for each } i\in\mathcal{I} \end{align} $$

with $0<\vert r_{i}\vert <1$ . We say that the IFS is (positive) equicontractive if each $r_{i}=r>0$ .

Each IFS generates a unique non-empty compact set K satisfying

$$ \begin{align*} K=\bigcup_{i\in\mathcal{I}}S_{i}(K). \end{align*} $$

This set K is known as the associated self-similar set. Throughout, we will assume K is not a singleton. By rescaling and translating the $d_{i}$ if necessary, without loss of generality, we may assume the convex hull of K is $[0,1]$ .

Given a probability vector $\boldsymbol {p}=(p_{i})_{i\in \mathcal {I}}$ , where $p_{i}>0$ and $\sum _{i\in \mathcal {I}}p_{i}=1$ , there exists a unique Borel measure ${\mu _{\boldsymbol {p}}}$ with $\operatorname {\mathrm {supp}}{\mu _{\boldsymbol {p}}}=K$ satisfying

(2.2) $$ \begin{align} {\mu_{\boldsymbol{p}}}(E) = \sum_{i\in\mathcal{I}}p_{i}{\mu_{\boldsymbol{p}}}(S_{i}^{-1}(E)) \end{align} $$

for any Borel set $E\subseteq K$ . This measure ${\mu _{\boldsymbol {p}}}$ is known as an associated self-similar measure.

Let $\mathcal {I}^{*}$ denote the set of all finite words on $\mathcal {I}$ . Given $\sigma =(\sigma _{1},\ldots ,\sigma _{j})\in \mathcal {I}^{*}$ , we denote

$$ \begin{align*} \sigma^{-}=(\sigma_{1},\ldots ,\sigma_{j-1})\text{, }S_{\sigma }=S_{\sigma_{1}}\circ \cdots \circ S_{\sigma_{j}}\quad\text{and}\quad r_{\sigma }=r_{\sigma_{1}}\cdots r_{\sigma_{j}}. \end{align*} $$

Given $t>0,$ put

$$ \begin{align*} \Lambda_{t}=\{\sigma \in \mathcal{I}^{\ast }:|r_{\sigma }|<t \leq |r_{\sigma^{-}}|\}. \end{align*} $$

We refer to the set of $\sigma \in \Lambda _{t}$ as the words of generation t. We remark that in the literature, it is more common to see this defined by the rule $|r_{\sigma }|\leq t <|r_{\sigma ^{-}}|$ . The two choices are essentially equivalent, but this choice is more convenient for our purposes.

2.2 Neighbour sets

The notions of net intervals and neighbour sets were introduced in [Reference Feng8, Reference Hare, Hare and Simms22]. In [Reference Hare, Hare and Rutar20], these notions were extended to an arbitrary IFS, and we present those definitions here. We then continue the discussion to define the children of a net interval, and show in Theorem 2.8 that the children depend only on the neighbour set of the parent.

Let $h_{1},\ldots ,h_{s(t)}$ be the collection of distinct elements of the set $\{S_{\sigma }(0),S_{\sigma }(1):\sigma \in \Lambda _{t}\}$ listed in strictly ascending order; we refer to this set as the endpoints of generation t. Set

$$ \begin{align*} \mathcal{F}_{t}=\{[h_{j},h_{j+1}]:1\leq j<s(t)\quad\text{and}\quad (h_{j},h_{j+1})\cap K\neq \emptyset \}. \end{align*} $$

Elements of $\mathcal {F}_{t}$ are called net intervals of generation t. Write $\mathcal {F}=\bigcup _{t>0}\mathcal {F}_{t}$ to denote the set of all possible net intervals.

Suppose $\Delta \in \mathcal {F}$ . We denote by $T_{\Delta }$ the unique contraction $T_{\Delta }(x)=rx+a$ with $r>0$ such that

$$ \begin{align*} T_{\Delta }([0,1])=\Delta. \end{align*} $$

Of course, $r=\operatorname {\mathrm {diam}}(\Delta )$ and a is the left endpoint of $\Delta $ .

Definition 2.1. We will say that a similarity $f(x)=Rx+a$ is a neighbour of $\Delta \in \mathcal {F}_{t}$ if there exists some $\sigma \in \Lambda _{t}$ such that $S_{\sigma }(K)\cap \Delta ^{\circ }\neq \emptyset $ and $f=T_{\Delta }^{-1}\circ S_{\sigma }$ . In this case, we also say that $S_{\sigma }$ generates the neighbour f. The neighbour set of $\Delta $ is the maximal set

$$ \begin{align*} {\mathcal{V}}_{t}(\Delta )=\{f_{1},\ldots ,f_{m}\}, \end{align*} $$

where each $f_{i}=T_{\Delta }^{-1}\circ S_{\sigma _{i}}$ is a distinct neighbour of $\Delta $ .

Since $K=\bigcup _{\sigma \in \Lambda _{t}}S_{\sigma }(K)$ , every net interval has a non-empty neighbour set.

If $\sigma $ generates a neighbour of $\Delta $ , then $S_{\sigma }([0,1])\supseteq \Delta $ . When the generation of $\Delta $ is implicit, we will simply write ${\mathcal {V}}(\Delta )$ . For notational convenience, we define the quantity ${{R_{\max }}}(\Delta )=\max \{|R|:\{x\mapsto Rx+a\}\in {\mathcal {V}}(\Delta )\}$ , which depends only on ${\mathcal {V}}(\Delta )$ .

Remark 2.2. For an IFS of the form $\{S_{i}(x)=r x+d_{i}\}_{i\in \mathcal {I}}$ where $0<r<1$ is fixed, the notion of a neighbour set is related to the characteristic vector of Feng [Reference Feng8]. We describe the equivalence here.

Let $\Delta =[a,b]\in \mathcal {F}_{t}$ be some net interval and let n be such that $r^{n}<t\leq r^{n-1}$ . Let $\sigma _{1},\ldots ,\sigma _{m}$ generate distinct neighbours of $\Delta $ , so that $r_{\sigma _{i}}=r^{n}$ for each $1\leq i\leq m$ . Then the (reduced) characteristic vector of $\Delta $ (see [Reference Feng8, §2] for notation) is determined by

$$ \begin{align*} \ell_{n}(\Delta) &= r^{-n}\operatorname{\mathrm{diam}}(\Delta) \quad V_{n}(\Delta) = \{r^{-n}(a-S_{\sigma_{i}}(0)):1\leq i\leq m\}, \end{align*} $$

whereas the neighbour set of $\Delta $ is given by

$$ \begin{align*} {\mathcal{V}}(\Delta) &= \{T_{\Delta}^{-1}\circ S_{\sigma_{i}}\} = \bigg\{x\mapsto \frac{S_{\sigma_{i}}(x)-a}{\operatorname{\mathrm{diam}}(\Delta)}\bigg\}\\[6pt] &= \bigg\{x\mapsto \frac{x}{r^{-n}\operatorname{\mathrm{diam}}(\Delta)}+\frac{S_{\sigma_{i}}(0)-a}{\operatorname{\mathrm{diam}}(\Delta)}\bigg\}. \end{align*} $$

Thus, when the IFS has a common positive contraction ratio, our neighbour set construction can be interpreted directly as a normalized version of Feng’s characteristic vector.

When the IFS has arbitrary contraction ratios, there is no clear choice of normalization factor analogous to $\ell _{n}(\Delta )$ that is uniform across all net intervals $\Delta \in \mathcal {F}_{t}$ . This issue is resolved by normalizing directly by $\operatorname {\mathrm {diam}}(\Delta )$ , but now it is no longer clear how to define the children of a net interval in a global way. Instead, a local definition for the children of net intervals and the analogue of [Reference Feng8, Lemma 2.1] are given in §2.3.

Neighbour sets of net intervals are relevant in the sense that they completely determine the local geometry of K in the net interval, as well as the behaviour of associated self-similar measures on Borel subsets of the net interval. To be precise, we have the following lemma.

Lemma 2.3. Let $\{S_{i}\}_{i\in \mathcal {I}}$ be an IFS as in equation (2.1) with attractor K and associated self-similar measure ${\mu _{\boldsymbol {p}}}$ . Suppose $\Delta _{1},\Delta _{2}$ are net intervals with ${\mathcal {V}}(\Delta _{1})={\mathcal {V}}(\Delta _{2})$ . Then there exists a surjective similarity $g:\Delta _{1}\cap K\to \Delta _{2}\cap K$ and constants $c_{1},c_{2}>0$ such that if $E\subseteq \Delta _{1}$ is any Borel set,

$$ \begin{align*} c_{1}{\mu_{\boldsymbol{p}}}(E)\leq {\mu_{\boldsymbol{p}}}(g(E))\leq c_{2}{\mu_{\boldsymbol{p}}}(E). \end{align*} $$

Proof. By definition of the neighbour set, if $\Delta $ is any net interval, we have

$$ \begin{align*} \Delta\cap K = \bigcup_{f\in{\mathcal{V}}(\Delta)}(T_{\Delta}\circ f(K))\cap\Delta. \end{align*} $$

Set $g=T_{\Delta _{2}}\circ T_{\Delta _{1}}^{-1}$ so that g is clearly a similarity, and applying this observation to $\Delta _{1}$ and $\Delta _{2}$ , we have

$$ \begin{align*} g(\Delta_{1}\cap K) &= \bigcup_{f\in{\mathcal{V}}(\Delta_{1})}g(T_{\Delta_{1}}\circ f(K)\cap\Delta_{1}) = \bigcup_{f\in{\mathcal{V}}(\Delta_{1})}(g\circ T_{\Delta_{1}}\circ f(K))\cap g(\Delta_{1})\\[3pt] &= \bigcup_{f\in{\mathcal{V}}(\Delta_{2})}(T_{\Delta_{2}}\circ f(K))\cap\Delta_{2} = \Delta_{2}\cap K. \end{align*} $$

Thus g is surjective with the correct image.

We now verify the measure property. By the invariant property of the self-similar measure in equation (2.2), if $\Delta \in \mathcal {F}_{t}$ is any net interval and $E\subseteq \Delta $ is any Borel set,

$$ \begin{align*} {\mu_{\boldsymbol{p}}}(E) &= \sum_{\sigma\in\Lambda_{t}}p_{\sigma{\mu_{\boldsymbol{p}}}}\circ S_{\sigma}^{-1}(E)= \sum_{f\in{\mathcal{V}}(\Delta)}{\mu_{\boldsymbol{p}}}(f^{-1}\circ T_{\Delta}^{-1}(E))\sum_{\substack{\sigma\in\Lambda_{t}\\ \sigma\text{ generates }f}}p_{\sigma}. \end{align*} $$

Since f is a neighbour of $\Delta $ , there is at least one $\sigma $ generating f. In particular, say $\Delta _{1}\in \mathcal {F}_{t_{1}}$ and $\Delta _{2}\in \mathcal {F}_{t_{2}}$ , write ${\mathcal {V}}(\Delta _{1})={\mathcal {V}}(\Delta _{2})=\{f_{1},\ldots ,f_{m}\}$ , and set for each $1\leq i\leq m$ and $j=1,2$

$$ \begin{align*} q_{i,j} := \sum_{\substack{\sigma\in\Lambda_{t_{j}}\\ \sigma\text{ generates }f_{i}}}p_{\sigma}>0. \end{align*} $$

Set $c_{1} = \min \{q_{i,2}/q_{i,1}:1\leq i\leq m\}$ . We then have for $E\subseteq \Delta _{1}$ that $g(E)\subseteq \Delta _{2}$ so that

$$ \begin{align*} {\mu_{\boldsymbol{p}}}(g(E)) &= \sum_{i=1}^{m}{\mu_{\boldsymbol{p}}}(f_{i}^{-1}\circ T_{\Delta_{2}}^{-1}\circ g(E))q_{i,2}\\[3pt] &\geq c_{1}\sum_{i=1}^{m} {\mu_{\boldsymbol{p}}}(f_{i}^{-1}\circ T_{\Delta_{1}}^{-1}(E)) q_{i,1}= c_{1}{\mu_{\boldsymbol{p}}}(E). \end{align*} $$

Similarly, we have ${\mu _{\boldsymbol {p}}}(g(E))\leq c_{2}{\mu _{\boldsymbol {p}}}(E)$ , where $c_{2}=\min \{q_{i,1}/q_{i,2}:1\leq i\leq m\}$ .

We will revisit these ideas in §2.5.

2.3 Children of net intervals

Let $\Delta \in \mathcal {F}$ have neighbour set $\{f_{1},\ldots ,f_{m}\}$ , and for each i, let $S_{\sigma _{i}}$ generate the neighbour $f_{i}$ (recall that this means that $S_{\sigma _{i}}(K)\cap \Delta ^{\circ }\neq \emptyset $ and $f_{i}=T_{\Delta }^{-1}\circ S_{\sigma _{i}}$ ).

Definition 2.4. We define the ancestral generation of $\Delta $ , denoted $\operatorname {\mathrm {ag}}(\Delta )$ , and the transition generation of $\Delta $ , denoted $\operatorname {\mathrm {tg}}(\Delta )$ , to be positive real values such that

$$ \begin{align*} \bigcap_{i=1}^{m} (|r_{\sigma_{i}}|,|r_{\sigma_{i}^{-}}|]=(\operatorname{\mathrm{tg}}(\Delta),\operatorname{\mathrm{ag}}(\Delta)]. \end{align*} $$

Note that $0<\operatorname {\mathrm {tg}}(\Delta )\leq 1$ ; if $\Delta =[0,1]$ , we say $\operatorname {\mathrm {ag}}(\Delta )=\infty $ . It is straightforward to verify that:

• • $\operatorname {\mathrm {tg}}(\Delta )={{R_{\max }}}(\Delta )\cdot \operatorname {\mathrm {diam}}(\Delta )$ ;

• • $t\in (\operatorname {\mathrm {tg}}(\Delta ),\operatorname {\mathrm {ag}}(\Delta )]$ ;

• • for any $s\in (\operatorname {\mathrm {tg}}(\Delta ),\operatorname {\mathrm {ag}}(\Delta )]$ , $\Delta \in \mathcal {F}_{s}$ and ${\mathcal {V}}_{s}(\Delta )={\mathcal {V}}_{t}(\Delta )$ ; and

• • if $s\notin (\operatorname {\mathrm {tg}}(\Delta ),\operatorname {\mathrm {ag}}(\Delta )]$ , either $\Delta \notin \mathcal {F}_{s}$ or ${\mathcal {V}}_{s}(\Delta )\neq {\mathcal {V}}_{t}(\Delta )$ .

Let $t>0$ and $\Delta \in \mathcal {F}_{t}$ . Let $(\Delta _{1},\ldots ,\Delta _{n})\in \mathcal {F}_{\operatorname {\mathrm {tg}}(\Delta )}$ be the distinct net intervals, ordered from left to right, of generation $\operatorname {\mathrm {tg}}(\Delta )$ contained in $\Delta $ . Note that either $n>1$ or if $n=1$ , then ${\mathcal {V}}(\Delta )\neq {\mathcal {V}}(\Delta _{1})$ . Then we call the tuple $(\Delta _{1},\ldots ,\Delta _{n})$ the children of $\Delta \in \mathcal {F}_{t}$ . Note that for any child $\Delta _{i}$ of $\Delta $ , $\operatorname {\mathrm {ag}}(\Delta _{i})=\operatorname {\mathrm {tg}}(\Delta )$ .

Similarly, we define the parent of $\Delta \in \mathcal {F}_{t}$ to be the net interval $\widehat \Delta \in \mathcal {F}_{s}$ with $s> t$ , where $\Delta $ is a child of $\widehat \Delta $ .

A key feature of the preceding definitions is that, in a sense that will be made precise, the neighbour set of some net interval $\Delta \in \mathcal {F}_{\alpha }$ completely determines the placement and the neighbour set of each child of the net interval.

One purpose of the position index is to distinguish the children of $\Delta $ which have the same neighbour set.

We have the following basic result. The insight behind this result is straightforward. The children of a net interval are determined precisely by the words which generate the neighbours of maximal length. Up to normalization by the position of $\Delta $ , these correspond uniquely to the neighbours of $\Delta $ with maximal contraction factor.

Proof. Given a map $f(x)=rx+d$ , we set $R(f)=|r|$ .

Write ${\mathcal {V}}(\Delta ^{\prime })={\mathcal {V}}(\Delta )=\{f_{1},\ldots ,f_{m}\}$ , and let

$$ \begin{align*} \mathcal{W}^{\prime} &= \{T_{\Delta^{\prime}}\circ f_{i}:R(f_{i})={{R_{\max}}}(\Delta^{\prime}),1\leq i\leq m\},\\ \mathcal{W} &= \{T_{\Delta}\circ f_{i}:R(f_{i})={{R_{\max}}}(\Delta),1\leq i\leq m\} \end{align*} $$

denote the corresponding sets of neighbours corresponding to functions with maximal contraction factor, where ${{R_{\max }}}(\Delta ^{\prime })={{R_{\max }}}(\Delta )$ . Then let

$$ \begin{align*} \mathcal{C}^{\prime} &= \{S_{\tau}:\tau\in\Lambda_{\operatorname{\mathrm{tg}}(\Delta^{\prime})},S_{\tau}(K)\cap (\Delta^{\prime})^{\circ}\neq\emptyset\},\\ \mathcal{C} &= \{S_{\tau}:\tau\in\Lambda_{\operatorname{\mathrm{tg}}(\Delta)},S_{\tau}(K)\cap\Delta^{\circ}\neq\emptyset\}. \end{align*} $$

In other words, $\mathcal {C}$ is the set of words of generation $\operatorname {\mathrm {tg}}(\Delta )$ which contribute to some child of $\Delta $ , and similarly for $\Delta ^{\prime }$ . Using the observation that the only new words are those which are one-level descendants of those which generate neighbours of maximal length, we have

(2.3) $$ \begin{align} \mathcal{C} &= \{f\circ S_{j}:f\in \mathcal{W},f\circ S_{j}(K)\cap\Delta^{\circ}\neq\emptyset\}\cup\{T_{\Delta}^{-1}\circ f_{i}:R(f_{i})\neq {{R_{\max}}}(\Delta)\}\nonumber\\ &= \{T_{\Delta}\circ T_{\Delta^{\prime}}^{-1}\circ f:f\in \mathcal{C}^{\prime}\}. \end{align} $$

Note that, in the above set of equalities, we use the fact that for $f\in \mathcal {W}$ ,

$$ \begin{align*} f\circ S_{j}(K)\cap\Delta^{\circ}\neq\emptyset &\iff T_{\Delta}^{-1}\circ f\circ S_{j}(K)\cap (0,1)\neq\emptyset\\ &\iff T_{\Delta^{\prime}}\circ T_{\Delta}^{-1}\circ f\circ S_{j}(K)\cap(\Delta^{\prime})^{\circ}\neq\emptyset, \end{align*} $$

where $T_{\Delta ^{\prime }}\circ T_{\Delta }^{-1}\circ f\in \mathcal {W}^{\prime }$ .

Write $\Delta =[a,b]$ and $\Delta ^{\prime }=[a^{\prime },b^{\prime }]$ . Now consider the set $H=\{a,b\}\cup \{f(0),f(1):f\in \mathcal {C}\}\cap \Delta $ so that H is the set of all endpoints of generation $\operatorname {\mathrm {tg}}(\Delta )$ contained in $\Delta $ . Then if $H^{\prime }=\{a^{\prime },b^{\prime }\}\cup \{f(0),f(1):f\in \mathcal {C}^{\prime }\}\cap \Delta ^{\prime }$ , we observe by equation (2.3) that $T_{\Delta ^{\prime }}^{-1}(H^{\prime })=T_{\Delta }^{-1}(H)$ . Let $a=h_{1}<\cdots <h_{k+1}=b$ denote the ordered elements of H and $a^{\prime }=h_{1}^{\prime }<\cdots <h_{k+1}^{\prime }=b^{\prime }$ the ordered elements of $H^{\prime }$ , where $k=|H|-1=|H^{\prime }|-1$ . By Lemma 2.3, $(h_{i},h_{i+1})\cap K\neq \emptyset $ if and only if $(h_{i}^{\prime },h_{i+1}^{\prime })\cap K\neq \emptyset $ . Thus the children of $\Delta $ are $\{[h_{i},h_{i+1}]:(h_{i},h_{i+1})\cap K\neq \emptyset \}$ and the children of $\Delta ^{\prime }$ are $\{T_{\Delta ^{\prime }}\circ T_{\Delta }^{-1}([h_{i},h_{i+1}]):(h_{i},h_{i+1})\cap K\neq \emptyset \}$ , so $k=n=n^{\prime }$ .

Now fix some $1\leq i\leq n$ . Note that $T_{\Delta }\circ T_{\Delta ^{\prime }}^{-1}(\Delta _{i}^{\prime })=\Delta _{i}$ so that $T_{\Delta _{i}}^{-1}\circ T_{\Delta }\circ T_{\Delta ^{\prime }}^{-1}=T_{\Delta _{i}^{\prime }}^{-1}$ .

1. (i) By direct computation,

   $$ \begin{align*} \hspace{-1.5pc} {\mathcal{V}}(\Delta_{i}) &= \{T_{\Delta_{i}}^{-1}\circ f:f\in\mathcal{C},f(K)\cap\Delta_{i}^{\circ}\neq\emptyset\}\\[3pt] &= \{T_{\Delta_{i}}^{-1}\circ T_{\Delta}\circ T_{\Delta^{\prime}}^{-1}\circ f:f\in\mathcal{C}^{\prime},T_{\Delta}\circ T_{\Delta^{\prime}}^{-1}\circ f(K)\cap(T_{\Delta}\circ T_{\Delta^{\prime}}^{-1}(\Delta_{i}^{\prime}))^{\circ}\neq\emptyset\}\\[3pt] &= \{T_{\Delta_{i}^{\prime}}^{-1}\circ f:f\in\mathcal{C}^{\prime},f(K)\cap(\Delta_{i}^{\prime})^{\circ}\neq\emptyset\}={\mathcal{V}}(\Delta_{i}^{\prime}). \end{align*} $$

2. (ii) Since the $T_{\Delta }$ are isometries, $q(\Delta ,\Delta _{i}) = ({h_{i}-h_{1}})/{\operatorname {\mathrm {diam}}(\Delta )}=T_{\Delta }^{-1}(h_{i})$ since $T_{\Delta }^{-1}(h_{1})=0$ . Then the result follows since $T_{\Delta }^{-1}(h_{i})=T_{\Delta ^{\prime }}^{-1}(h_{i}^{\prime })$ .

3. (iii) We have

   $$ \begin{align*} \frac{\operatorname{\mathrm{diam}}(\Delta_{i})}{\operatorname{\mathrm{diam}}(\Delta)}=\operatorname{\mathrm{diam}}(T_{\Delta}^{-1}(\Delta_{i}))=\operatorname{\mathrm{diam}}(T_{\Delta^{\prime}}^{-1}(\Delta_{i}^{\prime}))=\frac{\operatorname{\mathrm{diam}}(\Delta_{i}^{\prime})}{\operatorname{\mathrm{diam}}(\Delta^{\prime})}. \end{align*} $$

4. (iv) Recall that for an arbitrary net interval, $\operatorname {\mathrm {tg}}(\Delta _{0}) = {{R_{\max }}}(\Delta _{0})\cdot \operatorname {\mathrm {diam}}(\Delta _{0})$ where ${{R_{\max }}}(\Delta _{0})$ depends only on ${\mathcal {V}}(\Delta _{0})$ . Apply (i) and (iii).

We thus have the desired result.

2.4 The transition graph of an iterated function system

In the context of Theorem 2.8, to understand the behaviour of the IFS, it is in a sense sufficient to track the behaviour of the neighbour sets. Thus, we construct the transition graph of the IFS. The transition graph is a directed graph $\mathcal {G}(\{S_{i}\}_{i\in \mathcal {I}})$ , possibly with loops and multiple edges, (denoted by $\mathcal {G}$ when the IFS is clear from the context) defined as follows. The vertex set of $\mathcal {G}$ , denoted $V(\mathcal {G})$ , is $\{{\mathcal {V}}(\Delta ):\Delta \in \mathcal {F}\kern1.5pt\}$ , the set of distinct neighbour sets. The edge set of $\mathcal {G}$ , denoted $E(\mathcal {G})$ , is a set of triples $(v_{1},v_{2},q)$ , where $v_{1}$ is the source vertex, $v_{2}$ is the target vertex and q is the edge label to distinguish multiple edges. The edges are given as follows: for each net interval $\Delta \in \mathcal {F}_{t}$ with children $(\Delta _{1},\ldots ,\Delta _{m})$ and for each i, we introduce an edge $e=({\mathcal {V}}_{t}(\Delta ),{\mathcal {V}}_{\operatorname {\mathrm {tg}}(\Delta )}(\Delta _{i}),q(\Delta ,\Delta _{i}))$ . By Theorem 2.8, this construction is well defined since it depends only on the neighbour set of $\Delta $ .

An (admissible) path $\eta $ in $\mathcal {G}$ is a sequence of edges $\eta =(e_{1},\ldots ,e_{n})$ in $\mathcal {G}$ where the target of $e_{i}$ is the source of $e_{i+1}$ . A path in $\mathcal {G}$ is a cycle if the path begins and ends at the same vertex.

We can encode the behaviour of the IFS symbolically using the transition graph. Given $\Delta \in \mathcal {F}_{t}$ , consider the sequence $(\Delta _{0},\ldots ,\Delta _{n})$ , where $\Delta _{0}=[0,1]$ , $\Delta _{n}=\Delta $ and each $\Delta _{i}$ is a child of $\Delta _{i-1}$ . Then the symbolic representation of $\Delta $ is the path $\eta =(e_{1},\ldots ,e_{n})$ of G, where for each $1\leq i\leq n$

$$ \begin{align*} e_{i}=({\mathcal{V}}(\Delta_{i-1}),{\mathcal{V}}(\Delta_{i}),q(\Delta_{i-1},\Delta_{i})). \end{align*} $$

Conversely, if $\eta $ is any admissible path, we say that $(\Delta _{i})_{i=0}^{k}$ is a (net interval) realization of $\eta $ if:

• • each $\Delta _{i}$ is a child of $\Delta _{i-1}$ ; and

• • each $e_{i}=({\mathcal {V}}(\Delta _{i-1}),{\mathcal {V}}(\Delta _{i}),q(\Delta _{i-1},\Delta _{i}))$ .

By construction, every admissible path has a net interval realization.

Now let $x\in K$ be arbitrary and let $(\Delta _{i})_{i=0}^{\infty }$ be a sequence of nested intervals where $\Delta _{0}=[0,1]$ and $\Delta _{i+1}$ is a child of $\Delta _{i}$ and $\{x\}=\bigcap _{i=1}^{\infty }\Delta _{i}$ . The symbolic representation of x corresponding to sequence $(\Delta _{i})_{i=0}^{\infty }$ is the infinite path $(e_{i})_{i=1}^{\infty }$ , where for each n, $(e_{1},\ldots ,e_{n})$ is the symbolic representation of $\Delta _{n}$ . The symbolic representation uniquely determines x, but if x is an endpoint of some net interval, it can happen that there are two distinct symbolic representations.

Suppose $\{S_{i}\}_{i\in \mathcal {I}}$ is of the form $\{x\mapsto r x+d_{i}\}_{i\in \mathcal {I}}$ , where $0<r<1$ . Then if $\Delta \in \mathcal {F}_{t}$ is any net interval with symbolic representation $\eta =(e_{1},\ldots ,e_{n})$ , $\operatorname {\mathrm {tg}}(\Delta )=r^{n}$ and $r^{n}< t\leq r^{n-1}$ . In other words, given the symbolic representation, we can approximate the generation of $\Delta $ .

However, when the IFS is not of this form, paths with the same length can result in net intervals in substantially different generations, and if the contraction ratios are not logarithmically commensurable (that is, $\log r_{i}/\log r_{j}\in {\mathbb {Q}}$ for any $i,j\in \mathcal {I}$ ), there is no way to resolve this in a uniform way. Thus, to approximate the change in generation along a path in the transition graph, it is necessary to assign distinct values to the edges in the transition graph.

This function is well defined by Theorem 2.8. When $\eta =(e_{1},\ldots ,e_{n})$ is an admissible path, we say $L(\eta )=L(e_{1})\cdots L(e_{n})$ .

The main point here is that if $\Delta \in \mathcal {F}_{t}$ is any net interval with symbolic representation $\eta $ , then $L(\eta )\asymp t$ with constants of comparability not depending on $\Delta $ . While the above choice of the length for an edge is not unique with this property, a straightforward argument shows that any such function must agree with L on any cycle.

2.5 Encoding the invariant measure by the transition graph

Given an IFS $\{S_{i}\}_{i\in \mathcal {I}}$ with a corresponding invariant measure ${\mu _{\boldsymbol {p}}}$ , we are interested in formulae for computing or approximating ${\mu _{\boldsymbol {p}}}(E)$ , where $E\subseteq K$ is an arbitrary Borel set. When $\{S_{i}\}_{i\in \mathcal {I}}$ satisfies the strong separation condition (that is, for $i\neq j$ , $S_{i}(K)$ and $S_{j}(K)$ are disjoint), this is straightforward since ${\mu _{\boldsymbol {p}}}(S_{\sigma }(K))=p_{\sigma }$ . However, when images of K overlap, such a formula no longer holds.

The net interval construction can be thought of as a way of converting the behaviour of the IFS on overlapping images of K into behaviour on net intervals, which are disjoint except on a countable set (which has ${\mu _{\boldsymbol {p}}}$ -measure 0). It turns out that one may also encode the dynamics of the invariant measure ${\mu _{\boldsymbol {p}}}$ using products of matrices. This technique was developed in the equicontractive case for IFS of the form $\{x\mapsto rx+d_{i}\}_{i\in \mathcal {I}}$ with ${0<r<1}$ by Feng [Reference Feng8], and extended to IFS which satisfy the finite-type condition [Reference Hare, Hare and Simms22]. Using similar techniques, we describe here how to generalize this construction to an arbitrary IFS.

Let $\{S_{i}\}_{i\in \mathcal {I}}$ be an IFS and ${\mu _{\boldsymbol {p}}}$ the self-similar measure associated to probabilities $\{p_{i}\}_{i\in \mathcal {I}}$ . The main mechanism to compute the approximate measure of net intervals is through transition matrices. Recall that $\mathcal {G}$ has vertex set $V(\mathcal {G})=\{{\mathcal {V}}(\Delta ):\Delta \in \mathcal {F}\kern1.5pt\}$ . Fix some total ordering on the set of all neighbours $\{f:f\in {\mathcal {V}}(\Delta ),\Delta \in \mathcal {F}\kern1.5pt\}$ .

Let $e\in E(\mathcal {G})$ be a fixed edge with source $v_{1}$ and target $v_{2}$ . Suppose $\Delta _{1}\supseteq \Delta _{2}$ are net intervals such that $\Delta _{1}$ is the parent of $\Delta _{2}$ and $e=({\mathcal {V}}(\Delta _{1}),{\mathcal {V}}(\Delta _{2}),q(\Delta _{1},\Delta _{2}))$ . Suppose the neighbour sets are given by ${\mathcal {V}}(\Delta _{1})=\{f_{1},\ldots ,f_{m}\}$ and ${\mathcal {V}}(\Delta _{2})=\{g_{1},\ldots ,g_{n}\}$ , where $f_{1}<\cdots <f_{m}$ and $g_{1}<\cdots <g_{n}$ . We then define the transition matrix $T(e)$ as the non-negative $m\times n$ matrix given by

(2.4) $$ \begin{align} T(e)_{i,j}=\frac{{\mu_{\boldsymbol{p}}}(g_{j}^{-1}((0,1))}{{\mu_{\boldsymbol{p}}}(f_{i}^{-1}((0,1))}\cdot p_{\ell} \end{align} $$

if there exists an index $\ell \in \mathcal {I}$ such that $f_{i}$ is generated by $\sigma $ and $g_{j}$ is generated by $\sigma \ell $ ; otherwise, set $T(e)_{i,j}=0$ . This is well defined since a neighbour f has ${f^{-1}((0,1))\cap K\neq \emptyset }$ by definition. Recall that if $\sigma ^{\prime }$ generates any neighbour of $\Delta _{2}$ , then necessarily $\sigma ^{\prime }=\sigma \ell $ for some $\sigma $ which generates a neighbour of $\Delta _{1}$ ; thus, every column of $T(e)$ has a positive entry. However, it may not hold that each row of $T(e)$ has a positive entry.

It is clear from Theorem 2.8 that this definition depends only on the edge e. If $\eta =(e_{1},\ldots ,e_{n})$ is an admissible path, we define $T(\eta )=T(e_{1})\cdots T(e_{n})$ .

Throughout, we will use ${\lVert T\rVert }=\sum _{i,j}|T_{ij}|$ to denote the matrix $1$ -norm. Suppose $\Delta \in \mathcal {F}_{t}$ is an arbitrary net interval. From the defining identity of the self-similar measure,

$$ \begin{align*} {\mu_{\boldsymbol{p}}}(\Delta)=\sum_{\sigma\in\Lambda_{t}}p_{\sigma{\mu_{\boldsymbol{p}}}}(S_{\sigma}^{-1}(\Delta)) \end{align*} $$

where, since ${\mu _{\boldsymbol {p}}}$ is non-atomic, the summation may be taken over $\sigma $ such that $S_{\sigma }^{-1}(\Delta ^{\circ })\cap K$ is non-empty. Note that $S_{\sigma }^{-1}(\Delta ^{\circ })=S_{\sigma }^{-1}\circ T_{\Delta }((0,1))=f^{-1}((0,1))$ , where $f\in {\mathcal {V}}(\Delta )$ . We thus have

(2.5) $$ \begin{align} {\mu_{\boldsymbol{p}}}(\Delta) = \sum_{f\in {\mathcal{V}}(\Delta)}{\mu_{\boldsymbol{p}}}(f^{-1}((0,1)))\sum_{\substack{\sigma\in\Lambda_{t}\\ \sigma\text{ generates }f}}p_{\sigma}. \end{align} $$

Let ${\mathcal {V}}(\Delta )=\{f_{1},\ldots ,f_{m}\}$ with $f_{1}<\cdots <f_{m}$ ; then, we denote the vector form of ${\mu _{\boldsymbol {p}}}$ by ${\boldsymbol {Q}_{\boldsymbol {p}}}(\Delta )=(q_{1},\ldots ,q_{m})$ , where

$$ \begin{align*} q_{i} = {\mu_{\boldsymbol{p}}}(f_{i}^{-1}((0,1)))\sum_{\substack{\sigma\in\Lambda_{t}\\ \sigma\text{ generates }f_{i}}}p_{\sigma}. \end{align*} $$

In particular, ${\boldsymbol {Q}_{\boldsymbol {p}}}(\Delta )$ is a strictly positive vector for any $\Delta $ and ${\mu _{\boldsymbol {p}}}(\Delta )={\lVert {\boldsymbol {Q}_{\boldsymbol {p}}}(\Delta )\rVert }$ .

With this notation, we have the following theorem.

Theorem 2.12. Let $\{S_{i}\}_{i\in \mathcal {I}}$ have associated self-similar measure ${\mu _{\boldsymbol {p}}}$ . If $\eta $ is any admissible path realized by $(\Delta _{i})_{i=0}^{m}$ ,

$$ \begin{align*} {\boldsymbol{Q}_{\boldsymbol{p}}}(\Delta_{0})T(\eta)={\boldsymbol{Q}_{\boldsymbol{p}}}(\Delta_{m}). \end{align*} $$

Proof. Suppose $\Delta _{0}\in \mathcal {F}_{t}$ and $\Delta _{m}\in \mathcal {F}_{s}$ . Say ${\mathcal {V}}(\Delta _{0})=\{f_{1},\ldots ,f_{\ell }\}$ with $f_{1}<\cdots <f_{\ell }$ and ${\mathcal {V}}(\Delta _{m})=\{g_{1},\ldots ,g_{m}\}$ with $g_{1}<\cdots <g_{m}$ . For each i, assume $\tau _{i}$ generates the neighbour $f_{i}$ , and set $\mathcal {A}_{ij}=\{\omega :\tau _{i}\omega \in \Lambda _{s},\tau _{i}\omega \text { generates }g_{j}\}$ . Then for any $1\leq j\leq m$ , we have

$$ \begin{align*} ({\boldsymbol{Q}_{\boldsymbol{p}}}(\Delta_{0})T(\eta))_{j} &= \sum_{i=1}^{\ell} {\mu_{\boldsymbol{p}}}(f_{i}^{-1}((0,1)))\bigg(\!\sum_{\substack{\sigma\in\Lambda_{t}\\ \sigma\text{ generates }f_{i}}}p_{\sigma}\!\bigg)\cdot\bigg(\sum_{\omega\in\mathcal{A}_{ij}}\frac{{\mu_{\boldsymbol{p}}}(g_{j}^{-1}((0,1))}{{\mu_{\boldsymbol{p}}}(f_{i}^{-1}((0,1))}p_{\omega}\bigg)\\[3pt] &= {\mu_{\boldsymbol{p}}}(g_{j}^{-1}((0,1))) \sum_{i=1}^{\ell}\bigg(\!\sum_{\substack{\sigma\in\Lambda_{t}\\ \sigma\text{ generates }f_{i}}}p_{\sigma}\!\bigg)\cdot\bigg(\sum_{\omega\in\mathcal{A}_{ij}}p_{\omega}\bigg)\\[3pt] &= {\mu_{\boldsymbol{p}}}(g_{j}^{-1}((0,1)))\sum_{\substack{\omega\in\Lambda_{s}\\ \omega\text{ generates }g_{j}}}p_{\omega} \end{align*} $$

so that ${\boldsymbol {Q}_{\boldsymbol {p}}}(\Delta _{0})T(\eta )={\boldsymbol {Q}_{\boldsymbol {p}}}(\Delta _{m})$ .

3.1 The essential class of the transition graph

Let $\{S_{i}\}_{i\in \mathcal {I}}$ be an IFS with associated transition graph $\mathcal {G}$ . Recall that in a directed graph $\mathcal {G}$ , an induced subgraph $\mathcal {G}^{\prime }$ is a subgraph for which there exists some set of vertices $H\subseteq V(\mathcal {G})$ such that $\mathcal {G}^{\prime }$ has vertex set H and edge set composed of every outgoing edge from a vertex in H which connects to another vertex in H.

In a finite graph, there is always at least one essential class [Reference Seneta37, Lemma 1.1]. In an infinite graph, there need not be an essential class; moreover, the essential class, if it exists, need not be finite. When $\mathcal {G}$ has exactly one essential class, we denote it by $\mathcal {G}_{\operatorname {\mathrm {ess}}}$ .

We have the following basic observation. The proof of this result is similar to the idea in [Reference Hare, Hare and Rutar20, Lemma 4.2], but we reiterate the aspects of the proof that we need here for clarity.

Proof. It suffices to show that there exists some vertex v such that if w is any other vertex, there exists an admissible path from w to v. Then the essential class is the set of all vertices $v^{\prime }$ for which there is a path from v to $v^{\prime }$ . By Proposition 3.3, there exists some $t>0$ and net interval $\Delta _{0}\in \mathcal {F}_{t}$ such that $\#{\mathcal {V}^{c}}(\Delta _{0})$ is maximal; let $v:= {\mathcal {V}}(\Delta _{0})$ .

Now, let $w\in V(\mathcal {G})$ be arbitrary and $\Delta \in \mathcal {F}$ such that ${\mathcal {V}}(\Delta )=w$ . Since $\Delta ^{\circ }\cap K\neq \emptyset $ , there exists some $\sigma \in \mathcal {I}^{*}$ such that $S_{\sigma }(K)\subseteq \Delta $ and $r_{\sigma }>0$ . Set $\gamma =r_{\sigma }\cdot t$ and let $\Delta _{1}:= S_{\sigma }(\Delta _{0})$ .

Let $\Delta _{0}=[a,b]$ have covering neighbours generated by words $\{\omega _{1},\ldots ,\omega _{m}\}$ with $\omega _{i}\in \Lambda _{t}$ . By definition of $\gamma $ , $\{\sigma \omega _{1},\ldots ,\sigma \omega _{m}\}$ are words of generation $\Lambda _{\gamma }$ . Note that $(\Delta _{1})^{\circ }\cap K\neq \emptyset $ and that the endpoints of $\Delta _{1}$ are of the form $S_{\sigma \zeta }(z)$ , where $z\in \{0,1\}$ and $\zeta \in \Lambda _{t}$ , so that $\sigma \zeta \in \Lambda _{\gamma }$ . In particular, if $\Delta _{1}\notin \mathcal {F}_{\gamma }$ , then there exists some $\tau \in \Lambda _{\gamma }$ such that $S_{\tau }\notin \{S_{\sigma \omega _{1}},\ldots ,S_{\sigma \omega _{m}}\}$ and $S_{\tau }([0,1])\supseteq \Delta _{1}$ . However, then there exists some $\Delta _{2}\in \mathcal {F}_{\gamma }$ with $\Delta _{2}\subseteq \Delta _{1}\cap S_{\tau }([0,1])$ , where $\Delta _{2}$ has distinct covering neighbours generated by $\{\omega _{1},\ldots ,\omega _{m}\}\cup \{\tau \}$ , contradicting the maximality of $\#{\mathcal {V}^{c}}(\Delta _{0})$ .

Thus $\Delta _{1}$ is in fact a net interval of generation $\gamma $ . Moreover, since $r_{\sigma }>0$ , we have $T_{\Delta _{1}}=S_{\sigma }\circ T_{\Delta _{0}}$ , so that

$$ \begin{align*} {\mathcal{V}^{c}}(\Delta_{1})=\{T_{\Delta_{1}}^{-1}\circ S_{\sigma \omega_{i}}\}_{i=1}^{m}=\{T_{\Delta_{0}}^{-1}\circ S_{\sigma }^{-1}\circ S_{\sigma }\circ S_{\omega_{i}}\}_{i=1}^{m}={\mathcal{V}^{c}}(\Delta_{0}). \end{align*} $$

Thus by Remark 3.2, we have ${\mathcal {V}}(\Delta _{1})=v$ and $\Delta _{1}\subseteq \Delta $ , so that there exists a path from ${\mathcal {V}}(\Delta )$ to ${\mathcal {V}}(\Delta _{1})$ , as claimed.

If $\Delta $ is an essential net interval, then $\Delta ^{\circ }\cap K\subseteq K_{\operatorname {\mathrm {ess}}}$ . Of course, a given path $(e_{j})_{j=1}^{\infty }$ is eventually in the essential class if and only if a single edge is in the essential class. One may verify that the set of interior essential points is the topological interior of the set of essential points; in particular, the essential points form an open set in K. Interior essential points play an important role in the multifractal analysis of self-similar measures under the weak separation condition.

In the next proposition, we observe that interior essential points are abundant.

Proof. To see that $S_{\sigma }(U(x_{0},t_{0}))$ also attains the maximal value in equation (3.1), if

$$ \begin{align*} \mathcal{S}_{t_{0}}(U(x_{0},t_{0}))=\{S_{\phi_{1}},\ldots,S_{\phi_{m}}\}, \end{align*} $$

then $S_{\sigma \phi _{i}}\in \mathcal {S}_{|r_{\sigma }|t_{0}}(S_{\sigma }(U(x_{0},t_{0})))$ for each i and $\#\mathcal {S}_{|r_{\sigma }|t_{0}}(S_{\sigma }(U(x_{0},t_{0})))\geq m$ . Then equality holds by maximality of m.

We now see (ii). By definition of net intervals, we know that for any $t>0$ , $U(x_{0},t_{0})\cap K$ is contained in a finite union of net intervals of generation t. In particular, it suffices to show that there is some $t_{1}>0$ such that the set

$$ \begin{align*} \{\Delta\in\mathcal{F}_{t_{1}}:\Delta\cap U(x_{0},t_{0})\neq\emptyset\} \end{align*} $$

is composed only of essential net intervals. Let $\Delta _{0}$ be a fixed essential net interval and let $\sigma _{0}\in \mathcal {I}^{*}$ have $r_{\sigma _{0}}>0$ and $S_{\sigma _{0}}([0,1])\subseteq \Delta _{0}$ . As argued above, $S_{\sigma _{0}}(U(x_{0},t_{0}))$ also attains the maximal value in equation (3.1). Let

$$ \begin{align*} H=\{S_{\sigma}:\sigma\in\Lambda_{r_{\sigma_{0}}t_{0}},S_{\sigma}(K)\cap S_{\sigma_{0}}(U(x_{0},t_{0}))=\emptyset\}. \end{align*} $$

Since $S_{\sigma _{0}}(U(x_{0},t_{0}))$ is open, there exists some $\epsilon _{0}>0$ such that for any $\epsilon $ with $|\epsilon |<\epsilon _{0}$ , $S_{\sigma _{0}}(U(x_{0}+\epsilon ,t_{0}))$ also attains the maximal value in equation (3.1). In particular, if $S_{\sigma }\in H$ is arbitrary, we in fact have $S_{\sigma }(K)\cap S_{\sigma _{0}}(B(x_{0},t_{0}))=\emptyset $ . Since H is a finite set, take

$$ \begin{align*} t_{1} = \min\{\min\{\operatorname{\mathrm{dist}}(f(K),S_{\sigma_{0}}(B(x_{0},t_{0}))) : f\in H\},t_{0}\}>0. \end{align*} $$

It remains to show that such a $t_{1}$ works.

Write $\mathcal {S}_{t_{0}}(U(x_{0},t_{0}))=\{S_{\phi _{1}},\ldots ,S_{\phi _{m}}\}$ and set

$$ \begin{align*} F = \{\Delta\in\mathcal{F}_{t_{1}}:\Delta\cap U(x_{0},t_{0})\neq\emptyset\}. \end{align*} $$

Suppose for contradiction there is some $\Delta \in F$ that is not an essential net interval, and let $\Delta $ have neighbours generated by distinct functions $\{S_{\omega _{1}},\ldots ,S_{\omega _{k}}\}$ with $\omega _{i}\in \Lambda _{t_{1}}$ . As argued in Proposition 3.5, since $\Delta _{1}:= S_{\sigma _{0}}(\Delta )$ is not a net interval with neighbour set ${\mathcal {V}}(\Delta )$ (or $\Delta _{1}$ would be a descendant of $\Delta _{0}$ , and hence essential), there exists some $\tau \in \Lambda _{r_{\sigma _{0}}t_{1}}$ such that $S_{\tau }(K)\cap \Delta _{1}^{\circ }\neq \emptyset $ and $S_{\tau }\neq S_{\sigma _{0}\omega _{i}}$ for each $1\leq i\leq k$ . We also observe that

(3.2) $$ \begin{align} \{S_{\sigma_{0}\omega_{1}},\ldots,S_{\sigma_{0}\omega_{k}}\}=\{S_{\sigma_{0}\xi}:\xi\in\Lambda_{t_{1}},S_{\sigma_{0}\xi}(K)\cap\Delta_{1}^{\circ}\neq\emptyset\}. \end{align} $$

Since $t_{1}\leq t_{0}$ , let $\tau _{1}\preccurlyeq \tau $ be the unique prefix in $\Lambda _{r_{\sigma _{0}} t_{0}}$ . Suppose for contradiction $S_{\tau _{1}}(K)\cap S_{\sigma _{0}}(U(x_{0},t_{0}))\neq \emptyset $ . Since $S_{\sigma _{0}}(U(x_{0},t_{0}))$ attains the maximal value in equation (3.1), we have $S_{\tau _{1}}=S_{\sigma _{0}}\circ S_{\omega }$ for some $S_{\omega }\in \mathcal {S}_{r_{\sigma _{0}} t_{0}}(S_{\sigma }(U(x_{0},t_{0})))$ . Thus there exists some word $\xi $ such that $S_{\tau }=S_{\sigma _{0}}\circ S_{\xi }$ , which contradicts equation (3.2). We thus have that $S_{\tau _{1}}(K)\cap S_{\sigma _{0}}(U(x_{0},t_{0}))=\emptyset $ so that $S_{\tau _{1}}\in H$ .

However, by definition of $\Delta _{1}$ , we have that $\Delta _{1}\cap S_{\sigma _{0}}(U(x_{0},t_{0}))\neq \emptyset $ and $\Delta _{1}^{\circ }\cap S_{\tau _{1}}(K)\neq \emptyset $ , so

$$ \begin{align*} \operatorname{\mathrm{dist}}(S_{\tau_{1}}(K),S_{\sigma_{0}}(U(x_{0},t_{0})))<\operatorname{\mathrm{diam}}(\Delta_{1})\leq t_{1}, \end{align*} $$

contradicting the choice of $t_{1}$ . Thus every $\Delta \in F$ is in fact essential, as claimed.

3.2 An important measure approximation lemma

The following technical lemma is a key approximation property for measures satisfying the weak separation condition and the main factor behind the regularity of the measure on the essential class. Note the similarity of the result to the weak separation ‘counting’ results; see, for example, Feng and Lau [Reference Feng and Lau13, Proposition 4.1].

Proof. Since ${\mu _{\boldsymbol {p}}}(B(x,t))>0$ and ${\mu _{\boldsymbol {p}}}$ is non-atomic, $U(x,t)\cap K\neq \emptyset $ . From the weak separation condition, there exists some $\ell \in {\mathbb {N}}$ such that $\#\mathcal {S}_{t}(B(x,t))\leq \ell $ for any $x\in {\mathbb {R}}$ and $t>0$ . By the invariant property of ${\mu _{\boldsymbol {p}}}$ and since ${\mu _{\boldsymbol {p}}}$ is a probability measure, we have

$$ \begin{align*} {\mu_{\boldsymbol{p}}}(B(x, t)) &= \sum_{\sigma\in\Lambda_{t}(B(x, t))}p_{\sigma{\mu_{\boldsymbol{p}}}}\circ S_{\sigma}^{-1}((B(x, t)))\leq \sum_{\sigma\in\Lambda_{t}(B(x, t))}p_{\sigma}\\[6pt] &= \sum_{S_{\omega}\in\mathcal{S}_{t}(B(x, t))}\sum_{\substack{\sigma\in\Lambda_{t}(B(x, t))\\ S_{\sigma}=S_{\omega}}}p_{\sigma}. \end{align*} $$

In particular, since $\#\mathcal {S}_{t}(B(x, t))\leq \ell $ , get $\omega _{0}$ such that

(3.3) $$ \begin{align} \sum_{\substack{\sigma\in\Lambda_{t}(B(x, t))\\ S_{\sigma}=S_{\omega_{0}}}}p_{\sigma}\geq{\mu_{\boldsymbol{p}}}(B(x, t))/\ell. \end{align} $$

Note that $S_{\omega _{0}}(K)\cap B(x, t)\neq \emptyset $ , so that $S_{\omega _{0}}([0,1])\subseteq B(x,2 t)$ . If $r_{\omega _{0}}<0$ , get $k\in \mathcal {I}$ with $r_{k}<0$ and set $\omega _{1}=\omega _{0}k$ ; otherwise, take $\omega _{1}=\omega _{0}$ . Now, let $\Delta _{0}\in \mathcal {F}_{s_{0}}$ be such that $\#{\mathcal {V}}^{c}(\Delta _{0})$ is maximal. Exactly as argued in Proposition 3.5, $\Delta _{1}:= S_{\omega _{1}}(\Delta _{0})$ is a net interval in generation $r_{\omega _{1}}\cdot s_{0}$ with ${\mathcal {V}}(\Delta _{1})={\mathcal {V}}(\Delta _{0})$ . Moreover, we know that if $\sigma $ generates some neighbour f of $\Delta _{0}$ , then $\omega _{1}\sigma $ generates the same neighbour f of $\Delta _{1}$ . Fix some $1\leq j\leq \#{\mathcal {V}}(\Delta _{1})$ and let $f_{j}$ be the neighbour of $\Delta _{1}$ corresponding to the index j. We then have by using equation (3.3) and the above observation that

$$ \begin{align*} ({\boldsymbol{Q}_{\boldsymbol{p}}}(\Delta_{1}))_{j} &= \mu(f_{j}^{-1}((0,1)))\sum_{\substack{\sigma\in\Lambda_{ s_{0} r_{\omega_{1}}}\\ \sigma\text{ generates }f_{j}}}p_{\sigma}\\[3pt] &\geq p_{k}\bigg(\sum_{\substack{\sigma\in\Lambda_{t}(B(x, t))\\ S_{\sigma}=S_{\omega_{1}}}}p_{\sigma}\bigg)\cdot\mu(f_{j}^{-1}((0,1)))\cdot\sum_{\substack{\sigma\in\Lambda_{ s_{0}}\\ \sigma\text{ generates }f_{j}}}p_{\sigma}\\[3pt] &\geq {\mu_{\boldsymbol{p}}}(B(x, t))\cdot\frac{p_{k}\cdot ({\boldsymbol{Q}_{\boldsymbol{p}}}(\Delta_{0}))_{j}}{\ell}\geq{\mu_{\boldsymbol{p}}}(B(x, t))\cdot C_{1}, \end{align*} $$

where $C_{1}:= p_{k}\cdot \min _{j}({\boldsymbol {Q}_{\boldsymbol {p}}}(\Delta _{0}))_{j}/\ell $ , which depends only on the IFS and choice of probabilities.

Now let $\eta $ be any fixed path from ${\mathcal {V}}(\Delta _{0})$ to v and let $\epsilon $ be the smallest strictly positive entry of $T(\eta )$ . Let $\Delta $ be the unique net interval with symbolic $\gamma \eta $ , where $\gamma $ is the symbolic representation of $\Delta _{0}$ . Since $T(\eta )$ is non-negative and ${\boldsymbol {Q}_{\boldsymbol {p}}}(\Delta )={\boldsymbol {Q}_{\boldsymbol {p}}}(\Delta _{1})T(\eta )$ is a positive vector, we have that $({\boldsymbol {Q}_{\boldsymbol {p}}}(\Delta ))_{j}\geq {\mu _{\boldsymbol {p}}}(B(x, t))\cdot C_{1}\cdot \epsilon $ . Taking $C:= C_{1}\epsilon $ , we see that C satisfies the requirements. Moreover, since $\Delta _{0}\in \mathcal {F}_{r_{\omega _{0}} s_{0}}$ , taking $c= s_{0}L(\eta )\cdot r_{\min }^{2}$ and noting that $ t\cdot r_{\min }\leq |r_{\omega _{0}}|\leq t$ , we have that $\Delta \in \mathcal {F}_{s}$ , where $ s\geq c t$ . Finally, $\Delta \subseteq \Delta _{1}\subseteq S_{\omega _{0}}([0,1])\subseteq B(x,2 t)$ as required.

3.3 Measure properties of the essential class

As our first consequence of this lemma, we establish that the interior essential points form a large subset of K.