# On Constructions of Fractal Spaces Using Replacement and the Combinatorial
Loewner Property

Riku Anttila<EMAIL_ADDRESS>Department of Mathematics and Statistics
P.O. Box 35
FI-40014 University of Jyväskylä  and  Sylvester Eriksson-Bique
<EMAIL_ADDRESS>Department of Mathematics and Statistics
P.O. Box 35
FI-40014 University of Jyväskylä

###### Abstract.

The combinatorial Loewner property was introduced by Bourdon and Kleiner as a
quasismmetrically invariant substitute for the Loewner property for general
fractals and boundaries of hyperbolic groups. While the Loewner property is
somewhat restrictive, the combinatorial Loewner property is very generic –
Bourdon and Kleiner showed that many familiar fractals and group boundaries
satisfy it. If $X$ is quasisymmetric to a Loewner space, it has the
combinatorial Loewner property. Kleiner conjectured in 2006 that the converse
to this holds for self-similar fractals – the hope being that this would lead
to the existence of many exotic Loewner spaces. We disprove this conjecture
and give the first examples of spaces which are self-similar and
combinatorially Loewner and which are not quasisymmetric to Loewner spaces.

In the process we introduce a self-similar replacement rule, called a linear
replacement rule, which is inspired by the work of Laakso. This produces a new
rich class of fractal spaces, where closed form computations of potentials and
their conformal dimensions are possible. These spaces exhibit a rich class of
behaviors from analysis on fractals in regards to diffusions, Sobolev spaces,
energy measures and conformal dimensions. These behaviors expand on the known
examples of Cantor sets, gaskets, Vicsek sets, and the often too difficult
carpet-like spaces. Especially the counter examples to Kleiner’s conjecture
that arise from this construction are interesting, since they open up the
possibility to study the new realm of combinatorially Loewner spaces that are
not quasisymmetric to Loewner spaces.

###### Key words and phrases:

Conformal dimension, Ahlfors regular, combinatorially Loewner, quasisymmetric
mappings, self-similar space, conformal gauge, Loewner spaces, replacement
rule, graphs, Laakso spaces, attainment problem

###### 2020 Mathematics Subject Classification:

30L10, 20F65, 51F99, 53C23, 28A78

The first author was partially supported by the Eemil Aaltonen foundation. The
second author was supported by the Research Council of Finland grant 354241.
We especially thank Mario Bonk, Jeff Cheeger, Guy C. David, Bruce Kleiner,
Mathav Murugan and Ryosuke Shimizu for discussions on Kleiner’s conjecture. We
also thank Shimizu for comments on a draft of this paper.

##  1\. Introduction

###  1.1. Overview

In this paper we answer to the negative a conjecture posed by Kleiner on
whether approximately self-similar combinatorially Loewner spaces are
quasisymmetric to Loewner spaces, [52, Conjecture 7.5]. A concise overview of
the problem and its significance can be gleaned from [52] and [29, 19]. In the
process, we give a new and rich class of spaces, derived from linear
replacement rules, which exhibits curious behaviors and offers a tractable,
and unexplored, setting to study analysis on fractals, see [74, 51, 6] for
surveys. The precise terminology is presented in detail in Section 2, which a
non-expert reader may first wish to consult, before reading the statements of
the main results.

12345678 Figure 1.1. Figure of a replacement rule that produces a counter
example to Kleiner’s question. The example, which is described in more detail
in Example 3.8, is obtained by recursively replacing every edge in the graph
by the graph depicted in the middle, and the copies corresponding to adjacent
edges are glued along their left or right boundaries. The edges in the new
graph have length of the original edges divided by four. The labels in the
figure correspond to those given in Example 3.8. The figure shows two steps of
the replacement.

###  1.2. Main theorem

The theory of quasiconformal mappings was revealed in settings of increasing
generalities: first developed on surfaces [1], then higher dimensional
Euclidean spaces [70, 79], followed by certain Carnot groups [68] and
ultimately Loewner spaces [37]; see [11, 52, 35, 19, 46] for nice surveys.
This generalization was partly motivated by quasi-isometric rigidity problems
in geometric group theory (e.g. Mostow’s rigidity theorem, [62, 15, 46]).
Indeed, it was observed that whenever the boundary of a hyperbolic group
possessed a Loewner structure in its conformal gauge, this implied rigidity
properties for the group [52, 35, 19]. The conformal gauge consists of all
metrics quasisymmetric to a given (visual) reference metric. While successful,
this approach suffered a somewhat serious limitation: there are only very few
hyperbolic groups whose boundaries are known to possess a Loewner structure.
An essentially exhaustive list is given by the following families of examples:
boundaries of groups acting by isometries co-compactly on a rank-one symmetric
space [68, 61] or a Bourdon-Pajot building [17, 18]. Prompted by this, it was
asked by several authors [52, 35, 29, 20, 46] if more hyperbolic group
boundaries would admit a Loewner structure.

Towards an answer, Bourdon and Kleiner observed in [16] that a seemingly
similar combinatorial Loewner property (CLP) was much more generic than the
Loewner property, and was much easier to establish in given examples. Indeed,
they and Clais [30] showed that far more group boundaries and fractals satisfy
it: the Sierpińki carpet, the Menger curve, boundaries of certain higher
dimensional hyperbolic buildings, and boundaries of certain Coxeter groups.
None of these examples were known to possess a Loewner metric in their
conformal gauge. Given the amplitude of such examples, Kleiner conjectured in
[52] the following.

######  Conjecture 1.2.

All approximately self-similar (or: group boundary) combinatorially Loewner
metric spaces are quasisymmetric to a Loewner space.

Kleiner’s conjecture, and related problems, have attracted considerable
attention, see e.g. [29, 64, 43, 26, 55, 23], and its statement is quite
natural. Due to a lack of counter-examples, and hopes of finding new exotic
Loewner spaces, there have been several attempts at proving the conjecture
true, and an expectation by many that the conjecture would have a positive
answer. However, a negative answer also would be quite intriguing: it opens up
the possibility to study the combinatorial Loewner porperty as something
strictly weaker and more general than the Loewner property.

As we will discuss below, when we introduce the terminology in detail, this
problem is also equivalent to the attainment problem for the Ahlfors regular
conformal dimension. This attainment problem has also been studied
extensively, and is known to be quite difficult. In a different disguise, the
same problem has also been studied for conductively homogeneous spaces, see
[64, 51]. In that setting, it is closely related to the attainment problem of
other critical dimensions, such as the conformal walk dimension – whose
attainment problem is much better understood thanks to [43].

Our main contribution is to answer the main part of Kleiner’s conjecture to
the negative by giving the first examples of approximately self-similar
combinatorially Loewner metric spaces which are not quasisymmetric to Loewner
spaces.

######  Theorem 1.3.

There exists a compact metric space $X$, which is approximately self-similar
and combinatorially $Q$-Loewner for some $Q>1$, but which is not
quasisymmetric to a Loewner space. This space also does not attain its Ahlfors
regular conformal dimension.

The counter examples are obtained by a new construction of fractal spaces that
we call _linear replacement rules_. A linear replacement rule consists of a
given graph $G_{1}=(V_{1},E_{1})$ together with an iteration procedure that
produces a sequence of graphs $G_{m}=(V_{m},E_{m})$. The graphs $G_{m+1}$ for
$m\in\mathbb{N}$, are obtained recursively by replacing each edge of $G_{m}$
by a copy of $G_{1}$ and identifying neighboring copies by a given rule. The
resulting graphs $G_{n}$ are equipped with a re-scaled path metric $d_{n}$.
Under some easily checkable conditions the sequence of metric spaces
$(G_{n},d_{n})$ Gromov-Hausdorff converge to a limit space $X$. (At this
juncture, an eager reader may check Definition 3.2.)

With specific graphs, see e.g. Figure 1.5 and Example 4.23, such linear
replacement rules produce variants of fractals called Laakso-spaces [57],
although the construction is distinct from the quotient-construction of Laakso
and the identifications used are different. These can also be realized by an
inverse limit construction, see [28], and as we will see in Example 5.14 they
are Loewner spaces. Our key insight is that such constructions can be modified
by adding edges, which we call _removable edges_ , to produce spaces that do
not arise as inverse limits. Such spaces are still combinatorially Loewner. It
is one these modified examples that are depicted in Figure 1.1 which yields
one counter-example to Kleiner’s conjecture. Indeed, the example in this
figure shows the following explicit version of Theorem 1.3.

######  Proposition 1.4.

The space $X$ from Example 3.8 is $\log(9)/\log(4)$-Ahlfors regular,
$3/2$-combinatorially Loewner and approximately self-similar. Its conformal
dimension is equal to $3/2$, and it is not attained. In particular, the space
$X$ is not quasisymmetric to a Loewner space.

12345678 Figure 1.5. Figure of two steps of a replacement rule that produces a
variant of a Laakso space from [57]. In the figure, an edge is replaced by two
copies glued at points which lie 1/4 and 3/4 of their lengths. The graph
together with its labeling is described in more detail as part of Example 3.8.

Our result will be more general, and we will indicate which assumptions and
mechanisms lead to the failure of being quasisymmetric to a Loewner space. In
particular, we give a fairly large family of examples, all of which share
crucial features and all of which are fairly simple to describe using the
replacement procedure above. Indeed, by relatively minor assumptions, see
Assumptions 5.1, linear replacement rules produce approximately self-similar
and combinatorially Loewner limit spaces. While we do not resolve the full
question of Kleiner in all settings (in particular group boundaries or the
Sierpiński carpet), these examples indicate mechanisms and tools by which one
can prove that other examples are also not quasisymmetric to Loewner spaces.
Further work is needed to determine which combinatorially Loewner fractals
attain their conformal dimensions. Next, we shall briefly discuss
relationships to recent work.

###  1.3. Background

Until the present work, not much definitive was known about Kleiner’s
question. Due to the work of [48, 24, 72, 63, 50, 61] it was known how to
construct metrics in the conformal gauge of $X$. The construction was powerful
enough to lead to a characterization of the conformal dimension as a critical
exponent for certain discrete modulus problems, [16, 24]. It was also known,
that the hypothetical Loewner metric would be a minimizer for the Ahlfors
regular conformal dimension, see e.g. [26, Introduction]. These insights
suggested that, if a Loewner metric were to exist in the conformal gauge of a
metric space, it could be obtained by studying minimizers for discrete modulus
problems ${\rm Mod}_{Q}(\Gamma,G_{m})$ with $Q$ equal to the conformal
dimension.

By applying discretization, certain structures from this hypothetical Loewner
metric can be gleaned form the original undeformed space. Versions of Sobolev
spaces were constructed in [14, 64, 60], which would coincide with the Sobolev
space associated to the Loewner space in the sense of [25, 71, 3], if the
space was quasisymmetric to a Loewner space. Also, discrete modulus at the
critical exponent would be comparable to the continuous modulus of the
associated Loewner space [33, 67, 77]. Finally, Murugan and Shimizu
constructed an energy measure associated to the Sobolev spaces mentioned
before, that would yield a measure with respect to which the Hausdorff
$Q$-measure of the Loewner structure would be mutually absolutely continuous
[64]. Their work built upon the work of Kigami [51] and Shimizu [73].

Indeed, the work of Murugan and Shimizu presented a strategy by which the
existence of a Loewner metric in the conformal gauge could be determined in
general. This was inspired by earlier work of Kajino and Murugan [43], where
they observed that a similar strategy could be used for $p=2$ to resolve a
problem on the attainment of a related conformal walk dimension. In this
latter work the energy was associated with the exponent $p=2$, and in the
first, $p$ is chosen to coincide with the conformal dimension. In both works,
one observes that if the hypothesized structure were to exist, the measure and
metric could be obtained by considering Sobolev functions on the space and
their energy measures. Our argument also is based on computing minimizers for
modulus, but the logic is more direct, and we do not discuss energy measures
or the associated Sobolev space in this work.

###  1.4. New Fractal Spaces

Finally, we note that the fractal spaces that we construct using linear
replacement rules may have broader interest. Indeed, there has been much work
on analysis on fractals, especially regarding diffusions see e.g. [6, 49, 74,
75] and conformal geometry [61]. Some major examples in this realm have been
the Cantor sets, Sierpiński gasket, general gaskets, Sierpiński carpets,
Menger curve and the Vicsek set. Indeed, these classes of spaces are often
presented as the prototypical examples of fractals and all arise from iterated
function systems of Euclidean space. Often, the Cantor sets, gasket and Vicsek
cases have been solved first, and then (with considerably more work) the
Sierpiński carpet/Menger curve case is pursued. Cantor sets are quantitatively
totally disconnected and, in the gasket and Vicsek spaces, cut points give
significant simplifications for calculations. In the cases with local cut-
points many problems have been fully resolved, such as determining the
conformal dimension and its attainment [61, 78], and the construction of
diffusions [49]. For carpets, the lack of cut points is reflected in
significant difficulty in performing calculations. Consequently, most of the
same questions such as the attainment problem of conformal dimension are not
known for carpets (e.g. [61]), while others, such as the existence of a
diffusion [7, 8] and construction of a Dirichlet form [54] are quite involved.

The reason to study these example spaces is that they are “toy models” used to
develop tools to approach much more difficult problems, such as: the behavior
of random walks on percolation clusters (e.g. [6, Introduction]), or Cannon’s,
Kleiner’s and Kapovitch’s conjectures in geometric group theory [26,
Introduction]. This is natural for the reason that connected components of
supercritical percolation clusters in the plane, and planar group boundaries,
are homeomorphic to carpets (see e.g. [47]). It has been suggested that the
next step in difficulty from gasket-type spaces is carpet-like spaces.
However, at present, beyond some intriguing work in [16, 51, 73, 64], there is
not much progress in the study of carpets. Indeed, the difficulties in the
analysis of carpet-like spaces has stymied development of the theory.

The class of fractals that we present here are a new class of natural spaces
where to study analysis and which do not arise as attractors of iterated
function systems in Euclidian spaces. (Indeed, the metrics on these spaces
often do not bi-Lipschitz embed into Euclidean space, see e.g. [58] for an
argument for the example given in Figure 3.4). Similar to carpet’s, the
examples (often) do not have cut points. Despite this, the computations in
these examples are even somewhat simpler than those for gaskets. (This is
especially true for exponents $p\neq 2$. For $p=2$ the computations for
gaskets are also very tractable, see e.g. [51, 74] and compare these to [39,
76, 22], which showcase the $p\neq 2$ theory for gaskets.) Thus, in a sense,
the spaces lie in difficulty between gaskets and carpets, and offer a new
setting to explore prototypical behaviors of random walks and conformal
geometry. The examples have some common features: they all have topological
and Assouad-Nagata dimension equal to $1$, see Remark 3.38. Further, as
explained in Example 3.8 and as follows from [4, 5], the constructions often
yield spaces homeomorphic to the Menger curve.

Often analysis on fractals hinges upon potential theory and computing
minimizers of energy. In general this is quite difficult - especially for
carpet-like spaces (see e.g. [55]). The inability to perform exact
computations has left many interesting questions unanswered. The distinguished
feature of linear replacement rules is the ability to effectively compute
minimizers of energy. This explicit computability and richness of the family
suggest that studying the class further could yield crucial insights and may
allow one to answer further questions in analysis on fractals that have
heretofore resisted effort. Answering Conjecture 1.2 seems to be just the
first example of these insights. Another example is understanding when the so
called $p$-walk dimension $d_{w,p}$ is strictly grater than $p$, which we
define in our framework in Definition 6.10. It has been observed in multiple
different works (see e.g. [45, 64, 32]) that this problem is closely related
to the structure of Sobolev spaces. See also [51, Problem 4 in Section 6.3].
In Subsection 6.3 we present a simple characterisation in our framework of
when $d_{w,p}\geq p$ is a strict inequality and present examples for both
cases.

Next, we will go over each of the main concepts: conformal dimension, the
Loewner property and the combinatorial Loewner property. We will also explain
the proof of the counter examples using them. The linear replacement rules are
described and studied in Section 3. A reader mostly interested in the
construction of such fractal spaces and graphs can read this section
independent of most of the other sections, and this section does not require
most of the notions from quasiconformal geometry. Modulus on graphs is
presented in Section 4 and the combinatorial Loewner property for certain
linear replacement rules is proven in Section 5. In Section 6 we study
porosity and prove the main theorems. The concepts that we use are mostly
standard, and their classic theory can be read from textbooks such as [34, 61,
38].

##  2\. Core concepts

###  2.1. Conformal dimension

Let $Q>1$ and let $(X,d)$ be a metric space. A compact metric space $X$ is
said to be $Q$-_Ahlfors regular_ , if there exists a constant $C>1$ so that
the $Q$-dimensional Hausdorff measure satisfies
$C^{-1}r^{Q}\leq\mathcal{H}^{Q}(B(x,r))\leq Cr^{Q}$ for all $r\in(0,{\rm
diam}(X)]$ and all $x\in X$. Here, the _Hausdorff measure_ is given by

$\mathcal{H}^{Q}(A):=\lim_{\delta\to 0}\mathcal{H}^{Q}_{\delta}(A),$

where the _Hausdorff content_ ’s are defined by

$\mathcal{H}^{Q}_{\delta}(A):=\inf\\{\sum_{i=1}^{\infty}{\rm
diam}(A_{i})^{Q}:A\subset\bigcup_{i=1}^{\infty}A_{i},{\rm
diam}(A_{i})\leq\delta\\}.$

The value $Q$ is equal to the _Hausdorff dimension_ for a $Q$-Ahlfors regular
space. In general, if $\mu$ is any Radon measure on $X$, we say that
$(X,d,\mu)$ is $Q$-Ahlfors regular if $C^{-1}r^{Q}\leq\mu(B(x,r))\leq Cr^{Q}$
for all $r\in(0,{\rm diam}(X)]$ and $x\in X$. In this case there exists a
constant $C$ for which $\frac{1}{C}\mu\leq\mathcal{H}^{Q}\leq C\mu$, and the
metric space $X$ is also $Q$-Ahlfors regular.

Our focus will be on self-similar spaces with self-similairity defined as in
[16]. See [69] for a more detailed discussion on this, and related, notions.

######  Definition 2.1.

We say that a metric space $X$ is _approximately self-similar_ , if there
exists a constant $L\geq 1$ so that for every $x\in X$ and every $r\in(0,{\rm
diam}(X)]$, there exists an open set $U_{x,r}\subset X$ and a $L$-biLipschitz
map $f_{x,r}:(B(x,r),d/r)\to U_{x,r}$, where the ball $B(x,r)$ is equipped
with the scaled metric $d/r$.

The maps $f_{x,r}$ in the statement will be refered to as scaling maps. A map
$f:(X,d_{X})\to(Y,d_{Y})$ is a $L$-_biLipschitz map_ (or embedding) if for all
$x,y\in X$, we have

$\frac{1}{L}d_{X}(x,y)\leq d_{Y}(f(x),f(y))\leq Ld_{X}(x,y).$

Let $\eta:[0,\infty)\to[0,\infty)$ be a continuous homeomorphism. A
homeomorphism $f:(X,d_{X})\to(Y,d_{Y})$ is said to be $\eta$-_quasisymmetric_
if for every $x,y,z\in Z$ with $x\neq z$, we have

$\frac{d_{Y}(f(x),f(y))}{d_{Y}(f(x),f(z))}\leq\eta\left(\frac{d_{X}(x,y)}{d_{X}(x,z)}\right).$

We say that $f:X\to Y$ is a _quasisymmetry_ , if it is an $\eta$-quasisymmetry
for some $\eta$. In these cases, we say that $X$ is quasisymmetric to $Y$. For
more background on these mappings and the notions below, see [34]. At this
juncture, we note that we will usually omit the subscript of a metric $d$,
where the space is clear from context.

The _conformal gauge_ of a metric space $X$ is given by

$\mathcal{G}(X,d):=\\{\hat{d}:\hat{d}\text{ is a metric on }X\text{ s.t. }{\rm
id}:(X,d)\to(X,\hat{d})\text{ is a quasisymmetry}\\}.$

Here ${\rm id}$ is the identity map. This gauge is generally quite big – see
[24] for a description of metrics in this gauge (also, [63, 72]). We will
focus mostly on those metrics, which are Ahlfors regular. This gives the
_Ahlfors regular conformal gauge_ :

$\displaystyle\mathcal{G}_{\rm AR}(X,d):=\\{\hat{d}:$ $\displaystyle\ \ {\rm
id}:(X,d)\to(X,\hat{d})\text{ is a quasisymmetry and }$
$\displaystyle(X,\hat{d})\text{ is $\widehat{Q}$-Ahlfors regular for some
}\widehat{Q}>0\\}.$

Where the metric $d$ is clear from context, we will drop it in the notation
and simply write $\mathcal{G}(X),\mathcal{G}_{\rm AR}(X)$. Among all such
metrics, one may wish to find an “optimal metric”, which satisfies some nice
analytic or symmetry properties. A frequently considered problem is to
minimize the Hausdorff dimension among metrics in the (Ahlfors regular)
conformal gauge: the _(Ahlfors regular) conformal dimension_ is given by

${\rm dim}_{\rm AR}(X):=\inf\\{\widehat{Q}:\hat{d}\in\mathcal{G}_{\rm
AR}(X,d)\text{ and }(X,\hat{d})\text{ is $\widehat{Q}$-Ahlfors regular}\\}.$

A version of this invariant, the conformal Hausdorff dimension was first
considered by Pansu [67], and the present definition was given in [20].
Further variants of this conformal dimension, such as conformal Assouad, have
alse been studied. See [61] for an introduction to conformal dimension and
[26, 24] for some further work on these dimensions. The various definitions
are equivalent in self-similar settings, see [31]. Since all the spaces
considered in this paper are self-similar, we will mostly speak of just
conformal dimension to refer to these notions.

If the infimum in the definition of the (Ahlfors regular) conformal dimension
is attained, we say that $X$ attains its (Ahlfors regular) conformal
dimension. The question to determine if this happens for a given space, or a
class of spaces, is called the _attainment problem_. Determining the conformal
dimension and the attainment problem are notoriously hard. The value of the
conformal dimension can, however, be numerically estimated. Numerical
computations of the conformal dimension of the Sierpiński carpet were
performed in [55]. These calculations are based on characterizations of the
conformal dimension in terms of combinatorial modulus, see [24, 16] and the
related work [63, 72, 50]. In order to describe these characterizations, we
will next define combinatorial and continuous moduli.

###  2.2. Continuous and combinatorial modulus

Out of the two notions of moduli, the continuous one is easier to define, and
is much more classical, see e.g. [80, 38]. We will define it only in the case
when $X$ is $Q$-Ahlfors regular for some $Q>1$. While modulus can be defined
with respect to any measure on $X$, we will only consider Hausdorff
$Q$-measures in the present paper. Let $\Gamma$ be a family of curves. We say
that a Borel measurable $\rho:X\to[0,\infty]$ is admissible for $\Gamma$ if
for all rectifiable curves $\gamma\in\Gamma$ we have $\int_{\gamma}\rho ds\geq
1$, where the integration is with respect to the length measure. If
$p\in[1,\infty)$, the $p$-modulus of a curve family $\Gamma$ is defined as

${\rm Mod}_{p}(\Gamma):=\inf\left\\{\int\rho^{p}d\mathcal{H}^{Q}:\rho\text{ is
admissible for }\Gamma\right\\}.$

For a quick overview of modulus and its properties, see [80, 38]. A crucial
issue with this definition is that it is not a quasisymmetry invariant.
Indeed, different metric in the conformal gauge do not yield mutually
absolutely continuous Hausdorff-measures, nor do they agree on the class of
rectifiable curves. The only setting, where invariance can be obtained, is if
$f:X\to Y$ is a quasisymmetry between two $Q$-Ahlfors regular metric spaces –
see [77].

In order to get full invariance, the notion of modulus needs to be
discretized. There are a variety of approaches to do this, [33, 77, 36, 48,
51, 16, 31, 64, 60]. For some comparisons between different approaches, see
[31]. In each approach the space and curves are discretized in different ways.
We will follow [16] since the combinatorial Loewner property was originally
expressed in terms of it. (It may be possible to express the combinatorial
Loewner property also in terms of other notions of modulus and by employing
other graphical approximations, but this is not essential for the present
work.)

Let $\alpha,L_{*}>1$. An $\alpha$-approximation to a compact metric space $X$
is a sequence of incidence graphs
$\\{G_{m}=(V_{m},E_{m})\\}_{m\in\mathbb{N}}$, where for each $m\in\mathbb{N}$
the set of vertices $V_{m}$ is a collection of subsets of $X$, which forms a
covering of $X$, and which satisfies the following conditions

  1. (1)

For every $v\in V_{m}$ there exists a $z_{v}\in v$ for which

$B(z_{v},\alpha^{-1}L_{*}^{-m})\subset v\subset B(z_{v},\alpha L_{*}^{-m}).$

  2. (2)

For every pair of distinct $v,w\in V_{m}$ we have

$B(z_{v},\alpha^{-1}L_{*}^{-m})\cap B(z_{w},\alpha^{-1}L_{*}^{-m})=\emptyset.$

Here, an incidence graph $G=(V,E)$ of $X$ is a collection $V$ of subsets of
$X$, which cover $X$, where $\\{v,u\\}\in E$ is an edge if and only if $u\cap
v\neq\emptyset$. For a given subset $F\subseteq X$ we write

$G[F]:=\\{v\in V:X_{v}\cap F\neq\emptyset\\}.$

If $\Gamma$ is a family of subsets of $X$ we define

$G[\Gamma]:=\\{G[\gamma]:\gamma\in\Gamma\\}.$

In [16], the value $L_{*}=2$ is used exclusively, but the results apply just
the same with a different $L_{*}$. It will be useful to allow a different
$L_{*}>1$ to conform more directly with our examples. We note that in our
examples, the logic of [16] is a bit reversed. We construct a sequence of
graphs $G_{m}$, and $X$ is a limit space under Gromov-Hausdorff convergence. A
posteriori, the graphs $G_{m}$ can also be identified with an
$\alpha$-approximation in the sense of Bourdon and Kleiner. Fix
$m\in\mathbb{N},p\geq 1$ and let $\rho:V_{m}\to\mathbb{R}_{\geq 0}$, and
$\gamma\subset X$ be a continuous curve identified with its image. We define
the _$\rho$ -length_ of $\gamma$ as

$L_{\rho}(\gamma):=\sum_{v\in G_{m}[\gamma]}\rho(v).$

Let $p\in[1,\infty)$. We define the _$p$ -mass_ of $\rho$ as

$\mathcal{M}_{p}(\rho):=\sum_{v\in V_{m}}\rho(v)^{p}.$

Given a family of continuous curves $\Gamma$ in $X$, we say that
$\rho:V_{m}\to\mathbb{R}_{\geq 0}$ is _$\Gamma$ -admissible_ if
$L_{\rho}(\gamma)\geq 1$ for all $\gamma\in\Gamma$. We define the _discrete
modulus_ of $\Gamma$ with respect to $G_{m}$ by

${\rm
Mod}_{p}^{BK}(\Gamma,G_{m}):=\inf\\{\mathcal{M}_{p}(\rho):L_{\rho}(\gamma)\geq
1,\forall\gamma\in\Gamma,\rho:V_{m}\to\mathbb{R}_{\geq 0}\\}.$

Using the discrete modulus, Keith and Kleiner describe the conformal dimension
of self-similar spaces as a critical exponent – see [24] for a proof by
Carrasco, which was developed independently. For $\delta>0$ we write
$\Gamma_{\delta}=\\{\gamma:{\rm diam}(\gamma)>\delta\\}$ and

(2.2) $\mathcal{M}_{\delta,p}^{(m)}:={\rm
Mod}_{p}^{BK}(\Gamma_{\delta},G_{m}).$

######  Proposition 2.3 (Corollary 1.4 [24]).

Let $X$ be an approximately self-similar metric space. There exists a
$\delta_{0}>0$ so that for all $\delta\in(0,\delta_{0})$ we have

${\rm dim}_{\rm AR}(X)=\inf\left\\{p\geq
1:\lim_{k\to\infty}\mathcal{M}_{\delta,p}^{(k)}=0\right\\}.$

This characterization is very powerful, but difficult to use since the moduli
in question are quite challenging to compute. At present, for most spaces,
this can only be done via numerical approximation. See e.g. [55] for an
explicit numerical approximation of the conformal dimension of the Sierpiński
carpet based on this approach.

###  2.3. Loewner condition

Given the work of Heinonen and Koskela [37], and given its implications for
the study of quasisymmetric maps, one special property of a metric in the
conformal gauge is the Loewner property. This property is expressed using the
concept of a continuous modulus. Now, through its connection to capacity, see
e.g. [51, 80], it is particularly relevant to consider the moduli of path
families connecting sets. Let $E,F\subset X$ be closed sets, and let
$\Gamma(E,F)$ be the collection of rectifiable curves connecting $E$ to $F$.
The Loewner condition is given by a lower bound for the modulus of curves
between two continua. If $E,F\subset X$ are two non-degenerate continua (i.e.
compact connected subsets with ${\rm diam}(E),{\rm diam}(F)\neq 0$), then
their relative distance is given by $\Delta(E,F)=\frac{d(E,F)}{\min\\{{\rm
diam}(E),{\rm diam}(F)\\}}$.

A metric space $(X,d)$ is said to be $Q$-Loewner, if it is $Q$-Ahlfors regular
and

${\rm Mod}_{Q}(\Gamma(E,F))\geq\phi(\Delta(E,F)^{-1}),$

for some increasing function $\phi:[0,\infty)\to[0,\infty)$ and for all non-
degenerate continua $E,F\subset X$.

The problems of minimizing Hausdorff dimension in the conformal gauge, and
that of finding a Loewner metric in the conformal gauge are very closely
related. First, if $(X,d^{\prime})$ is $Q^{\prime}$-Loewner for some metric
$d^{\prime}\in\mathcal{G}_{\rm AR}(X)$, then $Q^{\prime}={\rm dim}_{\rm
AR}(X)$, and thus the metric $d^{\prime}$ attains the infimum in the
definition of the conformal dimension. This follows from [77, 67]. Indeed, the
Loewner property is somewhat excessive here, since it suffices that ${\rm
Mod}_{Q^{\prime}}(\Gamma)>0$ for any family of rectifiable curves.

Conversely, if $(X,d^{\prime})$ is $Q^{\prime}$-Ahlfors regular,
$d^{\prime}\in\mathcal{G}_{\rm AR}(X)$ and $Q^{\prime}={\rm dim}_{\rm AR}(X)$,
then $X$ almost admits a positive modulus family of curves. Indeed, Keith and
Laakso showed that a weak tangent of $X$ admits such a positive modulus family
of curves [48]. If $X$ is further assumed to be self-similar, then this family
of curves can be elevated to the space $X$, see e.g. [13, Proof of Corollary
1.6].

The Loewner property is, however, stronger than a mere positivity of modulus
for some family of curves. It involves a quantitative, and scale-invariant,
lower bound for the connecting modulus between any pair of continua. However,
it was already observed in [13], that further symmetry in the space elevates
having positive modulus to a Loewner property. Thus, the Loewner property can
be seen as involving three ingredients: positive modulus, scale invariance and
symmetry. Further, as a consequence, at least for group boundaries, the
attainment of conformal dimension is equivalent to the existence of a Loewner
metric in the conformal gauge, [13]. How can one then recognize the existence
of a Loewner metric in the conformal gauge? Bourdon and Kleiner proposed a
discrete version of the Loewner property, the combinatorial Loewner property
as a potential condition for this [16].

$\varepsilon$ Figure 2.4. The figure shows a deformation of the example in
Figure 1.1, where the central edge has a length $\varepsilon$, while each
other edge still has length $1/4$. It can be shown, that the map from the
original graph to this space is quasisymmetric

######  Example 2.5.

At this juncture it is helpful to return to the primary counter example from
Figure 1.1. We will be a bit informal and will not explain all details, since
we will later give a detailed proof of the main theorem using somewhat
different ideas. However, the argument given here gives a good intuition of
what is going on.

Let $Y$ be obtained from the replacement rule in Figure 1.5. It is self-
similar and consists of $8$ copies scaled by $1/4$. Thus this example has
Hausdorff dimension equal to $\log(8)/\log(4)=3/2$. The space $Y$ is also
$3/2$-Loewner space, and thus has conformal dimension equal to its Hausdorff
dimension. This can be seen in one of three ways. Two proofs are obtained by
recognizing the construction as a variant of the Laakso space construction
from [57], or the inverse limit construction of [27]. The proofs of the
relevant Poincaré inequality in these settings, and the equivalence of the
Poincaré inequality and Loewner condition from [37] implies the Loewner
condition. We will also give a third self-contained argument in Example 5.14,
which is based on the replacement rule construction and explicit discrete
modulus comutations.

Our counter-example $X$ which arises form the replacement rule in Figure 1.1
has Hausdorff dimension $\log(9)/\log(4)>3/2$. Since this space contains a
copy of $Y$ from the previous paragraph, we must have ${\rm dim}_{\rm
AR}(X)\geq 3/2.$ It is not minimal for the dimension, and Figure 2.4 shows how
the dimension can be reduced by iteratively reducing the length of the central
segment to $\varepsilon$. This reduction is repeated similar to a
multiplicative cascade: The central edge arising from replacing an edge $e$ at
some level will have length $\varepsilon$ times the length of the edge $e.$
For example, the edges arising from replacing the central edge at the first
level have length $\varepsilon/4$ and $\varepsilon^{2}$, where the latter is
the length of the central edge of the central edge. It can be shown, that for
every $\varepsilon>0$ such a cascade produces a new metric in the conformal
gauge of $X$.

As $\varepsilon\to 0$, the Hausdorff dimension of this new metric approaches
$3/2$, establishing that ${\rm dim}_{\rm AR}(X)=3/2$. This argument gives an
intuitive computation of the conformal dimension and an explanation for why it
is not attained. Attainment would require setting $\varepsilon=0$, which would
degenerate the space by collapsing the central edge. This is not allowed for
quasisymmetries. This argument can also be made rigorous, but our proof
involves a simpler and more general argument that uses the fact that $X$ is
combinatorially $3/2$-Loewner and Proposition 2.8.

The construction with $\varepsilon>0$ can be thought of as a construction of a
metric by using an admissible weight function for the left-right modulus
problem similar to [24] and [48], which comes from a modulus problem – see
also [50] for similar ideas and [61] for similar constructions. The fact that
the optimal weight function vanishes for the central edge corresponds with the
fact that attainement is not possible.

###  2.4. Combinatorial Loewner property

.

By discretizing the continuous modulus, Bourdon and Kleiner gave a
quasisymmetrically invariant combinatorial Loewner property, [16].

######  Definition 2.6 (CLP).

Let $Q>1$ and $L_{*}>1$. We say that a compact metric space satisfies the
_combinatorial $Q$-Loewner property_ if there exists two positive increasing
functions $\phi,\psi$ on $(0,\infty)$ with $\lim_{t\to 0}\psi(t)=0$ which
satisfies the following two conditions:

  1. (CLP1)

If $F_{1},F_{2}\subseteq X$ are two disjoint non-degenerate continua so that
$L_{*}^{-n}\leq{\rm diam}(F_{1})\land{\rm diam}(F_{2})$, then for all
$m\in\mathbb{N}$

$\phi(\Delta(F_{1},F_{2})^{-1})\leq{\rm
Mod}_{Q}^{BK}(\Gamma(F_{1},F_{2}),G_{n+m}).$

  2. (CLP2)

If $x\in X$, $r\geq L_{*}^{-n}$ and $C>0$ then for all $m\in\mathbb{N}$

${\rm Mod}_{Q}^{BK}\left(\Gamma\left(\overline{B(x,r)},X\setminus
B(x,Cr)\right),G_{n+m}\right)\leq\psi(C^{-1}).$

The definition does not depend on the choice of an $\alpha$-approximation, as
shown in [16, Proposition 2.2], and it is a quasisymmetry invariant [16,
Theorem 2.6]. The definition is slightly different from that in [16], and we
adopt the corrected version from [30]. As noted by [16], and shown in detail
by [30], the combinatorial Loewner property implies the following _LLC
property_ : For each $x,y\in X$, there exists a connected set $E_{x,y}\subset
X$ with ${\rm diam}(E)\leq Cd(x,y)$.

If $(X,d)$ is $Q$-Loewner, then it is also $Q$-combinatorially Loewner, see
[16, Theorem 2.6]. Moreover, by the quasisymmetric invariance, if
$\mathcal{G}_{\rm AR}(X)$ possesses a Loewner metric, then $(X,d)$ is also
combinatorially Loewner. Kleiner’s conjecture regards the converse to this
statement. We also note that, regardless of attainment, it follows from
Proposition 2.3 that if $X$ is combinatorially $Q$-Loewner for $Q>1$, then
$Q={\rm dim}_{\rm AR}(X)$; see also [31, Lemma 4.2].

For combonatorially Loewner spaces the existence of a Loewner metric in the
conformal gauge is equivalent to minimizing conformal dimenion. This follows
from work by Haïssinski on the comparability of discrete and continuous
modulus [33, Proposition B.2], and was explicitly stated in [26].

######  Proposition 2.7.

Let $Q>1$. If $X$ is a compact combinatorially $Q$-Loewner metric space, then
there exists a metric $d^{\prime}\in\mathcal{G}_{\rm AR}(X)$ which is
$Q$-Ahlfors regular with $Q={\rm dim}_{\rm AR}(X)$ if and only if there exists
a metric $d^{\prime}\in\mathcal{G}_{\rm AR}(X)$ which is $Q-$Loewner.

This equivalence is crucial for us, as we will answer Kleiner’s conjecture by
precluding the attainment of the conformal dimension.

###  2.5. Proof of counterexamples

There are two key insights that lead to our counter-examples.

  1. (1)

Explicit computability: There is a family of spaces,

$\\{X:X\text{ is a limit of a linear replacement rule}\\},$

similar to those introduced by Laakso in [57], for which discrete moduli are
“easy” to compute. Consequently for these examples, we can compute their
conformal dimensions explicitly, see Section 4 and Theorem 5.2.

  2. (2)

Porosity: There are pairs of spaces $(Y,X)$ in the given family where
$Y\subset X$ is a porous subset, and where using the first insight we can find
${\rm dim}_{\rm AR}(X)={\rm dim}_{\rm AR}(Y)$.

We say that $Y\subset X$ is a (uniformly) _porous_ subset of $X$, if there
exists an $r_{0}>0$ and $c>0$ so that for every $y\in Y$, and every
$r\in(0,r_{0})$ there exists $x\in B(y,r)\subset X$ for which $B(x,cr)\cap
Y=\emptyset$. Porous sets of Ahlfors regular spaces have strictly smaller
Assouad dimension, and the conformal dimension is bounded from above by the
Assouad dimension of any quasisymmetric image. These facts together with the
quasisymmetric invariance of porosity yield the following proposition. See
Section 6.1 for definitions and a detailed proof.

######  Proposition 2.8.

Let $Y\subset X$ be a porous subset. If ${\rm dim}_{\rm AR}(Y)={\rm dim}_{\rm
AR}(X)$, then $X$ does not attain its conformal dimension.

In our case, $X$ will be the space arising from the replacement rule in Figure
1.1, and $Y$ the Laakso-type space arising from Figure 1.5. A direct
computation will show that their conformal dimensions are equal. The fact that
the replacement rule for $Y$ arises from a sub-graph of that of $X$ yields
that $Y$ is porous in $X$. From these, we see that $X$ can not attain its
conformal dimension, and thus by Proposition 2.7 can not contain a Loewner
metric in its conformal gauge. This, together with the fact that $X$ is
combinatorially Loewner and approximately self-similar yields the proof of the
main theorem.

##  3\. Linear replacement rules

###  3.1. General notation

Simple graphs (without loops) are pairs $G=(V,E)$, where $V$ is some finite
set of vertices and $E\subset\\{\\{x,y\\}:x,y\in V,x\neq y\\}$ is a finite set
of edges. An edge $e$ with end points $x,y$ is denoted $\\{x,y\\}$. Notice
that we do not permit loops in our graphs. If $\\{v,w\\}\in E$ we say that $w$
is a neighbor of $v$. The degree of a vertex $v\in V$ is the number ${\rm
deg}(v)$ of edges to which it is adjacent. The degree of the graph $G$ is
defined as ${\rm deg}(G)=\max_{v\in V}{\rm deg}(v)$.

######  Remark 3.1.

For simplicity of notation and to avoid some tedious case analysis, we
restrict attention to simple graphs. Our results can also be stated for multi-
graphs, and these require mostly small changes to the statements and
arguments, e.g. in Proposition 3.11.

It will be convenient to identify $G$ with its geometric realization, which is
obtained by gluing copies of unit edges indexed by the edges of $G$ along
their end points, whenever two edges are adjacent. This makes $G$ into a
simplicial complex. If this geometric realization is connected, we call $G$
connected.

A vertex path in $G$ is a sequence $\theta=[v_{1},\dots,v_{k}]$, where
$\\{v_{i},v_{i+1}\\}\in E$ for each $i=1,\dots,k-1$. Here ${\rm
len}(\theta)=k-1$ is the length of the vertex path $\theta$, and we say that
the path connects $v_{1}$ to $v_{n}$. The path metric $d_{G}(v,w)$ in a
connected metric space $G$ between two vertices $v,w\in V$ is the smallest
$n\in\mathbb{N}$ such that there is a vertex path of length $n$ connecting $v$
to $w$. If $A,B\subseteq V$ then $\Theta(A,B)$ is the set of paths connecting
a point $v\in A$ to a point $u\in B$. Further if $d$ is a metric on $V$ and
$A\subseteq V$ is a set of vertices, then ${\rm diam}(A,d):=\max_{a,b\in
A}d_{G}(a,b)$ is the diameter in the metric $d$. Where the metric $d$ is clear
from context, we may drop it from the notation.

###  3.2. Linear replacement rules

######  Definition 3.2.

_A linear replacement rule_ consists of a graph $G_{1}=(V_{1},E_{1})$,
together with the following data.

  1. (1)

A finite set $I$ called the _gluing set_.

  2. (2)

Each edge $e=\\{v,w\\}$ and endpoint point $v\in e$ is associated with an
injective mapping $\phi_{v,e}:I\to V_{1}$, whose image
$I_{v,e}:=\phi_{v,e}(I)$ is an independent set of $V_{1}$ \- that is, no edges
between vertices in $I_{v,e}$.

  3. (3)

We have $I_{v,e}\cap I_{w,e}=\emptyset$ for each edge $e=\\{v,w\\}$.

The set $I$ together with the maps $\phi_{v,e}$ will be referred to as the
gluing rules. In general, a graph $G=(V,E)$ is said to be labeled by $G_{1}$
if for every edge $e$ of $G$ and any $v\in e$, there is an associated
injective mapping $\phi_{v,e}:I\to V_{1}$, whose image is an independent set.
Given a labeled graph $G$ be can form a replacement graph
$\hat{G}=(\hat{V},\hat{E})$, whose vertices are $\hat{V}=V_{1}\times E/\sim$,
where we identify

$(\phi_{x,e}(v),e)\sim(\phi_{x,f}(v),f)$

for every $e,f\in E$ which share an end point $x\in V$ and $v\in I$. Further,
we define edge sets

$\hat{E}=\\{[(v,e)],[(w,e)]:\\{v,w\\}\in E_{1},e\in E\\}.$

This amounts to replacing each edge in $G$ by a copy of $G_{1}$, which are
glued along the images of the mappings $\phi_{v,e}$. We can also define a
labeling for $\hat{G}$ by

$\phi_{[(v,e)],\\{[(v,e)],[(w,e)]\\}}=\phi_{v,\\{v,w\\}}$

This replacement rule can be applied recursively to $G_{1}$ to produce a
sequence of graphs $G_{k}$: Set $G_{k+1}:=\hat{G_{k}}$ for $k\in\mathbb{N}$.
The edges and vertices of $G_{k}=(V_{k},E_{k})$ can be described as follows.

  1. (1)

Let $V_{k+1}=V_{1}\times E_{k}/\sim$, where we identify vertices with the
relationships $(\phi_{x,e}(v),e)\sim(\phi_{x,f}(v),f)$ for every $e,f\in E$
which share an end point $x$ and $v\in I$.

  2. (2)

$\\{[v,e],[w,e]\\}\in E_{k+1}$ if $\\{v,w\\}\in E_{1}$.

  3. (3)

$\phi_{[v,e],\\{[v,e],[w,e]\\}}=\phi_{v,\\{v,w\\}}$.

Notice that in the notation of equivalence classes $[(v,e)]$, we drop the
parenthesis and write $[v,e]$. When $(\phi_{x,e}(v),e)\sim(\phi_{x,f}(v),f)$,
we say that the equivalence relation is given by the vertex $x$. We call the
graphs $G_{k}$ thus constructed _replacement graphs._

######  Remark 3.3.

We have chosen the above fairly general framework in order to encompass a wide
variety of examples. In all relevant examples of this paper, we will give
significantly simpler linear replacement rules. In our case, the graph
$G_{1}=(V_{1},E_{1})$ is a fixed graph, and we have just two maps
$\phi_{\pm}:I\to V_{1}$. The choice of mapping corresponds to a choice of
orientation for an edge. An orientation for edges is a choice of an ordered
pair $(v,w)$ (or $(w,v)$), for each edge $e=\\{v,w\\}$, in which case we say
that $e$ is positively oriented from $v$ and to $w$ (or $w$ to $v$). Given
such an orientation, we define $\phi_{v,e}=\phi_{-}$ if $e=\\{v,w\\}$ is
positively oriented from $v$, and $\phi_{w,e}=\phi_{+}$ if $e$ is positively
oriented towards $w$. We call these types of rules _oriented replacement
rules_. One simple way to obtain an orientation for edges is by ordering
vertices with a well-ordering and declaring $\\{v,u\\}$ positively ordered if
$v<u$. This will be the primary way we introduce an orientation in our
examples.

This simplification reduces the number of different maps $\phi_{v,e}$ one
would need to consider below. Another simplification, which will hold in many,
but not all, of our examples, is that there is a graph isomorphism
$\eta_{e}:G_{1}\to G_{1}$ with $\eta_{e}\circ\phi_{\mp}=\phi_{\pm}$. If an
oriented linear replacement rule satisfies this condition, we say that the
linear replacement rule is _symmetric_. For symmetric linear replacement
rules, it does not matter how the edges are oriented to define maps
$\phi_{v,e}$. Indeed, in such cases, the sequence of graphs produced with
different orientations are isomorphic.

Given this independence on the orientation for symmetric replacement rules, we
have striven to use notation that avoids the use of orientations wherever
possible. Indeed, the only instance where we need these orientations is in
defining the explicit maps $\phi_{v,e}$ in our examples. The desire to avoid
orientations leads to the framework above and considering most of the time the
maps $\phi_{v,e}$, instead of using the maps $\phi_{\pm}$ together with an
(arbitrary) orientation.

###  3.3. Examples

Next, we shall describe some important examples of linear replacement rules
and the fractals which they produce.

123456 Figure 3.4. Figure of a symmetric replacement rule that produces the
“Laakso diamond” space that first appeared in [57], and was studied e.g. in
[10, 58]. The figure shows two steps of the replacement.

######  Example 3.5.

The much studied Laakso diamond space also arises as a linear replacement
rule. This is shown in Figure 3.4, where the first two stages of the
replacement are shown. Here, an edge is replaced iteratively by graphs
$G_{1}=(V_{1},E_{1})$ where $V_{1}:=\\{1,2,3,4,5,6\\}$ and

$E_{1}:=\\{\\{1,2\\},\\{2,3\\},\\{2,4\\},\\{4,5\\},\\{3,5\\},\\{5,6\\}.$

The gluing rules for the graphs are given by the index set $I=\\{a\\}$, and
$\phi_{+}(a)=6$, and $\phi_{-}(a)=1$. We set $\phi_{v,\\{v,w\\}}=\phi_{-}$ if
$v<w$ and otherwise $\phi_{v,\\{v,w\\}}=\phi_{+}$. The construction yields a
sequence of planar graphs, which helps in drawing them and makes them simpler
to visualize. This family of graphs has cut points, since the gluing set $I$
is a singleton set. This implies that the conformal dimension is equal to one,
and these examples do not yield counter examples to Conjecture 1.2.

12345 Figure 3.6. Figure of a symmetric replacement rule that produces a
Laakso space in the sense of [56]. The figure shows two replacements

######  Example 3.7.

The Laakso spaces from [56], and its variants, can be obtained by replacement
rules, see Figures 1.5 and 3.6 for examples. Notice how in this example the
gluing set $I$ consists of two vertices, which prevents the existence of cut-
points. Whenever one employs such replacement rules, the resulting graph will
not be planar. In the example of Figure 3.6, an edge is replaced iteratively
by graphs $G_{1}=(V_{1},E_{1})$ where $V_{1}:=\\{1,2,3,4,5\\}$ and

$E_{1}:=\\{\\{1,3\\},\\{2,3\\},\\{3,4\\},\\{3,5\\}\\}.$

The gluing rules for the graphs are given by the index set $I=\\{a,b\\}$, and
$\phi_{+}(a)=4,\phi_{+}(b)=5$, and $\phi_{-}(a)=1,\phi_{-}(a)=2$. We set
$\phi_{v,\\{v,w\\}}=\phi_{-}$ if $v<w$ and otherwise
$\phi_{v,\\{v,w\\}}=\phi_{+}$. These examples yield Loewner spaces, and we
will see that one needs to add edges to these graphs to obtain counter-
examples to Conjecture 1.2. (Since our proof does not explicitly require this
Loewner property, we omit it.) By Remark 3.38, the spaces thus constructed
have topological dimension one. We note that by the results of [4, 5], any
topologically one dimensional compact connected set without local cut points,
and which does not have open sets homeomorphic to a subset of the plane, is
homeomorphic to the Menger curve. This is not essential to us, but this
characterization implies that this example, and more generally all the
examples which satisfy Assumption 5.1, which guarantees the combinatorial
Loewner property, are homeomorphic to Menger curves. For this, one simply
needs to observe that the graphs $G_{m}$ are not planar for $m$ large enough,
and that it follows form Proposition 3.11 that any open set contains a
homeomorphic copy of $G_{m}$. The non-planarity of $G_{m}$ can be easily
checked by the classical Kuratowski’s theorem [53] on forbidden graph minors.

######  Example 3.8.

We describe now in detail the graph from Figure 1.1. The labeling of vertices
is also shown in that figure. Let $\hat{G}_{1}=(\hat{V}_{1},\hat{E}_{1})$ be
the graph with eight vertices $\hat{V}_{1}:=\\{1,\dots,8\\}$ with the edges

$\displaystyle\hat{E}_{1}:=\\{$
$\displaystyle\\{1,3\\},\\{3,4\\},\\{4,6\\},\\{6,7\\}$
$\displaystyle\\{2,3\\},\\{3,5\\},\\{5,6\\},\\{6,8\\}$
$\displaystyle\\{4,5\\}\\}.$

Here, the first and second line yield the edges corresponding to the two
copies of an interval, and the final line gives the added central edge. We
also set $I=\\{a,b\\}$, and define two maps $\phi_{\pm}:I\to\hat{V}_{1}$ by
$\phi_{-}(a)=1,\phi_{-}(b)=2$ and $\phi_{+}(a)=7,\phi_{+}(b)=8$. These
correspond to the left and right end points of each edge. Then, the maps
$\phi_{v,e}$ are defined by $\phi_{v,\\{v,w\\}}=\phi_{-}$ if $v<w$ and
$\phi_{v,\\{v,w\\}}=\phi_{+}$ if $v>w$

If we remove the edge $\\{4,5\\}$ from $\hat{E}_{1}$ and keep all vertices, we
obtain a graph $G_{1}=(V,E)$, which yields the replacement rule for the
Laakso-type space in Figure 1.5.

###  3.4. Basic properties

Our assumptions on the replacement rule guarantee that equivalence relations
are simple to describe.

######  Lemma 3.9.

The relation $(\phi_{x,e}(v),e)\sim(\phi_{x,f}(v),f)$ for every $e,f\in E$
which share an end point $x$ is an equivalence relation.

###### Proof.

The relation is clearly reflexive and symmetric. We check transitivity. If
$(v_{1},e_{1})\sim(v_{2},e_{2})\sim(v_{3},e_{3})$, then $e_{1}$ and $e_{2}$
share an end point $x$, and $e_{2}$ and $e_{3}$ share an end point $y$. If
$x=y$, then $v_{i}=\phi_{x,e_{i}}(v)$ for $i=1,2,3$ and
$(v_{1},e_{1})\sim(v_{3},e_{3})$. If $x\neq y$, then $v_{2}=\phi_{x,e_{2}}(v)$
and $v_{2}=\phi_{y,e_{2}}(w)$ for some $v,w\in I$, but then $I_{x,e_{2}}\cap
I_{y,e_{2}}\neq\emptyset$, which by (3) in Definition 3.2 is a contradiction.
This completes the proof. ∎

We define maps $\pi_{k+1}:V_{k+1}\cup E_{k+1}\to V_{k}\cup E_{k}$ as follows.
For each vertex $[(v,e)]\in V_{k+1}$ define $\pi_{k+1}([(v,e)])=e$ if
$v\not\in I_{w,e}$ for any $w\in e$, and otherwise set $\pi_{k+1}([(v,e)])=w$
if $v\in I_{w,e}$. In Figure 3.4, the vertices marked with squares are mapped
to vertices, and vertices marked with circles are mapped to edges. In other
words, vertices in gluing sets are mapped to vertices, and all other vertices
are mapped to the edges which gave rise to them.

The map $\pi_{k+1}$ is easier to define for edges. Each edge in the replaced
graph arises from some edge. Indeed, we set
$\pi_{k+1}(\\{[(v,e)],[(w,e)]\\})=e$ for every edge $\\{[(v,e)],[(w,e)]\\}\in
E_{k+1}$. Because $I_{w,e}$ is an independent set for each $w\in e$, the
mapping $\pi_{k+1}$ is well defined.

######  Lemma 3.10.

The mapping $\pi_{k}$ is well-defined, and $\pi_{k}:E_{k}\to E_{k-1}$ for all
$k\geq 1$, and $\pi_{k}^{-1}(V_{k-1})\subset V_{k}$.

###### Proof.

First, we check the equivalence class for vertices. For the well-definedness
it suffices to verify that if $(v,e)\sim(w,f)$, for $f\neq e$, then the map is
well defined. In this case, $e$ and $f$ share a unique end point $x$. The
equivalence relation means that $v\in I_{x,e}$ and $w\in I_{x,f}$, and thus in
both cases $\pi_{k+1}([(v,e)])=\pi_{k+1}([(w,f)])=x$, as desired. The claim
about images of edges and preimages of vertices follows from the definition. ∎

For $n>m\geq 1$, define $\pi_{n,m}:=\pi_{m+1}\circ\cdots\circ\pi_{n}:V_{n}\cup
E_{n}\to V_{m}\cup E_{m}$. We call a vertex/edge $x\in G_{n}$ an ancestor of a
vertex/edge of $y\in G_{m}$ if $\pi_{n,m}(x)=y$. Let $e=\\{v,w\\}$ be an edge
in $G_{m}$. We call the set
$I_{e,n}=\pi_{n,m}^{-1}(e)\cup\pi_{n,m}^{-1}(v)\cup\pi_{n,m}^{-1}(w)$ a tile.
When $n=m+1$, then $I_{e,n}$ is equivalent to the copy of $e\times G_{1}$
which exists in $G_{m+1}$.

###  3.5. Scaling maps and paths

For each $k\in\mathbb{N}$ the graphs $G_{k+1}$ consist of copies of $G_{1}$
glued together, but we can also see $G_{n+m}$ as arising from copies of
$G_{m}$ glued along edges of $G_{n}$. The gluing sets can be described as
follows. For $m=1$, we define $I_{v,e}^{(m)}=\phi_{v,e}(I)$ and for $m>1$ we
define $I_{v,e}^{(m)}=\pi_{m,1}^{-1}(\phi_{v,e}(I))$, if $\phi_{v,e}$ is some
given map in a labeled graph $G_{n}$ for some $n\in\mathbb{N}$. The following
proposition makes this all precise and introduces maps $\sigma_{e,m}$, which
will be useful later. The proof is a straightforward induction which uses the
definition, but is a bit technical. Thus at first reading one may want to skip
the proof and to just focus on the conclusion, which is more intuitive.

######  Proposition 3.11.

For every $n,m\in\mathbb{N}$ and $e\in E_{n}$ there are maps
$\sigma_{e,m}:V_{m}\to V_{n+m}$, the image of $\sigma_{e,m}$ denoted as
$e\cdot G_{m}$ and the edges contained in this image as $e\cdot E_{m}$, with
the following properties.

  1. (SM1)

For every $e\in E_{n}$ the mapping $\sigma_{e,m}$ is injective and the
collection of subsets $\\{e\cdot G_{m}\\}_{e\in E_{n}}$ is a covering of
$V_{n+m}$. Moreover, if $v,u\in V_{m}$ then $\\{v,u\\}\in E_{m}$ if and only
if $\\{\sigma_{e,m}(v),\sigma_{e,m}(u)\\}\in E_{n+m}$ and

(3.12)
$\phi_{v,\\{v,u\\}}=\phi_{\sigma_{e,m}(v),\\{\sigma_{e,m}(v),\sigma_{e,m}(u)\\}}.$

  2. (SM2)

For distinct edges $e,f\in E_{n}$ the subsets $e\cdot G_{m}$ and $f\cdot
G_{m}$ intersect if and only if $e,f$ have a common vertex $v$. Moreover,
their intersection is

$\sigma_{e,m}\left(I_{v,e}^{(m)}\right)=\pi_{n+m,n}^{-1}(v)=\sigma_{f,m}\left(I_{v,f}^{(m)}\right)$

the set of ancestors of $v$.

  3. (SM3)

For every $e\in E_{n}$ we have $e\cdot E_{m}=\pi_{n+m,n}^{-1}(e)$. In
particular, $\\{e\cdot E_{m}\\}_{e\in E_{n}}$ is a partition of $E_{n+m}$.

###### Proof.

Fix $n\in\mathbb{N}$ and we will prove the existence of the mapping
$\sigma_{e,m}$ by induction on $m$. The case $m=1$ follows by setting
$\sigma_{e,1}:V_{1}\to V_{n+1},z\mapsto[z,e]$ for all $z\in V_{1}$ and $e\in
E_{n}$.

Assume that $m\geq 1$ and we have constructed $\sigma_{e,m}:V_{m}\to V_{n+m}$
for all $e\in E_{n}$ that satisfies (SM1), (SM2) and (SM3). We define
$\sigma_{e,m+1}:V_{m+1}\to V_{n+m+1}$ by

(3.13)
$\sigma_{e,m+1}([z,\\{v,u\\}])=[z,\\{\sigma_{e,m}(v),\sigma_{e,m}(u)\\}],$

where each vertex of $V_{m+1}$ can be represented by an equivalence class
$[z,\\{v,u\\}]$ for some edge $\\{v,u\\}\in E_{m}$ and some vertex $z\in
V_{1}$. By the induction hypothesis (IH) we get
$\phi_{v,\\{v,u\\}}=\phi_{\sigma_{e,m}(v),\\{\sigma_{e,m}(v),\sigma_{e,m}(u)\\}}$
and therefore $\sigma_{e,m+1}$ is a well-defined. The fact that
$\sigma_{e,m+1}$ is an injection can be seen as follows. Suppose that
$\sigma_{e,m+1}([z_{1},\\{v_{1},u_{1}\\}])=\sigma_{e,m+1}([z_{2},\\{v_{2},u_{2}\\}])$.
Then the edges $\\{\sigma_{e,m}(v_{1}),\sigma_{e,m}(u_{1})\\}$ and
$\\{\sigma_{e,m}(v_{2}),\sigma_{e,m}(u_{2})\\}$ must be either equal or
adjacent. But, since $\sigma_{e,m}$ is an injection that preserves edges, this
means that $\\{v_{1},u_{1}\\}$ and $\\{v_{2},u_{2}\\}$ must be either equal or
adjacent. In the first case, $z_{1}=z_{2}$, and that the map is an injection
follows. If they are adjacent, and (without loss of generality) the
equivalence relation is given by the vertex $v_{1}=v_{2}$, we get
$\phi_{\sigma_{e,m}(v_{1}),\\{\sigma_{e,m}(v_{1}),\sigma_{e,m}(u_{1})\\}}(z_{1})=\phi_{\sigma_{e,m}(v_{2}),\\{\sigma_{e,m}(v_{2}),\sigma_{e,m}(u_{2})\\}}(z_{2})$.
From (3.12) we get
$\phi_{v_{1},\\{v_{1},u_{1}\\}}(z_{1})=\phi_{v_{2},\\{v_{2},u_{2}\\}}(z_{2})$
and $[z_{1},\\{v_{1},u_{1}\\}]=[z_{2},\\{v_{2},u_{2}\\}]$. This concludes
injectivity.

Also,

$\displaystyle V_{n+m+1}$ $\displaystyle=\bigcup_{\\{v,u\\}\in
E_{n+m}}\\{[z,\\{v,u\\}]:z\in V_{1}\\}$
$\displaystyle\stackrel{{\scriptstyle\text{IH}}}{{=}}\bigcup_{e\in
E_{n}}\bigcup_{\\{v,u\\}\in
E_{m}}\\{[z,\\{\sigma_{e,m}(v),\sigma_{e,m}(u)\\}]:z\in V_{1}\\}$
$\displaystyle=\bigcup_{e\in E_{n}}e\cdot G_{m+1}.$

Furthermore,

$\displaystyle\\{\sigma_{e,m+1}([z_{1},\\{v_{1},u_{1}\\}]),\sigma_{e,m+1}([z_{2},\\{v_{2},u_{2}\\}])\\}\in
E_{n+m+1}$ $\displaystyle\stackrel{{\scriptstyle(\ref{eq:sigmadef})}}{{\iff}}$
$\displaystyle\\{\sigma_{e,m}(v_{1}),\sigma_{e,m}(u_{1})\\}=\\{\sigma_{e,m}(v_{2}),\sigma_{e,m}(u_{2})\\}\text{
and }\\{z_{1},z_{2}\\}\in E_{1}$
$\displaystyle\stackrel{{\scriptstyle\text{IH}}}{{\iff}}$
$\displaystyle\\{v_{1},u_{1}\\}=\\{v_{2},u_{2}\\}\text{ and
}\\{z_{1},z_{2}\\}\in E_{1}$ $\displaystyle\iff$
$\displaystyle\\{[z_{1},\\{v_{1},u_{1}\\}],[z_{2},\\{v_{2},u_{2}\\}]\\}\in
E_{m+1}.$

From these, it is also direct to verify (3.12). Next, let
$\\{[z_{1},\\{v,u\\}],[z_{2},\\{v,u\\}]\\}\in E_{m+1}$ for some $\\{v,u\\}\in
E_{m}$, and compute:

$\displaystyle\phi_{\sigma_{e,m+1}([z_{1},\\{v,u\\}]),\\{\sigma_{e,m+1}([z_{1},\\{v,u\\}]),\sigma_{e,m+1}([z_{2},\\{v,u\\}])\\}}$
$\displaystyle\quad=\phi_{[z_{1},\\{\sigma_{e,m}(v),\sigma_{e,m}(u)\\}],\\{[z_{1},\\{\sigma_{e,m}(v),\sigma_{e,m}(u)\\})],[z_{2},\\{\sigma_{e,m}(v),\sigma_{e,m}(u)\\}]\\}}$
$\displaystyle\quad=\phi_{\\{z_{1},\\{z_{1},z_{2}\\}\\}}$
$\displaystyle\quad=\phi_{[z_{1},\\{v,u\\}],\\{[z_{1},\\{v,u\\}],[z_{2},\\{v,u\\}]},$

where in the last two lines we used (3) from Definition 3.2. This covers
(SM1).

We move on to (SM2). Fix distinct edges $e,f\in E_{n}$. Let
$\\{v_{1},u_{1}\\},\\{v_{2},u_{2}\\}\in E_{m}$ and $z_{1},z_{2}\in V_{1}$.
Consider the following chain of equivalences.

$\displaystyle\sigma_{e,m+1}([z_{1}\\{v_{1},u_{1}\\}])=\sigma_{f,m+1}([z_{2},\\{v_{2},u_{2}\\}])$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:sigmadef})}}{{\iff}}$
$\displaystyle[z_{1},\\{\sigma_{e,m}(v_{1}),\sigma_{e,m}(u_{1})\\}]=[z_{2},\\{\sigma_{f,m}(v_{2}),\sigma_{f,m}(u_{2})\\}]$
$\displaystyle\iff$ $\displaystyle\exists
u\in\\{\sigma_{e,m}(v_{1}),\sigma_{e,m}(u_{1})\\}\cap\\{\sigma_{f,m}(v_{2}),\sigma_{f,m}(u_{2})\\}\text{
and }$ $\displaystyle\exists a\in
I,z_{1}=\phi_{u,\\{\sigma_{e,m}(v_{1}),\sigma_{e,m}(u_{1})\\}}(a),z_{2}=\phi_{u,\\{\sigma_{f,m}(v_{2}),\sigma_{f,m}(u_{2})\\}}(a)$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:phieq})}}{{\iff}}$
$\displaystyle\exists
u\in\\{\sigma_{e,m}(v_{1}),\sigma_{e,m}(u_{1})\\}\cap\\{\sigma_{f,m}(v_{2}),\sigma_{f,m}(u_{2})\\}\text{
and }$ $\displaystyle\exists a\in
I,z_{1}=\phi_{\sigma_{e,m}^{-1}(u),\\{v_{1},u_{1}\\}}(a),z_{2}=\phi_{\sigma_{f,m}^{-1}(u),\\{v_{2},u_{2}\\}}(a).$

Notice that in the equivalences above $u\in e\cdot G_{m}\cap f\cdot G_{m}$.
Hence, by (IH), it follows that $e\cdot G_{m+1}\cap f\cdot
G_{m+1}\neq\emptyset$ if and only if $e$ and $f$ have a common vertex. Indeed
the ”only if” part follows by choosing arbitrary $u\in e\cdot G_{m}\cap f\cdot
G_{m}$ and $a\in I$. Then assume that $e$ and $f$ indeed has a common vertex
$u$ and we prove the experssion. Again, from the equivalences, we see that
$e\cdot G_{m+1}\cap f\cdot G_{m+1}$ is the set of ancestors of the vertices in
$e\cdot G_{m}\cap f\cdot G_{m}$. This yields

$\displaystyle e\cdot G_{m+1}\cap f\cdot G_{m+1}$
$\displaystyle=\pi_{n+m+1,n+m}^{-1}(e\cdot G_{m}\cap f\cdot G_{m})$
$\displaystyle\stackrel{{\scriptstyle\text{IH}}}{{=}}\pi_{n+m+1,n+m}^{-1}(\pi_{n+m,n}^{-1}(u))$
$\displaystyle=\pi_{n+m+1,n}^{-1}(u).$

Furthermore,

$\displaystyle\pi_{n+m+1,n}^{-1}(u)$
$\displaystyle\stackrel{{\scriptstyle\text{IH}}}{{=}}\pi_{n+m+1,n+m}^{-1}\left(\sigma_{e,m}\left(I_{u,e}^{(m)}\right)\right)$
$\displaystyle=\pi_{n+m+1,n+m}^{-1}\left(\sigma_{e,m}\left(\pi_{m,1}^{-1}(I_{u,e})\right)\right)$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:phieq})}}{{=}}\sigma_{e,m+1}\left(\pi_{m+1,m}^{-1}\left(\pi_{m,1}^{-1}(I_{u,e}\right)\right)$
$\displaystyle=\sigma_{e,m+1}\left(I_{u,e}^{(m+1)}\right).$

The second last equality follows from the following equivalence.

$\displaystyle
v_{1}\in\pi_{n+m+1,n+m}^{-1}(\sigma_{e,m}\left(\pi_{m,1}^{-1}(I_{u,e})\right))$
$\displaystyle\iff$ $\displaystyle\exists
v_{2}\in\sigma_{e,m}\left(\pi_{m,1}^{-1}(I_{u,e})\right),\,\exists u_{2}\in
e\cdot G_{m},\,\exists a\in I,\text{ so that }$ $\displaystyle
v_{1}=[\phi_{v_{2},\\{v_{2},u_{2}\\}}(a),\\{v_{2},u_{2}\\}]$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:phieq})}}{{\iff}}$
$\displaystyle\exists
v_{2}\in\sigma_{e,m}\left(\pi_{m,1}^{-1}(I_{u,e})\right),\,\exists u_{2}\in
e\cdot G_{m},\,\exists a\in I,\text{ so that }$ $\displaystyle
v_{1}=[\phi_{\sigma_{e,m}^{-1}(v_{2}),\\{\sigma_{e,m}^{-1}(v_{2}),\sigma_{e,m}^{-1}(u_{2})\\}}(a),\\{v_{2},u_{2}\\}]$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:sigmadef})}}{{\iff}}$
$\displaystyle\exists
v_{2}\in\sigma_{e,m}\left(\pi_{m,1}^{-1}(I_{u,e})\right),\,\exists u_{2}\in
e\cdot G_{m},\,\exists a\in I,\text{ so that }$ $\displaystyle
v_{1}=\sigma_{e,m+1}([\phi_{\sigma_{e,m}^{-1}(v_{2}),\\{\sigma_{e,m}^{-1}(v_{2}),\sigma_{e,m}^{-1}(u_{2})\\}}(a),\\{\sigma_{e,m}^{-1}(v_{2}),\sigma_{e,m}^{-1}(u_{2})\\}])$
$\displaystyle\stackrel{{\scriptstyle(\ref{SM1})}}{{\iff}}$
$\displaystyle\exists v_{3}\in\pi_{m,1}^{-1}(I_{u,e})\,\exists u_{3}\in
V_{m},\,\exists a\in I\text{ so that }$ $\displaystyle
v_{1}=\sigma_{e,m+1}([\phi_{v_{3},\\{v_{3},u_{3}\\}}(a),\\{v_{3},u_{3}\\}])$
$\displaystyle\iff$ $\displaystyle
v_{1}\in\sigma_{e,m+1}(\pi_{m+1,m}^{-1}(\pi_{m,1}^{-1}(I_{u,e}))).$

Hence (SM2) is clear.

Lastly we argue (SM3) fix $e\in E_{n}$. Choose an edge $\hat{e}\in e\cdot
E_{m+1}$. By (SM1) we have the expression

$\displaystyle\hat{e}$
$\displaystyle=\\{\sigma_{e,m+1}([z_{1},\\{v_{1},u_{1}\\}]),\sigma_{e,m+1}([z_{2},\\{v_{1},u_{1}\\}])\\}$
$\displaystyle=\\{[z_{1},\\{\sigma_{e,m}(v_{1}),\sigma_{e,m}(v_{2})\\}],[z_{2},\\{\sigma_{e,m}(v_{1}),\sigma_{e,m}(v_{2})\\}]\\}$

so $\pi_{n+m,n}(\hat{e})\in e\cdot
E_{m}\stackrel{{\scriptstyle\text{IH}}}{{=}}\pi_{n+m,n}^{-1}(e)$. Hence
$e\in\pi_{n+m+1,n}^{-1}(e)$ and $e\cdot
E_{m+1}\subseteq\pi_{n+m+1,n}^{-1}(e)$. To prove the other inclusion, choose
$\widehat{e}\in\pi_{n+m+1,n}^{-1}(e)$. By construction, $\hat{e}$ has a unique
presentation as of form $\widehat{e}=\\{[z_{1},f],[z_{2},f]\\}$ for $f\in
E_{n+m}$. Then $\pi_{n+m+1,n+m}(\widehat{e})=f\in e\cdot E_{m}$ so
$f=\\{\sigma_{e,m}([z_{3},f^{\prime}]),\sigma_{e,m}([z_{4},f^{\prime}])\\}$
and

$\displaystyle\hat{e}$
$\displaystyle=\\{\sigma_{e,m+1}([z_{1},\\{[z_{3},f^{\prime}],[z_{4},f^{\prime}]\\}]),\sigma_{e,m+1}([z_{2},\\{[z_{3},f^{\prime}],[z_{4},f^{\prime}]\\}])\\}\in
e\cdot E_{m+1}.$

Hence $e\cdot E_{m+1}=\pi_{n+m+1,n}^{-1}(e)$. ∎

######  Corollary 3.14.

Let $\theta=\left[v_{1},\dots,v_{k}\right]$ be a path in $G_{n+m}$. Suppose
there are distinct vertices $v,u\in V_{n}$ and indices $1\leq j_{1},j_{2}\leq
k$ so that

$v_{j_{1}}\in\pi_{n+m,n}^{-1}(v)\text{ and }v_{j_{2}}\in\pi_{n+m,n}^{-1}(u).$

Then there is an edge $\hat{e}=\\{v,\hat{u}\\}\in E_{n}$ and a sub-path
$\hat{\theta}$ of $\theta$ from $\pi_{n+m,n}^{-1}(v)$ to
$\pi_{n+m,n}^{-1}(\hat{u})$ contained in $\hat{e}\cdot G_{m}$.

###### Proof.

We shall write $e_{j}=\pi_{n+m,n}(\\{v_{j+1},v_{j}\\})$ for $j=1,\dots,k-1$.
For simplicity, we assume that $j_{1}=1$ and $j_{2}=k$. Furthermore, by
possibly taking a sub-path, we may assume that if $u^{\prime}\neq v$ then
$v_{j}\notin\pi_{n+m,n}^{-1}(u^{\prime})$ for all $1\leq j\leq k-1$. Indeed
then, by (SM3), we must have $v\in e_{j}$ for all $1\leq j\leq k-1$.

We set $\hat{e}:=\\{v,\hat{u}\\}=e_{k-1}$ and

$l:=\min\\{1\leq j\leq k-1:e_{s}=\hat{e}\text{ for all }j\leq s\leq k-1\\}.$

By definition, $e_{j}=\hat{e}$ for all $l\leq j\leq k-1$. By (SM1) and (SM3),
the vertices on the sub-path $\hat{\theta}:=\left[v_{l},\dots,v_{k}\right]$
are contained in $\hat{e}\cdot G_{m}$. Now if $l=1$, we are done. Assume
$l>1$. By construction, $e_{l-1}\neq\hat{e}$, which, by (SM2) yields

$v_{l}\in(e_{l-1}\cdot G_{m})\cap(\hat{e}\cdot G_{m})=\pi_{n+m,n}^{-1}(v).$

∎

######  Proposition 3.15.

Let $\theta=\left[v_{1},\dots,v_{k}\right]$ be a path in $G_{n+m}$. Suppose
there are edges $e,f\in E_{n}$ with no common vertex, and indices $1\leq
j_{1},j_{2}\leq k$ so that

$v_{j_{1}}\in e\cdot G_{m}\text{ and }v_{j_{2}}\in f\cdot G_{m}.$

Then there is an edge $\hat{e}=\\{v,u\\}\in E_{n}$ so that $v\in e$ and a
subpath $\hat{\theta}$ of $\theta$ from $\pi_{n+m,n}^{-1}(v)$ to
$\pi_{n+m,n}^{-1}(u)$ contained in $\hat{e}\cdot G_{m}$.

###### Proof.

By (SM2) and (SM3) there are vertices $v^{\prime}\in e$ and $u^{\prime}\in f$
so that $v_{j_{1}}\in\pi_{n+m,n}^{-1}(v^{\prime})$ and
$v_{j_{2}}\in\pi_{n+m,n}^{-1}(u^{\prime})$ for some $1\leq j_{1},j_{2}\leq k$.
By assumption, $v^{\prime}\neq u^{\prime}$, so the claim follows from
Corollary 3.14. ∎

######  Proposition 3.16.

Let $\theta=\left[v_{1},\dots,v_{k}\right]$ be a path in $G_{n+m}$ and
$A,B\subseteq V_{n}$ be non-empty disjoint sets. If
$v_{1}\in\pi_{n+m,m}^{-1}(A)$ and $v_{k}\in\pi_{n+m,m}^{-1}(B)$ then there are
numbers $1\leq k_{1}<s_{1}\leq k_{2}<s_{2}\leq\dots,\leq k_{l}<s_{l}\leq k$
and vertices $u_{1},\dots,u_{l}\in V_{n}$ so that

  1. (1)

$[u_{1},\dots,u_{l}]\in\Theta(A,B)$, and

  2. (2)

the sub-path $\theta_{i}=[v_{k_{i}},\dots,v_{s_{i}}]$ of $\theta$ connects
$\pi_{n+m,n}^{-1}(u_{i})$ to $\pi_{n+m,n}^{-1}(u_{i+1})$ and is contained in
$e_{i}\cdot G_{m}$ where $e_{i}=\\{u_{i},u_{i+1}\\}\in E_{n}$.

###### Proof.

We will define the numbers $k_{i}<s_{i}$ inductively as follows: First we
apply Corollary 3.14 to $\theta$ and obtain an edge $e_{1}=\\{u_{1},u_{2}\\}$
so that $\pi_{n+m,n}(v_{1})=u_{1}$ and a sub-path $\theta_{1}$ of $\theta$
contained in $e_{1}\cdot G_{m}$. We set $k_{1}$ to be the first index, and
$s_{1}$ to be the last index of this sub-path, so we have
$\theta_{1}=[v_{k_{1}},\dots,v_{s_{1}}]$. If $v_{s_{1}}\in B$ then we
terminate the process. Otherwise, we apply Corollary 3.14 to
$[v_{s_{1}},v_{s_{1}+1},\dots,v_{k}]$ and obtain an edge
$e_{2}=\\{u_{2},u_{3}\\}$ and a sub-path $\theta_{2}$ contained in $e_{1}\cdot
G_{m}$. We contiunue this process until for some $l\geq 1$ we have
$v_{s_{l}}\in B$. ∎

###  3.6. Metrics on linear replacement graphs

We will next consider the problem of defining natural metrics on the sequence
of graphs produced by a linear replacement rule. There are many ways to do
this, but with the following assumption the construction becomes easier to
describe.

######  Definition 3.17.

For $L_{*}\geq 2$, we say that a linear replacement rule satisfies _$L_{*}$
-uniform scaling property_ if for all $e=\\{v,u\\}\in E_{1}$ and for every
$x\in I_{v,e},y\in I_{u,e}$ we have $d_{G_{1}}(x,y)=L_{*}$.

It is easy to construct examples of replacement rules with this property.
Indeed, all examples in this paper satisfy it, and the value of $L_{*}$ is
readily computable.

Throughout this paper, $C_{{\rm diam}}$ denotes the diameter of the graph
$G_{1}$ equipped with the path metric $d_{G_{1}}$ and $C_{\deg}=\deg(G_{1})$
denotes the maximum degree the graph $G_{1}$. Clearly $L_{*}$-uniform scaling
property implies $L_{*}\leq C_{{\rm diam}}$, however, in general, we may have
$L_{*}<C_{{\rm diam}}$. We equip the set $V_{n}$ with the metric
$d_{n}:=L_{*}^{-n}\cdot d_{G_{n}}$ which is the path metric on $G_{n}$ scaled
by $L_{*}^{-n}$. Furthermore define the semi-metric $d_{n}$ on $E_{n}$ by

$d_{n}(e,f):=\min_{x\in e,y\in f}d_{n}(x,y).$

We also write $d_{n}(e,y)=d_{n}(y,e):=\min_{x\in e}d_{n}(x,y)$ for $y\in
V_{n}$ and $e\in E_{n}$.

######  Lemma 3.18.

Assume that the linear replacement rule satisfies the $L_{*}$-uniform scaling
property and fix $n\in\mathbb{N},m\geq 0$. If $x_{n+m},y_{n+m}\in V_{n+m}$ are
ancestors of vertices $x_{n},y_{n}\in V_{n}$ respectively, then the following
hold.

  1. (DL1)

If $x_{n}\neq y_{n}$ then $d_{n+m}(x_{n+m},y_{n+m})=d_{n}(x_{n},y_{n})$.

  2. (DL2)

If $x_{n}=y_{n}$ then $d_{n+m}(x_{n+m},y_{n+m})\leq C_{{\rm diam}}\cdot
L_{*}^{-n}$.

  3. (DL3)

Suppose $e_{n}\in E_{n}$ so that $x\in e_{n}\cdot G_{m}$. If $x_{n}\in e_{n}$
and $\widehat{x}_{n}\in V_{n+m}$ is an ancestor of $x_{n}\in V_{n}$ then
$d_{n+m}(x,x_{n+m})\leq C_{{\rm diam}}\cdot L_{*}^{-n}$.

###### Proof.

We will first consider (DL1). It is sufficient to prove the case $m=1$. Let
$\theta_{n+1}=[v_{1},\dots,v_{k}]$ be a path in $G_{n+1}$ from $x_{n+1}$ to
$y_{n+1}$. By Proposition 3.16, there is a path $[u_{1},\dots u_{l}]$ from
$x_{n}$ to $y_{n}$ and for $i=1,\dots,l-1$ pairwise disjoint sub-paths
$\hat{\theta}_{i}$ from $\pi_{n}^{-1}(u_{i})$ to $\pi_{n}^{-1}(u_{i+1})$ which
are contained in $\\{u_{i},u_{i+1}\\}\cdot G_{1}$. By unifrom scaling
property, ${\rm len}(\hat{\theta}_{i})\geq L_{*}$ for all $i$, which gives

${\rm len}(\theta)\geq\sum_{i=1}^{l-1}{\rm
len}\left(\hat{\theta}_{i}\right)\geq L_{*}(l-1)\geq L_{*}^{1+n}\cdot
d_{n}(x_{n},y_{n}).$

Since the path $\theta$ was arbitary, we have $d_{n+1}(x_{n+1},y_{n+1})\geq
d_{n}(x_{n},y_{n})$.

To prove the other inequality, let $\theta_{n}=[u_{1},\dots,u_{l}]$ be a
shortest path from $x_{n}$ to $y_{n}$. For every $1\leq i\leq l$ fix an
ancestor $v^{(i)}$ of $u_{i}$ so that $v^{(1)}=x_{n+1}$ and $v^{(l)}=y_{n+1}$.
By uniform scaling property, for each $1\leq i\leq l-1$, there is a path
$\theta^{(i)}$ from $v^{(i)}$ to $v^{(i+1)}$ of length $L_{*}$. By joining
these paths, we obtain a path $\theta_{n+1}$ from $x_{n+1}$ to $y_{n+1}$ of
length

${\rm len}(\theta_{n+1})=\sum_{i=1}^{l-1}{\rm
len}\left(\theta^{(i)}\right)=L_{*}\cdot(l-1)=L_{*}^{n+1}\cdot
d_{n}(x_{n},y_{n})$

which yields $d_{n+1}(x_{n+1},y_{n+1})\leq d_{n}(x_{n},y_{n})$.

Next we move to (DL2). Without loss of generality assume $x_{n+m}\neq
y_{n+m}$. Let $1\leq k\leq m$ be the smallest integer so that

$x_{n+k}:=\pi_{n+m,n+k}(x_{n+m})\neq y_{m+k}:=\pi_{n+m,n+k}(y_{n+m}).$

Since $\pi_{n+k}(x_{n+k})=\pi_{n+k}(y_{n+k})$, $x_{n+k},y_{n+k}\in e\cdot
G_{1}$ for some $e\in E_{n+k-1}$. Hence

$d_{n+m}(x_{n+m},y_{n+m})\stackrel{{\scriptstyle\ref{lemma: Distance of
ancestors
(1)}}}{{=}}d_{n+k}(x_{n+k},y_{n+k})\stackrel{{\scriptstyle\ref{SM1}}}{{\leq}}C_{{\rm
diam}}\cdot L_{*}^{-(n+k)}\leq C_{{\rm diam}}\cdot L_{*}^{-n}$

Lastly we prove (DL3). We choose edges $e_{n+i}\in E_{n+i}$ for $0\leq i\leq
m-1$ so that $x\in e_{n+i}\cdot G_{m-i}$ and $\pi_{n+i+1}(e_{n+i+1})=e_{n+i}$.
For every $0\leq i\leq m-1$ we also choose $x_{n+i}\in e_{n+i}$. Then we fix
the ancestors $\widetilde{x}_{n+i}\in V_{n+i+1}$ of $x_{n+i}$ and their
ancestors $\widehat{x}_{n+i}\in V_{n+m}$. For every $0\leq i\leq m-1$, as
$x_{n+i+1},\widetilde{x}_{n+i}\in e_{n+i}\cdot G_{1}$, there is a path of
length at most $C_{{\rm diam}}$ between these two vertices. Therefore
$d_{n+i+1}(x_{n+i+1},\widetilde{x}_{n+i})\leq C_{{\rm diam}}\cdot
L_{*}^{-(n+i+1)}$. By (DL1) and (DL2) we have

$d_{n+m}(\widehat{x}_{n+i},\widehat{x}_{n+i+1})\leq C_{{\rm diam}}\cdot
L_{*}^{(-n+i+1)}.$

Similarly $d_{n+m}(x,\widehat{x}_{n+m-1})\leq C_{{\rm diam}}\cdot
L_{*}^{-(n+m)}$. We now obtain

$\displaystyle d_{n+m}(x,\widehat{x}_{n})$ $\displaystyle\leq
d_{n+m}(x,\widehat{x}_{n+m-1})+\sum_{i=0}^{m-2}d_{n+m}(\widehat{x}_{n+i},\widehat{x}_{n+i+1})$
$\displaystyle\leq\sum_{i=0}^{m-1}C_{{\rm diam}}\cdot L_{*}^{-(n+i+1)}$
$\displaystyle\leq C_{{\rm diam}}\cdot L_{*}^{-n}.$

∎

######  Corollary 3.19.

If $n>m\in\mathbb{N}$ then the following hold.

  1. (DL4)

$d_{m}(\pi_{n,m}(e),\pi_{n,m}(f))\leq d_{n}(e,f)$.

  2. (DL5)

$d_{n}(e,f)\leq d_{m}(\pi_{n,m}(e),\pi_{n,m}(f))+2C_{{\rm diam}}\cdot
L_{*}^{-m}$.

  3. (DL6)

$C_{{\rm diam}}\leq{\rm diam}(G_{n},d_{n})\leq 2\cdot C_{{\rm diam}}$.

###### Proof.

(DL4) is clear if $\pi_{n,m}(e),\pi_{n,m}(f)$ share a common vertex.
Otherwise, let $x\in e$ and $y\in f$ be such that $d_{n}(e,f)=d_{n}(x,y).$ If
$\theta$ is a path connecting $x$ to $y$, then by 3.16, $\theta$ contains a
sub-path from an ancestors $\widehat{x}$ and $\widehat{y}$ of some
$x_{e}\in\pi_{n,m}(e)$ and $x_{f}\in\pi_{n,m}(f)$ respectively. Hence

$d_{n}(e,f)=d_{n}(x,y)\geq
d_{n}(\widehat{x},\widehat{y})\stackrel{{\scriptstyle\ref{lemma: Distance of
ancestors (1)}}}{{=}}d_{m}(x_{e},x_{f})\geq d_{m}(\pi_{n,m}(e),\pi_{n,m}(f)).$

To prove the latter, suppose $x\in\pi_{n,m}(e),y\in\pi_{n,m}(f)$ so that

$d_{m}(\pi_{n,m}(e),\pi_{n,m}(f))=d_{m}(x,y).$

If $x=y$, then the claim follows from (DL3). Otherwise, pick
$\widehat{x},\widehat{y}$ ancestors of $x,y$ respectively. Then

$\displaystyle d_{n}(e,f)$ $\displaystyle\leq
d_{n}(e,\widehat{x})+d_{n}(\widehat{x},\widehat{y})+d_{n}(\widehat{y},f)$
$\displaystyle\stackrel{{\scriptstyle\ref{lemma: Distance of ancestors
(3)}}}{{\leq}}2C_{{\rm diam}}\cdot L_{*}^{-m}+d_{n}(\widehat{x},\widehat{y})$
$\displaystyle\stackrel{{\scriptstyle\ref{lemma: Distance of ancestors
(1)}}}{{=}}2C_{{\rm diam}}\cdot L_{*}^{-m}+d_{m}(x,y)$ $\displaystyle=2C_{{\rm
diam}}\cdot L_{*}^{-m}+d_{m}(\pi_{n,m}(e),\pi_{n,m}(f)).$

∎

###  3.7. Doubling, regularity and limit space

A metric space $(X,d)$ is said to be _metrically doubling_ if there exists a
constant $N$ so that for each $x\in X$ and every $r>0$, there exists points
$x_{1},\dots,x_{N}$ so that $B(x,r)\subset\bigcup_{i=1}^{N}B(x_{i},r/2)$. We
say that a sequence $\\{X_{k}\\}_{k\in\mathbb{N}}$ is _uniformly doubling_ ,
if each space in the sequence if $N$-doubling for a fixed $N$. The following
definition gives a simple necessary and sufficient condition for a linear
replacement rule to produce a uniformly doubling sequence of graphs.

######  Definition 3.20.

A linear replacement rule is _doubling_ , if for every $e=\\{v,u\\}\in E_{1}$
and $z\in I_{v,e}$ we have ${\rm deg}(z)=1$. For such $z\in I_{v,e}$ we set
$\mathfrak{n}(z)$ to be the unique neighbour of $z$ and for the corresponding
edge we use the notation $\mathfrak{e}_{z}$.

######  Lemma 3.21.

If the linear replacement rule is doubling, then $\deg(G_{n})=C_{\deg}$ for
all $n$. Furthermore, if the linear replacement rule also satisfies the
$L_{*}$-uniform scaling property, then the metric spaces $(G_{n},d_{n})$ are
doubling with the doubling constant $N_{D}=N_{D}(C_{\deg},C_{{\rm
diam}},L_{*})$.

###### Proof.

It follows from (SM1) that $\deg(G_{n})\geq C_{\deg}$ for all
$n\in\mathbb{N}$. We prove by induction, that $\deg(G_{n})=C_{\deg}$. The base
case $n=1$ is trivial so we assume it holds for $n$. If $x\in V_{n+1}$ and $x$
is not an ancestor of any $x_{n}\in V_{n}$, by (SM1) and (SM2),
$\deg(x)\leq\deg(G_{1})=C_{\deg}$. On the other hand, if $x$ is an anscestor
of $x_{n}\in V_{n}$, then, by the doubling property of the replacement rule,
$\deg(x)=\deg(x_{n})\leq C_{\deg}$.

Next we prove the metric doubling property of $d_{n}$. Fix $x\in V_{n}$ and
$r>0$. First assume $r\leq 4C_{{\rm diam}}\cdot L_{*}^{-n}$. Then every vertex
$y\in B(x,r)$ can be reached from $x$ by a path of length at most $4C_{{\rm
diam}}$. Hence $\lvert B(x,r)\rvert\leq(C_{\deg})^{4C_{{\rm diam}}}$ so in
this case we may set $N_{D}=(C_{\deg})^{4C_{{\rm diam}}}$. On the other hand,
if $r\geq 4C_{{\rm diam}}$, then $B(x,r/2)=V_{n}$ so we can choose $N_{D}=1$.
Hence we may assume that

$4C_{{\rm diam}}\cdot L_{*}^{-m}<r\leq 4C_{{\rm diam}}\cdot L_{*}^{-m+1}$

for some $m=1,\dots,n$. Fix $\widetilde{x}\in V_{n}$ so that it has an
ancestor in $V_{m}$ and

$d_{n}(x,\widetilde{x})\stackrel{{\scriptstyle\ref{lemma: Distance of
ancestors (3)}}}{{\leq}}C_{{\rm diam}}\cdot L_{*}^{-m}<r/4.$

Similarly, for each $y\in B(x,r)$, we choose $\widehat{y}\in V_{n}$ having an
ancestor in $V_{m}$ and

$d_{n}(y,\widehat{y})\stackrel{{\scriptstyle\ref{lemma: Distance of ancestors
(3)}}}{{\leq}}C_{{\rm diam}}\cdot L_{*}^{-m}<r/4.$

Then

$d_{m}(\pi_{n,m}(\widetilde{x}),\pi_{n,m}(\widehat{y}))\stackrel{{\scriptstyle\ref{lemma:
Distance of ancestors (1)}}}{{\leq}}d_{n}(\widetilde{x},\widehat{y})\leq
r/2+d_{n}(x,y)<2r$

which gives

$S:=\\{\pi_{n,m}(\widehat{y})\in V_{m}:y\in B(x,r)\\}\subseteq
B(\pi_{n,m}(\widetilde{x}),2r).$

Since $2r\leq 8C_{{\rm diam}}L_{*}\cdot L_{*}^{-m}$, $S$ contains at most
$(C_{\deg})^{8C_{{\rm diam}}L_{*}}$ vertices. Lastly, for every $v\in S$ we
choose $x_{v}\in\pi_{n+m}^{-1}(v)$. Then for any $y\in B(x,r)$, we have

$d_{n}(\widehat{y},x_{\pi_{n,m}({\widehat{y}})})\stackrel{{\scriptstyle\ref{lemma:
Distance of ancestors (2)}}}{{\leq}}C_{{\rm diam}}\cdot L_{*}^{-m}<r/4$

and we finally obtain

$B(x,r)\subseteq\bigcup_{y\in B(x,r)}B(\widehat{y},r/4)\subseteq\bigcup_{v\in
S}B(x_{v},r/2)$

so we may set $N_{D}=(C_{\deg})^{8C_{{\rm diam}}L_{*}}$. ∎

Next, we will show that there is a fractal space $X$ which can be obtained as
a Gromov-Hausdorff limit of the sequence $(G_{n},d_{n})$. This limit space has
a symbolic description, which we now give.

######  Definition 3.22.

Given a linear replacement rule, we define

$\Sigma:=\\{(e_{i})_{i=1}^{\infty}:e_{i}\in E_{i}\text{ and
}\pi_{m}(e_{m})=e_{m-1}\\}.$

For $e\in E_{n}$, we define the subsets $\Sigma_{e}\subseteq\Sigma$ by

$\Sigma_{e}:=\left\\{(e_{i})_{i=1}^{\infty}\in\Sigma:e_{n}=e\right\\}.$

We give $\Sigma$ the natural metric

$d_{\Sigma}((e_{i})_{i=1}^{\infty},(f_{i})_{i=1}^{\infty}):=\begin{cases}0&\text{
if }e_{i}=f_{i}\text{ for all }i\in\mathbb{N}\\\ 2^{-\max\left\\{k\ :\
(e_{i})_{i=1}^{k}=(f_{i})_{i=1}^{k}\right\\}}&\text{ otherwise}\end{cases}$

and the Radon measure $\mu_{\Sigma}$, for which

$\mu_{\Sigma}(\Sigma_{e})=\frac{1}{\lvert E_{1}\rvert^{n}}$

for all $e\in E_{n}$.

######  Remark 3.23.

The measure is well-defined by Kolmogorov’s extension theorem and Proposition
3.11. There is also an alternative way to construct $\mu_{\Sigma}$. The space
$\Sigma$ and can be identified as $(E_{1})^{\mathbb{N}}$ by identifying
$(e_{i})_{i=1}^{\infty}\in\Sigma$, with
$\left(\hat{e}_{i}\right)_{i=1}^{\infty}\in(E_{1})^{\mathbb{N}}$, where

  1. (1)

$e_{1}=\hat{e}_{1}$

  2. (2)

$\hat{e}_{n+1}=\\{z_{1},z_{2}\\}$, where
$e_{n+1}=\\{[z_{1},e_{n}],[z_{2},e_{n}]\\}$.

Let $T:(E_{1})^{\mathbb{N}}\to\Sigma$ be the map
$T(\left(\hat{e}_{i}\right)_{i=1}^{\infty})=(e_{i})_{i=1}^{\infty}$, and let
$\nu$ be the Bernoulli measure on $(E_{1})^{\mathbb{N}}$ where each element in
$E_{1}$ has equal measure $\lvert E_{1}\rvert^{-1}$. Then the measure can also
be constructed as the push-forward $\mu_{\Sigma}=(T)_{*}(\nu)$, where $T_{*}$
is the push-forward operation for measures.

######  Definition 3.24.

The _limit space_ of a linear replacement rule is

$X:=\Sigma/\sim,\text{ where
}(e_{i})_{i=1}^{\infty}\sim(f_{i})_{i=1}^{\infty}\iff e_{i}\cap
f_{i}\neq\emptyset\text{ for all }i$

and is given the metric

(3.25)
$d_{X}([(e_{i})_{i=1}^{\infty}],[(f_{i})_{i=1}^{\infty}]):=\lim_{n\to\infty}d_{n}(e_{n},f_{n}).$

The canonical projection $\Sigma\to X$ is denoted by $\chi$ and the push-
forward measure on $X$ is denoted as $\mu_{X}:=\chi_{*}(\mu_{\Sigma})$. We
also denote $X_{e}:=\chi(\Sigma_{e})$.

The limit in (3.25) exists and it defines a metric by the following lemma.

######  Lemma 3.26.

The function $d_{X}$ is a well-defined metric on $X$ and
$\chi:(\Sigma,d_{\Sigma})\to(X,d_{X})$ is a continuous function.

###### Proof.

Given any pair of sequences
$(e_{i})_{i=1}^{\infty},(f_{i})_{i=1}^{\infty}\in\Sigma$, by (DL4) and (DL6),
the sequence $d_{n}(e_{n},f_{n})$ converges as it is increasing and bounded.
Moreover, $d_{n}$ defines a semi-metric on $E_{n}$ so that $d_{n}(e,f)=0$ if
and only if $e$ and $f$ share a vertex. Therefore $d_{X}$ is a well-defined
metric on $X$. The continuity of $\chi$ follows from (DL5). ∎

The following lemma is immediate from (DL4) and (DL5).

######  Lemma 3.27.

Let $x=[(e_{i})_{i=1}^{\infty}],y=[(f_{i})_{i=1}^{\infty}]$ be points in $X$.
Then the following hold.

  1. (DL7)

$d_{X}(x,y)\geq d_{n}(e_{n},f_{n})$ for all $n\in\mathbb{N}$.

  2. (DL8)

If $e_{m}$ and $f_{m}$ have a common vertex then

$d_{X}(x,y)\leq 2C_{{\rm diam}}\cdot L_{*}^{-m}.$

In particular, ${\rm diam}(X_{e})\leq 2C_{{\rm diam}}\cdot L_{*}^{-m}$ for all
$e\in E_{m}$.

In summary, we can now show that the spaces $(G_{n},d_{n})$ Gromov-Hausdorff
converge to $X$. First, we give the relevant definitions. An
$\epsilon$-isometry is an isometry (i.e. a map which preserves distances)
between metric spaces $f:X\to Z$, for which $Z\subset\bigcup_{x\in
X}B(\iota_{X}(x),\epsilon)$. The _Gromov-Hausdorff metric_ between $X,Y$ is
defined as

$\displaystyle d_{GH}(X,Y):=\inf\\{\epsilon>0:$ $\displaystyle\exists\text{ a
metric space }Z$ $\displaystyle\text{ and $\epsilon$-isometries
}\iota_{X}:X\to Z,\iota_{Y}:Y\to Z\\}.$

A metric space $X$ is said to be _geodesic_ , if for all $x,y\in X$ there
exists a rectifiable curve $\gamma$ connecting $x$ to $y$ with
$d(x,y)=\ell(\gamma)$, where $\ell(\gamma)$ is the length of $\gamma$. See
[21] for background on these notions.

######  Proposition 3.28.

If the linear replacement rule satisfies the $L_{*}$-uniform scaling- and the
doubling property, then the sequence of metric spaces
$\\{(G_{n},d_{n})\\}_{n\in\mathbb{N}}$ Gromov-Hausdorff converges to $X$.
Moreover, $X$ is compact, geodesic and satisfies the metric-doubling property.

###### Proof.

Consider the metric spaces $(G_{n},d_{n})$ together with their geometric
realizations, which are obtained by gluing edges of length $L_{*}^{-n}$
between adjacent vertices. Notice that the Gromov-Hausdorff distance of
$(G_{n},d_{n})$ to its geometric realization is $L_{*}^{-n}$, since we can
choose $Z$ as the geometric realization and use the inclusion maps in the
definition of the Gromov-Hausdorff metric. Thus, if the metric space
$(G_{n},d_{n})$ converge to $X$, so do the geometric realizations. On the
other hand, the geometric realizations are compact, geodesic and uniformly
doubling (by Lemma 3.21), so it follows that if
$\\{(G_{n},d_{n})\\}_{n\in\mathbb{N}}$ Gromov-Hausdorff converges to $X$, then
$X$ is compact, geodesic, and satisfies metric doubling property; see e.g.
[21, Theorem 7.5.1].

We will construct $\epsilon(n)$-isometries $i_{G_{n}}:V_{n}\to X$ so that
$\epsilon(n)\to 0$ as $n\to\infty$. First for every $x\in V_{m}$ we choose
ancestors $\widehat{x}\in V_{m}$ of $x$ and define the map $i_{m,n}:V_{m}\to
V_{n},i_{m,n}(x)=\widehat{x}$. By (DL1) $i_{m,n}$ is an isometry and,
moreover, we can choose the ancestors so that for all $m>n>l$ we have

(3.29) $i_{n,l}\circ i_{m,n}=i_{m,l}\text{ and }\pi_{l,n}\circ
i_{m,l}=i_{m,n}.$

Next we construct maps $i_{m,n}^{E}:V_{m}\to E_{n}$ for all
$m,n\in\mathbb{N}$. For $n\leq m$ we define $i_{n,m}^{E}:V_{n}\to E_{m}$ by
choosing an edge $e_{x}$ for every $x\in V_{n}$ containing $i_{n,m}(x)$. For
$n>m$ we define $i_{n,m}^{E}:V_{n}\to E_{m}$ by

$i_{n,m}^{E}(x)=\begin{cases}\pi_{n,m}(x)&\text{ if }\pi_{n,m}(x)\in E_{m}\\\
e_{x}&\text{ if }\pi_{n,m}(x)\in V_{m},\end{cases}$

where $e_{x}\in E_{m}$ is an edge containing $\pi_{n,m}(x)\in V_{m}$. By
(3.29) we may choose the edges $e_{x}$ so that

(3.30) $\pi_{n,k}\circ i_{m,n}^{E}=i_{m,k}^{E}.$

We now define the mappings $i_{G_{n}}:V_{n}\to X$ by
$i(x)=[(i_{n,i}(x))_{i=1}^{\infty}]$. Indeed, by (3.30),
$(i_{n,i}(x))_{i=1}^{\infty}\in\Sigma$ for all $x\in V_{n}$ so $i_{G_{n}}$ is
well-defined. We will prove that $i_{G_{n}}$ is a $2C_{{\rm diam}}\cdot
L_{*}^{-n}$-isometry. Choose distinct vertices $x,y\in V_{n}$. Since the edges
$i_{n,m}^{E}(x),i_{n,m}^{E}(y)\in E_{m}$ contain the ancestors
$i_{n,m}(x),i_{n,m}(y)\in V_{m}$ of $x,y$ respectively, we have by (DL3)

$\displaystyle d_{m}(i_{n,m}(x),i_{n,m}(y))-2C_{\rm diam}\cdot L_{*}^{-m}$
$\displaystyle\leq d_{m}(i_{n,m}^{E}(x),i_{n,m}^{E}(y))$ $\displaystyle\leq
d_{m}(i_{n,m}(x),i_{n,m}(y))$ $\displaystyle=d_{n}(x,y).$

By letting $m\to\infty$, we have

$d_{X}(i_{G_{n}}(x),i_{G_{n}}(y))=\lim_{m\to\infty}d_{m}(i_{n,m}(x),i_{n,m}(y))=d_{n}(x,y).$

Lastly, if $x=[(e_{i})_{i=1}^{\infty}]\in X$ and $x_{n}\in e_{n}$, then

$d_{X}(x,i_{G_{n}}(x_{n}))\stackrel{{\scriptstyle(\ref{lemma: properties of
d_X (2)})}}{{\leq}}2C_{{\rm diam}}\cdot L_{*}^{-n}.$

Hence $i_{G_{n}}$ is a $2C_{{\rm diam}}\cdot L_{*}^{-n}$-isometry. This gives
$d_{GH}((G_{n},d_{n}),X)\leq 2C_{{\rm diam}}\cdot L_{*}^{-n}$ and thus $X$ is
the Gromov-Hausdorff limit of the spaces $G_{n}$. ∎

We can also compute that $X$ is Ahlfors regular, and identify the Hausdorff
dimension of $X$.

######  Lemma 3.31.

If the linear replacement rule satisfies the $L_{*}$-uniform scaling- and
doubling properties, then the metric measure space $(X,d_{X},\mu_{X})$ is
$Q$-Ahlfors regular with

$Q=\frac{\log(\lvert E_{1}\rvert)}{\log(L_{*})}.$

###### Proof.

Fix $x=[(e_{i})_{i=1}^{\infty}]\in X$ and $r\in(0,{\rm diam}(X)]$. Then

$r\leq{\rm diam}(X)\stackrel{{\scriptstyle\ref{corollary: Distance between
edges (3)}}}{{\leq}}2C_{{\rm diam}}$

so that for some $n\in\mathbb{N}$ we have $2C_{{\rm diam}}\cdot
L_{*}^{-n}<r\leq 2C_{{\rm diam}}\cdot L_{*}^{-(n-1)}$. Now if
$y=[(f_{i})_{i=1}^{\infty}]$ so that $f_{n}=e_{n}$, then, by (DL8),
$d_{X}(x,y)\leq 2C_{{\rm diam}}\cdot L_{*}^{-n}$. Hence $X_{e_{n}}\subseteq
B(x,r)$ which yields

$\mu_{X}(B(x,r))\geq\mu_{X}(X_{e_{n}})\geq\mu_{\Sigma}(\Sigma_{e_{n}})=\frac{1}{\lvert
E_{1}\rvert^{n}}\geq(2C_{{\rm diam}}L_{*})^{-Q}\cdot r^{Q}.$

Next we prove the upper bound. Let $f_{n}^{(1)},\dots,f_{n}^{(k)}\in E_{n}$ be
the edges $f\in E_{n}$ so that $\chi^{-1}(B(x,r))\cap\Sigma_{f}\neq\emptyset$.
Choose $y=[(f_{i})_{i=1}^{\infty}]\in B(x,r)$. Then

$d_{n}(e_{n},f_{n})\stackrel{{\scriptstyle(\ref{lemma: properties of d_X
(1)})}}{{\leq}}d_{X}(x,y)<r\leq 2C_{{\rm diam}}\cdot L_{*}^{-(n-1)}=2C_{{\rm
diam}}L_{*}\cdot L_{*}^{-n}.$

Hence $f_{n}$ can be reached from $e_{n}$ by a path of length at most
$2C_{{\rm diam}}L_{*}$. This yields $k\leq 2(C_{\deg})^{2C_{{\rm diam}}L_{*}}$
and further

$\displaystyle\mu_{X}(B(x,r))$
$\displaystyle\leq\mu_{\Sigma}\left(\bigcup_{i=1}^{k}\Sigma_{f_{n}^{(i)}}\right)$
$\displaystyle\leq(C_{\deg})^{2C_{{\rm diam}}L_{*}}\cdot\frac{1}{\lvert
E\rvert^{n}}$ $\displaystyle\leq(C_{\deg})^{2C_{{\rm diam}}L_{*}}(2C_{{\rm
diam}})^{Q}\cdot r^{Q}.$

∎

###  3.8. Approximate Self-similarity

Our spaces are constructed by a self-similar procedure. Thus, beyond a minor
combinatorial issue, it is not too difficult to obtain approximate self-
similarity; recall Definition 2.1. Our proof utilizes a condition similar to
that of the finite type condition of an iterated function system. This
condition is also called a finite intersection condition.

######  Definition 3.32.

Suppose the linear replacement rule satisfies the $L_{*}$-uniform scaling- and
doubling properties. The _fundamental neighbourhood_ of $e\in E_{n}$ is the
induced sub-graph ${\rm\mathcal{N}}(e)=(V(e),E(e))\subseteq G_{n}$ so that

$V(e)=\left\\{x\in V_{n}:d_{G_{n}}(x,e)\leq 2\right\\}.$

Notice that in above the distance is the un-scaled path metric. Furthermore,
we define the _extended fundamental neighbourhood_ of $e\in E_{n}$ to be the
induced sub-graph
$\widehat{{\rm\mathcal{N}}}(e)=\left(\widehat{V}(e),\widehat{E}(e)\right)\subseteq
G_{n}$ so that

$\widehat{V}(e)=\left\\{x\in V_{n}:d_{G_{n}}(x,e)\leq 10C_{{\rm
diam}}\right\\}.$

We say that the fundamental neighbourhoods
${\rm\mathcal{N}}(e),{\rm\mathcal{N}}(f)$ of $e=\\{x_{1},y_{1}\\}\in E_{n}$
and $f=\\{x_{2},y_{2}\\}\in E_{m}$ are _equivalent_ if there is a graph
isomorphism
$\beta:\widehat{{\rm\mathcal{N}}}(e)\to\widehat{{\rm\mathcal{N}}}(f)$ between
the extended fundamental neighbourhoods satisfying

  1. (1)

$\\{x_{2},y_{2}\\}=\\{\beta(x_{1}),\beta(y_{1})\\}$ and

  2. (2)

$\phi_{x,\\{x,y\\}}=\phi_{\beta(x),\\{\beta(x),\beta(y)\\}}$ for all
$\\{x,y\\}\in\widehat{E}(e)$.

We also say that $\beta$ is an isomorphism between the fundamental
neighbourhoods ${\rm\mathcal{N}}(e)$ and ${\rm\mathcal{N}}(f)$. The set of
equivalence classes of equivalent fundamental neighbourhoods is denoted as
$\mathcal{N}$.

The idea behind equivalent fundamental neighborhoods is their ”similarity” on
all scales. Indeed, if $e\in V_{n}$, we define the sub-graphs
$\mathcal{N}_{k}(e)=(V_{k}(e),E_{k}(e))$ and
$\widehat{\mathcal{N}}_{k}(e)=\left(\widehat{V}_{k}(e),\widehat{E}_{k}(e)\right)$
of $G_{n+k}$ where

$E_{k}(e)=\pi_{n+k,n}^{-1}(E(e))\text{ and }V_{n}(e)=\bigcup_{f\in V(e)}f\cdot
G_{k}$

and similarly

$\widehat{E}_{k}(e)=\pi_{n+k,n}^{-1}\left(\widehat{E}(e)\right)\text{ and
}\widehat{V}_{k}(e)=\bigcup_{f\in\widehat{V}(e)}f\cdot G_{k}.$

Now given an isomorphism
$\beta:\widehat{{\rm\mathcal{N}}}(e)\to\widehat{{\rm\mathcal{N}}}(f)$ between
the fundamental neighbourhoods of $e$ and $f$, we define the mappings
$\beta_{k}:\widehat{{\rm\mathcal{N}}}_{k}(e)\to\widehat{{\rm\mathcal{N}}}_{k}(f)$
inductively:

  1. (1)

$\beta_{0}=\beta$.

  2. (2)

If $\beta_{k}$ is defined then
$\beta_{k+1}([z,\\{x,y\\}])=[z,\\{\beta_{k}(x),\beta_{k}(y)\\}]$.

The corresponding map between edges $\widehat{E}_{k}(e)\to\widehat{E}_{k}(f)$
is denoted as $\beta_{k}^{E}$.

In the following lemma, recall that $d_{G_{n}}$ was the un-scaled path metric
on $G_{n}$, and $d_{n}=L_{*}^{-n}d_{G_{n}}$.

######  Lemma 3.33.

If $\beta:\widehat{{\rm\mathcal{N}}}(e)\to\widehat{{\rm\mathcal{N}}}(f)$ is an
isomorphism between the fundamental neighbourhoods of $e\in E_{n}$ and $f\in
E_{m}$ then the mappings $\beta_{k}$ are graph isomorphisms. These maps
satisfy

$\pi_{m+k+1}\circ\beta_{k+1}^{E}=\beta_{k}^{E}\circ\pi_{n+k+1}$

for all $k\geq 0$. Furthermore, if the replacement rule satisfies the
$L_{*}$-uniform scaling property and the doubling property, then the
restriction $\beta_{k}:{\rm\mathcal{N}}_{k}(e)\to{\rm\mathcal{N}}_{k}(f)$ are
isometries with respect to the (un-scaled) path metrics $d_{G_{n+k}}$ and
$d_{G_{m+k}}$ respectively.

###### Proof.

As in the proof of Proposition 3.11, it is a straightforward inductive
argument that
$\phi_{x,\\{x,y\\}}=\phi_{\beta_{k}(x),\\{\beta_{k}(x),\beta_{k}(y)\\}}$ for
all $k\in\mathbb{N}$ and $\\{x,y\\}\in E_{k}(e)$. This observation yields that
$\beta_{k}$ is a graph isomorphism. Moreover, the commutativity with the
projections follows from

$\displaystyle\pi_{m+k+1}(\beta_{k+1}^{E}(\\{[z_{1},e],[z_{2},e]\\}))$
$\displaystyle=\pi_{m+k+1}(\\{[z_{1},\beta_{k}^{E}(e)],[z_{2},\beta_{k}^{E}(e)]\\})$
$\displaystyle=\beta_{k}^{E}(e)$
$\displaystyle=\beta_{k}^{E}(\pi_{n+k+1}(\\{[z_{1},e],[z_{2},e]\\})).$

To prove that its restritcions
$\beta_{k}:{\rm\mathcal{N}}_{k}(e)\to{\rm\mathcal{N}}_{k}(f)$ are isometries,
we will show that a shortest path in $G_{n+k}$ between vertices in
${\rm\mathcal{N}}_{k}(e)$ is contained in $\widehat{{\rm\mathcal{N}}}_{k}(e)$.
By applying the same argument to the vertices in ${\rm\mathcal{N}}_{k}(f)$, we
would be done.

First, if $e^{\prime}\in E(e)$, then $d_{G_{n}}(e,e^{\prime})\leq 2$, which
gives ${\rm diam}(\mathcal{N}_{k}(e),d_{G_{n+k}})\leq 8C_{{\rm
diam}}L_{*}^{k}$ by (DL5) and the fact that
$d_{n+k}=L_{*}^{-(n+k)}d_{G_{n+k}}$. Next, suppose $\theta$ is a path in
$G_{n+k}$ containing a vertex $x\in\mathcal{N}_{k}(e)$ with ${\rm
diam}(\theta)\leq 8C_{{\rm diam}}\cdot L_{*}^{k}$. We will show that indeed
$\theta$ is contained in $\widehat{{\rm\mathcal{N}}}(e)$. Suppose not. Then
$\theta$ contains a vertex in $e^{\prime}\cdot G_{k}$ where
$d_{G_{n}}(e,e^{\prime})\geq 10C_{{\rm diam}}$. This, by Proposition 3.16,
gives that there are $y\in e,y^{\prime}\in e^{\prime}$ and a sub-path
$\theta^{\prime}$ from $\pi_{n+k,n}^{-1}(y)$ to
$\pi_{n+k,n}^{-1}(y^{\prime})$. Hence

${\rm len}(\theta)\geq{\rm
len}(\theta^{\prime})\stackrel{{\scriptstyle\ref{lemma: Distance of ancestors
(1)}}}{{\geq}}L_{*}^{n+k}\cdot d_{n}(y,y^{\prime})=L_{*}^{k}\cdot
d_{G_{n}}(y,y^{\prime})\geq 10C_{{\rm diam}}\cdot L_{*}^{k}>{\rm
diam}(\theta).$

This yields the contadiction so we are done. ∎

The previous lemma has the following immediate corollary. A map
$h:(X,d_{X})\to(Y,d_{Y})$ between metric spaces is called a homothety if there
exists a constant $s$ so that $d_{Y}(h(x),h(y))=s\cdot d_{X}(x,y)$ for all
$x,y\in X$.

######  Corollary 3.34.

Let $\beta:\widehat{{\rm\mathcal{N}}}(e)\to\widehat{{\rm\mathcal{N}}}(f)$ be
an isomorphism between the fundamental neighbourhoods of $e\in E_{n}$ and
$f\in E_{m}$. Then the mapping

$\beta_{\infty}:\bigcup_{e^{\prime}\in{\rm\mathcal{N}}(e)}X_{e^{\prime}}\to\bigcup_{f^{\prime}\in{\rm\mathcal{N}}(f)}X_{f^{\prime}},\quad[(e_{i})_{i=1}^{\infty}]\mapsto[(f_{i})_{i=1}^{\infty}],$

where

$f_{i}=\begin{cases}\beta_{k}^{E}(e_{n+k})&\text{ if }i=m+k\text{ for }k\geq
0,\\\ \pi_{m,i}(\beta(e_{n}))&\text{ if }i<m\end{cases}$

is a homothety with the constant $L_{*}^{n-m}$.

###### Proof.

Let
$\left(e_{i}\right)_{i=1}^{\infty},\left(e_{i}^{\prime}\right)_{i=1}^{\infty}\in\Sigma$
so that $e_{n},e_{n}^{\prime}\in{\rm\mathcal{N}}(e)$. By Lemma 3.33 we have

$d_{n+k}(e_{n+k},e_{n+k}^{\prime})\cdot
L_{*}^{n+k}=d_{m+k}(\beta_{k}^{E}(e_{n+k}),\beta(e_{n+k}^{\prime}))\cdot
L_{*}^{m+k}$

which proves that $\beta_{\infty}$ is a homothety. ∎

######  Proposition 3.35.

If the linear replacement rule satisfies the $L_{*}$-uniform scaling- and
doubling properties, then the limit space $X$ is approximately self-similar.

###### Proof.

It follows from the doubling property of the linear replacement rule that the
set ${\rm\mathcal{N}}$ of equivalence classes of equivalent fundamental
neighbourhoods is finite. In particular, there is $m_{*}\in\mathbb{N}$ so that
for all $n\in\mathbb{N}$ and $e\in E_{n}$, the fundamental neighbourhood
${\rm\mathcal{N}}(e)$ is equivalent to ${\rm\mathcal{N}}(f)$ for some $f\in
E_{m}$ so that $m\leq m_{*}$.

Let $x=[(e_{i})_{i=1}^{\infty}]\in X,r>0$ and set $B=B(x,r)$. If $r\geq
L_{*}^{-(m_{*}+1)}$ then we choose the identity ${\rm id}_{B}:B(x,r)\to
B(x,r)$ as our scaling map. Otherwise, suppose $L_{*}^{-(n+1)}<r\leq
L_{*}^{-n}$ where $n>m_{*}$. Let $f\in E_{m}$ so that $m\leq m_{*}$ and
$\beta:{\rm\mathcal{N}}(e_{n})\to{\rm\mathcal{N}}(f)$ an isomorphism between
fundamental neighbourhoods of $e_{n}$ and $f$. If $\beta_{\infty}$ is as in
Corollary 3.34 then, by (DL7)-(DL8), we have
$B(x,r)\subseteq\bigcup_{e^{\prime}\in{\rm\mathcal{N}}(e_{n})}X_{e^{\prime}}$
and $\beta_{\infty}(B(x,r))=B(\beta_{\infty}(x),L_{*}^{n-m}\cdot r)$.
Furthermore, $\beta_{\infty}$ is a homothety with constant $s=L_{*}^{n-m}$,
which is comparable to $r^{-1}$ by

$L_{*}^{-(m_{*}+1)}\cdot r^{-1}\leq s\leq r^{-1}.$

Hence approximate self-similarity follows and $\beta_{\infty}$ is the desired
scaling map. ∎

######  Definition 3.36.

The _approximation_ of the limit space $X$ _at scale_ $m\in\mathbb{N}$ is the
incidence graph $\mathbb{G}_{m}=(\\{X_{v}\\}_{v\in V_{m}},\mathbb{E}_{m})$,
where the covering of $X$ consists of the open subsets

$X_{v}:={\rm int}\\{[(e_{i})_{i=1}^{\infty}]:e_{m}\text{ contains }v\\}$

for all $v\in V_{m}$.

######  Proposition 3.37.

For all $m\in\mathbb{N}$ and $v\in V_{m}$ there are points $z_{v}\in X_{v}$ so
that

$B(z_{v},L_{*}^{-m})\subseteq X_{v}\subseteq B(z_{v},C_{{\rm diam}}\cdot
L_{*}^{-m})$

and $d_{X}(z_{v},z_{u})=d_{m}(v,u)$ for all $v,u\in V_{m}$. In particular,
$\mathbb{G}_{m}$ is an $\alpha$-approximation of $X$ at scale $m$ for
$\alpha=C_{{\rm diam}}$. Moreover, for distinct $v,u\in V_{m}$, we have
$X_{v}\cap X_{u}\neq\emptyset$ if and only if $\\{v,u\\}\in E_{m}$.

###### Proof.

Recall the functions $i_{G_{m}}$ defined in Proposition 3.28. We define
$z_{v}=i_{G_{m}}(v)$. It follows from (DL7) and (DL1) that
$B(z_{v},L_{*}^{-m})\subseteq X_{v}$ and $d_{X}(z_{v},z_{u})=d_{m}(v,u)$ for
all $v,u\in V_{n}$. The other inclusion follows from (DL3) and the last
assertion from (DL7). ∎

As the graphs $\mathbb{G}_{m}$ and $G_{m}$ are naturally isomorphic, we
hereafter identify the replacement graphs $G_{m}$ with the incidence graphs
$\mathbb{G}_{m}$.

######  Remark 3.38.

The collection $\\{X_{v}\\}_{v\in V_{m}}$ is an open cover of $X$ with sets
with diameter less than $2C_{\rm diam}L_{*}^{-m}$ (by (DL8)). The nerve
complex is given by the graph $G_{m}$, and thus $X$ has topological dimension
$1$, [65, p. 126]. Indeed, by a slight modification of $X_{v}$, it is direct
to see that $X$ has Assouad-Nagata dimension equal to $1$; see e.g. [59,
Definition 1.1.].

##  4\. Modulus on graphs

###  4.1. Edge- and vertex modulus

Ultimately, we will be interested in establishing the combinatorial Loewner
property for the limit spaces $X$. We already defined in Subsection 2.2 a
notion combinatorial moduli for $X$ with respect to an
$\alpha$-approximations. In Proposition 3.37 we saw that $G_{m}$ can be
identified with such an $\alpha$-approximation. For computational reasons, it
will be useful to consider two additional notions of modulus on $G_{m}$: the
edge modulus and the vertex modulus. We will shortly see that these notions of
moduli are comparable.

######  Definition 4.1.

Let $p\in[1,\infty)$ and $G=(V,E)$ a graph. If $\theta=[v_{1},\dots,v_{n}]$ is
a non-constant path in $G$ and $\rho:E\to\mathbb{R}_{\geq 0}$, we write

$L_{\rho}(\theta):=\sum_{i=1}^{n-1}\rho(\\{v_{i},v_{i+1}\\})\text{ and
}\mathcal{M}_{p}(\rho):=\sum_{e\in E}\rho(e)^{p}.$

Given a family of paths $\Theta$ in $G$, we say that $\rho$ is _$\Theta$
-admissible_ if $L_{\rho}(\theta)\geq 1$ for all $\theta\in\Theta$.
Furthermore, we denote the _(edge) modulus_ of $\Theta$ by

$\displaystyle{\rm Mod}_{p}(\Theta,G):=\inf_{\rho}\mathcal{M}_{p}(\rho)$

where the infimum is over all $\Theta$-admissible densities
$\rho:E\to\mathbb{R}_{\geq 0}$. For any density $\rho:E\to\mathbb{R}_{\geq 0}$
the _support_ of $\rho$ is ${\rm supp}(\rho):=\\{e\in E:\rho(e)\neq 0\\}$.

In the following definition, we say that a subset $\theta\subset G$ is
connected, if its induced sub-graph is connected. Most of the time we will
perform calculations for the edge modulus. However, in relating the estimates
to the modulus of Bourdon and Kleiner, we need to momentarily also discuss
vertex modulus.

######  Definition 4.2.

Let $p\in[1,\infty)$ and $G=(V,E)$ a graph. If $\theta\subseteq V$ is a non-
empty connected set in $G$ and $\rho:V\to\mathbb{R}_{\geq 0}$, we write

$L_{\rho}(\theta):=\sum_{v\in\theta}\rho(v)\text{ and
}\mathcal{M}_{p}(\rho):=\sum_{v\in V}\rho(v)^{p}.$

Given a family of paths $\Theta$ in $G$, we say that $\rho$ is _$\Theta$
-admissible_ if $L_{\rho}(\theta)\geq 1$ for all $\theta\in\Theta$.
Furthermore, we denote the _(vertex) modulus_ of $\Theta$ by

$\displaystyle{\rm Mod}_{p}^{V}(\Theta,G):=\inf_{\rho}\mathcal{M}_{p}(\rho)$

where the infimum is over all $\Theta$-admissible densities
$\rho:V\to\mathbb{R}_{\geq 0}$.

Given a density $\rho^{E}:E\to\mathbb{R}_{\geq 0}$, one can easily define a
density $\rho^{V}:\to\mathbb{R}_{\geq 0}$ by considering the density of the
edges adjacent to a given vertex and vice versa. This standard argument gives
the comparability of edge- and vertex modulus. Given a path
$\theta=[v_{1},\dots,v_{n}]$ in $G$, we call $\\{v_{1},\dots,v_{n}\\}$ the
corresponding set in $V$. For simple paths $\theta$ we use $\theta$ also to
denote the corresponding set.

######  Lemma 4.3.

Let $G=(V,E)$ be a graph and $\Theta$ a non-empty family of non-constant paths
so that each path in $\Theta$ contains a simple sub-path in $\Theta$, and let
$\hat{\Theta}$ be the corresponding subsets for each path. Then

(4.4) $C^{-1}\cdot{\rm Mod}_{p}(\Theta,G)\leq{\rm
Mod}_{p}^{V}(\hat{\Theta},G)\leq C^{-1}\cdot{\rm Mod}_{p}(\Theta,G)$

where $C=C(p,\deg(G))$. In particular, (4.4) holds if $\Theta=\Theta(A,B)$ for
non-empty disjoint sets $A,B\subseteq V$.

###### Proof.

Let $\rho^{E}:E\to\mathbb{R}_{\geq 0}$ be $\Theta$-admissible. Define
$\widehat{\rho}^{V}:V\to\mathbb{R}_{\geq 0}$ by

$\widehat{\rho}^{V}(v)=\max_{\begin{subarray}{c}e\in E\\\ v\in
e\end{subarray}}\rho^{E}(e).$

Let $\theta\in\hat{\Theta}$ and let
$\theta^{\prime}=\left[v_{1},\dots,v_{n}\right]\in\Theta$ be a simple sub-path
of $\theta$. Then

$\sum_{v\in\theta}\widehat{\rho}^{V}(v)\geq\sum_{v\in\theta^{\prime}}\widehat{\rho}^{V}(v)=\sum_{i=1}^{n}\widehat{\rho}^{V}(v_{i})\geq\sum_{i=1}^{n-1}\rho^{E}(\\{v_{i},v_{i+1}\\})\geq
1$

and

$\sum_{v\in V}\widehat{\rho}^{V}(v)^{p}\leq\sum_{v\in
V}\sum_{\begin{subarray}{c}e\in E\\\ v\in e\end{subarray}}\rho^{E}(e)^{p}\leq
2\cdot\sum_{e\in E}\rho^{E}(e)^{p}.$

Then let $\rho^{V}:V\to\mathbb{R}_{\geq 0}$ be $\hat{\Theta}$-admissible.
Define

$\widehat{\rho}^{E}(\\{v,u\\})=2\cdot\max\\{\rho^{V}(v),\rho^{V}(u)\\}.$

Let $\theta\in\Theta$, and let $\hat{\theta}$ be the corresponding subset.
Then

$\sum_{i=1}^{n-1}\widehat{\rho}^{E}(\\{v_{i},v_{i+1}\\})\geq\sum_{i=1}^{n}\rho^{V}(v_{i})\geq\sum_{v\in\hat{\theta}}\rho^{V}(v)\geq
1$

and

$\sum_{\\{v,u\\}\in E}\widehat{\rho}^{E}(\\{v,u\\})^{p}\leq
2^{p}\cdot\sum_{\\{v,u\\}\in
E}\left(\rho^{V}(v)^{p}+\rho^{V}(u)^{p}\right)\leq 2^{p}\deg(G)\cdot\sum_{x\in
V}\rho^{V}(x)^{p}.$

∎

Roughly dual to the notion of edge modulus is the notion of a flow. We will
use the exponent $p\in[1,\infty)$ for modulus, and the exponent
$q\in[1,\infty)$ for flows. Here, $q$ is called _the dual exponent_ of $p$ if
$\frac{1}{p}+\frac{1}{q}=1$. In all that follows, $q$ will always denote this
dual exponent. _A flow_ from $A\subset V$ to $B\subset V$ is a map
$\mathcal{F}:V\times V\to\mathbb{R}$ for which

  1. (1)

$\mathcal{F}(x,y)=-\mathcal{F}(y,x)$ and $\mathcal{F}(x,y)=0$ unless
$\\{x,y\\}\in E$;

  2. (2)

$\sum_{\\{x,y\\}\in E}\mathcal{F}(x,y)=0$ unless $x\in A\cup B$.

If $\mathcal{F}$ is a flow from $A$ to $B$ we also write that $\mathcal{F}$ is
a flow $A\to B$. The _support_ of a flow $\mathcal{F}$ is ${\rm
supp}(\mathcal{F}):=\\{\\{x,y\\}\in E:\mathcal{F}(x,y)\neq 0\\}$. The _total
flow_ is given by

$I(\mathcal{F}):=\sum_{x\in A}\sum_{\\{x,y\\}\in E}\mathcal{F}(x,y).$

A _unit flow_ from $\mathcal{F}$ is one for which $I(\mathcal{F})=1$. We also
write the amount of flow from $x$ as

${\rm div}\mathcal{F}(x)=\sum_{\\{x,y\\}\in E}\mathcal{F}(x,y).$

If $e=\\{x,y\\}\in E$, we also write $|\mathcal{F}(e)|=|\mathcal{F}(x,y)|$,
since this does not depend on the orientation of the edge. Define the energy
of the flow by

$\mathcal{E}_{q}(\mathcal{F}):=\sum_{e\in E}|\mathcal{F}(e)|^{q}.$

######  Definition 4.5.

Let $q\in[1,\infty)$. If $A,B\subset V$, then let

$\mathcal{E}_{q}(E,F,G):=\inf\\{\mathcal{E}_{q}(\mathcal{F}):\mathcal{F}\text{
is a unit flow from }E\text{ to }F\\}.$

Both optimal flows and optimal admissible functions are unique. We will use
this lemma frequently without additional mention.

######  Lemma 4.6.

Let $E,F\subset G$. If $p>1$, and $q$ is the dual exponent of $p$, then there
is a unique $\mathcal{E}_{q}$-minimizing flow $\mathcal{F}$ for
$\mathcal{E}_{q}(E,F,G)$. Further for all collections $\Theta$ of paths (or
connected subsets) in Definition 4.1 (or in 4.2) there is a unique optimal
admissible function $\rho$.

###### Proof.

The proof follows immediately from the fact that the optimization problem is
finite dimensional and strictly uniformly convex. ∎

The following proposition for edge modulus is fairly well known, and was first
shown in [66, Theorem 5.1]. More recently, a variant involving probability
measures on curves was shown in [2, Theorem 2.8]. Such probability measures
can be identified with unit flows via superposition and by using the flow
decomposition theorem.

######  Proposition 4.7.

Let $A,B\subseteq V$ be non-empty disjoint subsets. Then for every
$p\in(1,\infty)$ and its dual exponent $q\in(1,\infty)$

${\rm
Mod}_{p}(\Theta(A,B),G)^{\frac{1}{p}}\cdot\mathcal{E}_{q}(A,B,G)^{\frac{1}{q}}=1.$

Moreover, if $\mathcal{F}$ is a unit flow from $A$ to $B$ and $\rho$ is
$\Theta(A,B)$-admissible so that

(4.8)
$\mathcal{M}_{p}(\rho)^{\frac{1}{p}}\cdot\mathcal{E}_{q}(\mathcal{F})^{\frac{1}{q}}=1,$

then

${\rm Mod}_{p}(\Theta(A,B),G)=\mathcal{M}_{p}(\rho)\text{ and
}\mathcal{E}_{q}(A,B,G)=\mathcal{E}_{q}(\mathcal{F}).$

We introduce one more useful quantity that is equivalent to the edge modulus:
the capacity.

######  Definition 4.9.

Let $p>1,G=(V,E)$ be a graph and $\emptyset\neq A,B\subseteq V$ be disjoint
subsets of $V$. The _$p$ -capacity_ from $A$ to $B$ is

${\rm Cap}_{p}(A,B,G):=\inf\left\\{\sum_{\\{x,y\\}\in E}\lvert
U(x)-U(y)\rvert^{p}:U_{\mid A}=0\text{ and }U_{\mid B}=1\right\\}$

We now state a few classical definitions and lemmas. The minimizers for
capacity exist and are $p$-harmonic.

######  Definition 4.10.

Let $p>1,G=(V,E)$ be a graph and $\emptyset\neq V^{\prime}\subseteq V$. For
$p>1$ we say that $U:V\to\mathbb{R}$ is _$p$ -harmonic in $V^{\prime}$_ if for
every $x\in V^{\prime}$

$\sum_{\\{x,y\\}\in E}{\rm sgn}(U(x)-U(y))\lvert U(x)-U(y)\rvert^{p-1}=0.$

######  Lemma 4.11.

Let $p>1,G=(V,E)$ be a connected graph and $\emptyset\neq A,B\subseteq V$ be
disjoint subsets of $V$. There is a unique function $U:V\to\mathbb{R}$ so that
$U|_{A}=0,U|_{B}=1$ and

${\rm Cap}_{p}(A,B,G)=\sum_{\\{x,y\\}\in E}\lvert U(x)-U(y)\rvert^{p}.$

Furthermore, this function is $p$-harmonic in $V\setminus(A\cup B)$.

###### Proof.

The proof follows from uniform convexity and the variational principle, see
e.g. [40, Theorems 3.5 and 3.11] for details. ∎

######  Lemma 4.12.

Let $p>1,G=(V,E)$ be a finite graph and $\emptyset\neq A\subsetneq V$ a
connected subset. If $U:V\to\mathbb{R}$ is $p$-harmonic in $A$ then

$\max_{x\in\overline{A}}U(x)=\max_{x\in\partial{A}}U(x).$

where

$\partial{A}=\\{x\in V\setminus A:\\{x,y\\}\in E\text{ for some }y\in
A\\}\text{ and }\overline{A}=A\cup\partial A.$

Moreover, if $U(y)=\max_{x\in\overline{A}}U(x)$ for some $y\in A$ then $U$ is
constant in $\overline{A}$.

###### Proof.

The definition of $p$-harmonic functions yields by a standard argument the
maximum principle, for details see [40, Theorem 3.14]. ∎

######  Lemma 4.13.

Let $p>1,G=(V,E)$ be a finite connected graph and $\emptyset\neq A,B\subseteq
V$ be disjoint subsets of $V$. Then

(4.14) ${\rm Mod}_{p}(A,B,G)={\rm Cap}_{p}(A,B,G).$

Moreover, if $\widehat{\rho}:E\to\mathbb{R}_{\geq 0}$ is the optimal
$\Theta(A,B)$-admissible density, $\mathcal{F}:V\times V\to\mathbb{R}$ is the
optimal flow from $A$ to $B$ with respect to $\mathcal{E}_{q}$ and
$\widehat{U}:V\to\mathbb{R}$ so that $\widehat{U}|_{A}=0,\widehat{U}|_{B}=1$
and

${\rm Cap}_{p}(A,B,G)=\sum_{\\{x,y\\}\in
E}\lvert\widehat{U}(x)-\widehat{U}(y)\rvert^{p}.$

then

$\widehat{\rho}(\\{x,y\\})=\lvert\widehat{U}(x)-\widehat{U}(y)\rvert$

and

$\mathcal{F}(x,y)={\rm Mod}_{p}(A,B,G)^{-1}{\rm
sgn}(\widehat{U}(y)-\widehat{U}(x))\lvert\widehat{U}(y)-\widehat{U}(x)\rvert^{p-1}$

for all $\\{x,y\\}\in E$.

###### Proof.

Suppose $\rho:E\to\mathbb{R}$ is $\Theta(A,B)$-admissible and we define
$U:V\to\mathbb{R}$ by

$U(x)=\min_{\gamma}\sum_{i=1}^{n-1}\rho(\\{x_{i},x_{i+1}\\})$

where the minimum is over all paths $\gamma=[x_{1},\dots,x_{n}]$ so that
$x_{1}=x$ and $x_{n}\in A$. For $U^{\prime}=U\land 1$, we have
$U^{\prime}|_{A}=0,U^{\prime}|_{B}=1$ and

$\lvert U^{\prime}(x)-U^{\prime}(y)\rvert\leq\lvert
U(x)-U(y)\rvert\leq\rho(\\{x,y\\})$

for all $\\{x,y\\}\in E$. This proves

${\rm Cap}_{p}(A,B,G)\leq{\rm Mod}_{p}(A,B,G).$

On the other hand, if $U|_{A}=0,U|_{B}=1$, then $\rho(\\{x,y\\})=\lvert
U(x)-U(y)\rvert$ is, by the triangle inequality, $\Theta(A,B)$-admissible.
Hence (4.14) holds. Furthermore, if we define the density
$\widetilde{\rho}(\\{x,y\\})=\lvert\widehat{U}(x)-\widehat{U}(y)\rvert$, then

$\mathcal{M}_{p}(\widetilde{\rho})={\rm Mod}_{p}(A,B,G).$

By the uniqueness of the optimal density, $\widehat{\rho}=\widetilde{\rho}$.

Finally, let $\mathcal{F}$ be the optimal flow function from $A$ to $B$. By
[66, Lemma 3.1], we have

$1=\sum_{\\{x,y\\}\in E}(U(y)-U(x))\mathcal{F}(x,y).$

Thus, by Hölder’s inequality, we have

$1\leq\left(\sum_{\\{x,y\\}\in
E}|U(y)-U(x)|^{p}\right)^{\frac{1}{p}}\left(\sum_{\\{x,y\\}\in
E}\mathcal{F}(x,y)^{q}\right)^{\frac{1}{q}}.$

By duality, Proposition 4.7, the inequality is an equality. This is only
possible if $\mathcal{F}(x,y)$ is proportional to ${\rm
sgn}(\widehat{U}(y)-\widehat{U}(x))\lvert\widehat{U}(y)-\widehat{U}(x)\rvert^{p-1}$.
The propotionality constant can be computed using Proposition 4.7 again. ∎

The previous proposition also encodes the so called Ohm’s law for optimal
potential functions and flows; see [55, Section 2].

In what follows, we will often drop the graph $G$ from the notation of moduli,
capacity and flows, where it is clear from context and write e.g. ${\rm
Mod}_{p}(A,B)$, ${\rm Cap}_{p}(A,B)$ and $\mathcal{E}_{q}(A,B)$.

###  4.2. Moduli computations on replacement graphs

The biggest advantage of linear replacement rules is the simplicity of edge
modulus computations. Indeed, all modulus problems that we study in this paper
can be reduced to the modulus problem over the paths between the gluing sets
$\Theta(I_{v,\\{v,u\\}},I_{u,\\{v,u\\}})$, for $\\{u,v\\}\in E_{1}$. These
modulus problems are explicitly and effectively computable.

First we introduce the basic notations. For $e=\\{v,u\\}\in E_{n}$ and
$m\in\mathbb{N}$ we will write
$\Theta_{v,e}^{(m)}=\Theta\left(I_{v,e}^{(m)},I_{u,e}^{(m)}\right)$ for the
path family in $G_{m}$. We denote the optimal $\Theta_{v,e}^{(m)}$-admissible
density (for both edge- and vertex $p$-modulus) as $\rho_{e,m}$. Observe that,
by symmetry, $\rho_{e,m}$ is the optimal $\Theta^{(m)}_{u,e}$-density. Hence
we omit the vertex $v,u$ from the notation of the optimal function
$\rho_{e,m}$. For $m=1$ we simply write $\Theta_{v,e}^{(1)}=\Theta_{v,e}$ and
$\rho_{e,1}=\rho_{e}$. We denote the $\mathcal{E}_{q}$-minimizing unit flow
from $I_{v,e}$ to $I_{u,e}$ as $\mathcal{F}_{v,e}$. Lastly we define the
family of paths

$\Theta_{m}:=\bigcup_{\begin{subarray}{c}e\in E_{1}\\\ v\in
e\end{subarray}}\Theta_{v,e}^{(m)}$

and the $\Theta_{m}$-admissible density
$\widetilde{\rho}_{m}:V_{m}\to\mathbb{R}_{\geq 0}$ by

(4.15) $\widetilde{\rho}_{m}(v):=\max_{e\in E_{1}}\rho_{e,m}(v).$

######  Remark 4.16.

Observe that, in general, $\rho_{e}$ and $\mathcal{F}_{v,e}$ depend on $p$ and
$q$. Moreover, for $p=1$, $\rho_{e}$ is not unique. Since in most arguments we
deal with only one $p>1$ and $q>1$ is the dual exponent of $p$, we omit $p,q$
from the notation.

Next we introduce conductively uniform-condition, under which the moduli
computations are extremely simple and straightforward.

######  Definition 4.17 (Conductively uniform).

We say that a linear replacement rule satisfying doubling property is
_conductively uniform_ if for every $e=\\{v,u\\}\in E_{n}$ the value

(4.18)
$\mathcal{F}_{a}:=\mathcal{F}_{v,e}(\phi_{v,e}(a),\mathfrak{n}(\phi_{v,e}(a)))$

only depends on $a\in I$ and the value

(4.19) $\mathcal{R}_{p}:=\mathcal{E}_{q}(\mathcal{F}_{v,e})$

does not depend on $e$ or $v$.

The first of these means that the optimal flows in $G_{1}$ between the gluing
sets are distributed in a way that is independent of the gluing used. This is
particularly important when defining flows at higher level graphs, since we
need compatibility across the gluing sets. The second of these properties
means that the modulus “accross” an edge is independent of the edge. This
allows us to compute optimal moduli at the $n$’th level in a simple way. As we
will shortly see, optimality of such flows and potential functions is shown
using Proposition 4.7. But, first, we shall give a few examples.

######  Example 4.20.

Recall the symmetric replacement rules from Remark 3.3. These are all
conductively uniform by the first part of Lemma 6.7. Indeed, it seems that the
symmetric case is the most natural setting to obtain uniformity. Consequently,
since Figures 1.1 and 1.5 are symmetric, they also are conductively uniform.

Symmetry is, however, not always needed for conductive uniformity, see e.g.
Figures 4.21 and 4.22.

123456789 Figure 4.21. This non-symmetric oriented replacement rule is
obtained from the one in Figure 1.5 by adding one vertex, and an edge
connecting the vertex $3$ to this added vertex. The vertices are re-ordered
left-to-right. A direct calculation shows that this yields a conductively
uniform replacement rule, and that the edge $\\{3,4\\}$ is removable in the
sense of Definition 6.2. In general, taking any conductively uniform
replacement rule, and gluing to it an edge along one of its end points
produces another conductively uniform replacement rule. In this example,
$I=\\{a,b\\}$ and $\phi_{v,\\{v,u\\}}(a)=1,\phi_{v,\\{v,u\\}}(b)=2$ if $v<u$,
and $\phi_{v,\\{v,u\\}}(a)=8,\phi_{v,\\{v,u\\}}(b)=9$ if $u<v$.
123456789101112 Figure 4.22.  The solid lines in the figure shows one of the
non-symmetric spaces, with $N=3,L_{*}=4$ constructed in the fashion of Example
4.23, together with the ordering of vertices described. The dashed line
indicates an edge, which can be added to produce another conductively uniform
oriented replacement rule with the $L_{*}$-uniform scaling property, where the
dashed edge is removable in the sense of Definition 6.2. In general, any edge
which connects vertically aligned points is removable.

######  Example 4.23.

Let $N,L_{*}\geq 2$. A lot of (non-symmetric) conductively uniform oriented
replacement rules that satisfy the $L_{*}$-uniform scaling property can be
obtained as follows. Take $N$ line segments and subdivide each of them to
$L_{*}$ equal parts. Along each line segment, there are $N$ subdivision points
$v_{(k,i)}$, where $v_{(k,i)}$ is the $k$’th subdivision point in the $i$’th
copy of the line segment. We then form a new graph by identifying, for
$k=2,\dots,L_{*}-1$, pairs of these subdivision points $v_{(k,i)}\sim
v_{(k,j)}$ for $(i,j)\in S_{k}$, where $S_{k}$ are some given equivalence
relations on $\\{1,\dots,N\\}$. Given enough such identifications, we obtain a
connected linear replacement rule $G_{1}=(V_{1},E_{1})$. The gluing rules are
then described as follows. First, we give an order for the equivalence classes
$[v_{(k,i)}]$, which are ordered so that if $k<l$, then
$[v_{(k,j)}]<[v_{(l,m)}]$ for all $k<l$ and all $j\leq m$. Let $v_{i}$,
$i=1,\dots,M$ be the ordered list of equivalence classes, where $M$ is the
total number of vertices. The (oriented) gluing rules can then be given by
$I=\\{1,\dots,N\\}$ and $\phi_{v,\\{v,u\\}}(i)=\phi_{-}(i)=i$ if $v<u$, and
$\phi_{u,\\{v,u\\}}(i)=\phi_{+}(i)=M-N+i$ if $v<u$. In these examples,
conductive uniformity can be verified by using duality and using the weight
function $\rho(e)=\frac{1}{L_{*}}$ and flow function
$\mathcal{F}(x,y)=\frac{1}{N}$ if $x<y$ and $\\{x,y\\}\in E$. A direct
calculation shows that
$\mathcal{E}_{q}(\mathcal{F})^{\frac{1}{q}}\mathcal{M}_{p}(\rho)^{\frac{1}{p}}=1$
for all $p\in(1,\infty)$, and thus $\mathcal{F}$ and $\rho$ are optimal by
Proposition 4.7. See Figure 4.22, which shows one example of a replacement
rule obtained in this fashion that is non-symmetric and that satisfies the
$L_{*}$-uniform scaling property. These examples are also all Loewner, and
they attain their conformal dimensions, see Example 5.14. By adding “removable
edges” to them, we obtain counter-examples to Kleiner’s conjecture, see
Section 6.

######  Lemma 4.24.

For every $e=\\{v,u\\}\in E_{n}$ we have

(4.25) $\mathcal{F}_{v,e}=-\mathcal{F}_{u,e}.$

###### Proof.

As $\mathcal{E}_{q}(-\mathcal{F}_{v,e})=\mathcal{E}_{q}(\mathcal{F}_{v,e})$
and $-\mathcal{F}_{v,e}$ is a unit flow from $I_{u,e}$ to $I_{v,e}$, the claim
follows from the uniqueness of the optimal flow. ∎

######  Lemma 4.26.

If the linear replacement rule is doubling and conductively uniform then

(4.27) $\mathcal{M}_{p}:=\mathcal{M}_{p}(\rho_{e})$

does not depend on $e=\\{v,u\\}\in E_{n}$. In particular,

(4.28)
$(\mathcal{M}_{p})^{\frac{1}{p}}\cdot(\mathcal{R}_{p})^{\frac{1}{q}}=1.$

###### Proof.

Directly follows from Proposition 4.7 and (4.19). ∎

######  Proposition 4.29.

If the linear replacement rule satisfies the $L_{*}$-uniform scaling property
for $L_{*}\geq 2$, the doubling property and is conductively uniform, then
$\mathcal{M}_{p}$ is continuously strictly decreasing in $p\in[1,\infty)$.
Furthermore, if $G_{1}$ contains two edge-wise disjoint paths
$\theta_{1},\theta_{2}\in\Theta_{v,e}$ for some $e=\\{v,u\\}\in E_{1}$, then
$\mathcal{M}_{1}>1$ and there is a unique $Q\in(1,\infty)$ so that
$\mathcal{M}_{Q}=1$.

###### Proof.

In order to avoid confusion with the $p$-dependence of the optimal density, we
will write $\rho_{e,p}$ as the optimal $\Theta_{v,e}$-admissible density.

If $\rho$ is admissible for $\Theta$, then $\min(\rho,1)$ is also admissible.
Thus, for the optimal admissible density function we have $\rho_{e,p}(f)\leq
1$ for all $f\in E.$ From this, it is direct to verify that for any non-empty
family of paths $\Theta$ in a finite graph $G$, ${\rm Mod}_{p}(\Theta,G)$ is
continuously decreasing in $p\in[1,\infty)$. To prove that
$\mathcal{M}_{p}={\rm Mod}_{p}(\Theta_{v,e})$ is strictly decreasing, it is
sufficient to prove that for all $p\in(1,\infty)$ and $f\in E_{1}$ we have
$\rho_{e,p}(f)<1$. Indeed then

$\mathcal{M}_{p}=\mathcal{M}_{p}(\rho_{e,p})>\mathcal{M}_{p+\epsilon}(\rho_{e,p})\geq\mathcal{M}_{p+\epsilon}(\rho_{e,p+\epsilon})=\mathcal{M}_{p+\epsilon}$

for any $\epsilon>0$.

Fix $p>1$. By Lemmas 4.11 and 4.13 there is a function $U:V_{1}\to\mathbb{R}$,
which is $p$-harmonic in $V_{1}\setminus(I_{v,e}\cup I_{u,e})$ and
$\rho_{e,p}(\\{x,y\\})=\lvert U(x)-U(y)\rvert$ for all $\\{x,y\\}\in E_{1}$.
Now if $\\{x,y\\}\in E_{1}$, by the uniform scaling property and the fact that
$I_{v,e},I_{u,e}$ are independent sets, at most one of the vertices $x,y$ is
contained in $I_{v,e}\cup I_{u,e}$. Furthermore, by the doubling property,
$V_{1}\setminus(I_{v,e}\cup I_{u,e})$ is connected and
$\overline{V_{1}\setminus(I_{v,e}\cup I_{u,e})}=V_{1}$. By the strong maximum
principle, Lemma 4.12, we have

$\rho_{e,p}(\\{x,y\\})=\lvert U(x)-U(y)\rvert<1$

for all $\\{x,y\\}\in E_{1}$ and so we conclude that $\mathcal{M}_{p}$ is
strictly decreasing.

We proceed to the second part. Let
$\theta_{1}=[v_{1},\dots,v_{k}],\theta_{2}=[u_{1},\dots,u_{l}]\in\Theta_{v,e}$
be two edge-wise disjoint paths. By taking a sub-path, we may assume that both
are simple paths. Now if $\rho$ is $\Theta_{v,e}$-admissible we have

$\mathcal{M}_{1}(\rho)=\sum_{e\in
E_{1}}\rho(e)\geq\sum_{i=1}^{k-1}\rho(\\{v_{i},v_{i+1}\\})+\sum_{i=1}^{l-1}\rho(\\{u_{i},u_{i+1}\\})\geq
2$

Therefore $\mathcal{M}_{1}>1$. On the other hand, by the $L_{*}$-uniform
scaling property, the density $\widetilde{\rho}\equiv L_{*}^{-1}$ is
$\Theta_{v,e}$-admissible. Hence

$\mathcal{M}_{p}\leq\mathcal{M}_{p}(\widetilde{\rho})=\lvert
E_{1}\rvert\cdot\frac{1}{L_{*}^{p}}<1$

for large enough $p>1$. By continuity of $\mathcal{M}_{p}$, there must be
$Q\in(1,\infty)$ so that $\mathcal{M}_{Q}=1$. Since $\mathcal{M}_{p}$ is
strictly decreasing, this $Q$ is unique. ∎

For the rest of this section we assume that the linear replacement rule is
doubling and conductively uniform.

######  Proposition 4.30.

Let $A,B\subseteq V_{n}$ be non-empty disjoint subsets and $m\in\mathbb{N}$.
Given a $\Theta(A,B)$-admissible density $\rho_{n}:E_{n}\to\mathbb{R}_{\geq
0}$ there is a
$\Theta\left(\pi_{n+m,n}^{-1}(A),\pi_{n+m,n}^{-1}(B)\right)$-admissible
density $\rho_{n+m}:E_{n+m}\to\mathbb{R}_{\geq 0}$ with

(4.31)
$\mathcal{M}_{p}(\rho_{n+m})=\mathcal{M}_{p}(\rho_{n})\cdot(\mathcal{M}_{p})^{m}\text{
and }{\rm supp}(\rho_{n+m})\subseteq\pi_{n+m,n}^{-1}({\rm supp}(\rho_{n})).$

###### Proof.

It is sufficient to prove the case $m=1$. Define

$\rho_{n+1}(\\{[z_{1},e],[z_{2},e]\\})=\rho_{n}(e)\cdot\rho_{e}(\\{z_{1},z_{2}\\}).$

By (SM3) $\rho_{n+1}$ is well-defined and, by (4.27), it is sufficient to
prove that $\rho_{n+1}$ is
$\Theta\left(\pi_{n+1,n}^{-1}(A),\pi_{n+1,n}^{-1}(B)\right)$-admissible. Let
$\theta\in\Theta\left(\pi_{n+1,n}^{-1}(A),\pi_{n+1,n}^{-1}(B)\right)$ and
$\theta=[v_{1},\dots,v_{k}]$. By Proposition 3.16 there is a path
$\hat{\theta}=\left[u_{1},\dots,u_{l}\right]\in\Theta(A,B)$ and disjoint sub-
paths $\theta_{1},\dots,\theta_{l-1}$ of $\theta$ where

$\theta_{i}\in\Theta(\pi_{n+1,n}^{-1}(u_{i}),\pi_{n+1,n}^{-1}(u_{i+1}))\text{
and }e_{i}=\\{u_{i},u_{i+1}\\}$

and $\theta_{i}$ is contained in $e_{i}\cdot G_{1}$ where
$e_{i}=\\{u_{i},u_{i+1}\\}$ for all $i=1,\dots,l-1$. By (SM1),
$\hat{\theta}_{i}:=\sigma_{e_{i}}^{-1}(\theta_{i})\in\Theta_{e_{i}}$, so we
have

$\displaystyle L_{\rho_{n+1}}(\theta)$
$\displaystyle\geq\sum_{i=1}^{l-1}L_{\rho_{n+1}}(\theta_{i})$
$\displaystyle=\sum_{i=1}^{l-1}\rho_{n}\left(e_{i}\right)\cdot
L_{\rho_{e_{i}}}\left(\hat{\theta}_{i}\right)$
$\displaystyle\geq\sum_{i=1}^{l-1}\rho_{n}\left(e_{i}\right)\geq 1.$

∎

######  Proposition 4.32.

Let $A,B\subseteq V_{n}$ be non-empty disjoint subsets. Given any unit flow
$\mathcal{F}_{n}$ from $A$ to $B$ there is a unit flow $\mathcal{F}_{n+m}$
from $\pi_{n+m,n}^{-1}(A)$ to $\pi_{n+m,n}^{-1}(B)$ with

(4.33)
$\mathcal{E}_{q}(\mathcal{F}_{n+m})=\mathcal{E}_{q}(\mathcal{F}_{n})\cdot(\mathcal{R}_{p})^{m}\text{
and }{\rm supp}(\mathcal{F}_{n+m})\subseteq\pi_{n+m,n}^{-1}({\rm
supp}(\mathcal{F}_{n})).$

###### Proof.

Sufficient to prove the case $m=1$. We define $\mathcal{F}_{n+1}$ as follows:
For $e=\\{v,u\\}\in E_{n}$ and $z_{1},z_{2}\in V$ we set

$\mathcal{F}_{n+1}([z_{1},\\{v,u\\}],[z_{2},\\{v,u\\}])=\mathcal{F}_{n}(v,u)\cdot\mathcal{F}_{v,e}(z_{1},z_{2}).$

The above definition is well-defined in the sense that it does not depend on
the orientation of $e=\\{v,u\\}$. Indeed, by Lemma 4.24 we have

$\displaystyle\mathcal{F}_{n+1}([z_{1},\\{u,v\\}],[z_{2},\\{u,v\\}])$
$\displaystyle=\mathcal{F}_{n}(u,v)\cdot\mathcal{F}_{u,e}(z_{1},z_{2})$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq: Reversing
flow})}}{{=}}(-\mathcal{F}_{n}(v,u))\cdot(-\mathcal{F}_{v,e}(z_{1},z_{2}))$
$\displaystyle=\mathcal{F}_{n}(v,u)\cdot\mathcal{F}_{v,e}(z_{1},z_{2})$
$\displaystyle=\mathcal{F}_{n+1}([z_{1},\\{v,u\\}],[z_{2},\\{v,u\\}]).$

Also (4.33) follows from (4.19) and the the definition of $\mathcal{F}_{n+1}$
so we only need to prove that $\mathcal{F}_{n+1}$ is a unit flow from
$\pi_{n+1,n}^{-1}(A)$ to $\pi_{n+1,n}^{-1}(B)$. Let $[z,\\{v,u\\}]\in V_{n+1}$
for some $e=\\{v,u\\}\in E_{n}$ and $z\in V$. Assume that
$[z,\\{v,u\\}]\not\in\pi_{n+1,n}^{-1}(A)\cup\pi_{n+1,n}^{-1}(B)$. There are
two cases to consider. First, assume that $z\notin I_{v,e}\cup I_{u,e}$. Then,
by (SM1) and (SM2), we have

${\rm div}(\mathcal{F}_{n+1})([z,\\{v,u\\}])={\rm
div}(\mathcal{F}_{v,e})(z)\cdot\mathcal{F}_{n}(v,u)=0.$

Next consider $z\in I_{v,e}\cup I_{u,e}$. By symmetry we may assume
$z=\phi_{v,e}(a)\in I_{v,e}$ for some $a\in I$. Let $e_{1},\dots,e_{k}$ be the
edges in $E_{n+1}$ containing $[z,\\{v,u\\}]$ and write those as
$e_{i}=\\{[z,\\{v,u\\}],[z_{i},\\{v_{i},u_{i}\\}]\\}$. Then, by definition,
$\\{v,u\\}$ and $\\{v_{i},u_{i}\\}$ have a common vertex, which must be $v$
since otherwise $z\in I_{v,e}\cap I_{u,e}$. Up to relabeling, we may assume
$v_{i}=v$ so we have $z_{i}=\mathfrak{n}(\phi_{v_{i},e_{i}}(a))$ and
$v=v_{i}=\phi_{v_{i},\\{v_{i},u_{i}\\}}(a)$. We get

$\displaystyle{\rm div}(\mathcal{F}_{n+1})([z,\\{v,u\\}])$
$\displaystyle=\sum_{i=1}^{k}\mathcal{F}_{n+1}([z,\\{v,u\\}],[z_{i},\\{v_{i},u_{i}\\}])$
$\displaystyle=\sum_{i=1}^{k}\mathcal{F}_{n+1}([\phi_{v_{i},\\{v_{i},u_{i}\\}}(a),\\{v_{i},u_{i}\\}],[z_{i},\\{v_{i},u_{i}\\}])$
$\displaystyle=\sum_{i=1}^{k}\mathcal{F}_{n}(v_{i},u_{i})\cdot\mathcal{F}_{v_{i},\\{v_{i},u_{i}\\}}(\phi_{v_{i},\\{v_{i},u_{i}\\}}(a),z_{i})$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq: Flows agree on
boundary})}}{{=}}\mathcal{F}_{a}\cdot\sum_{i=1}^{k}F_{n}(v_{i},u_{i})$
$\displaystyle=\mathcal{F}_{a}\cdot{\rm div}(F_{n})(v).$

Recall that $[z,\\{v,u\\}]\not\in\pi_{n+1,n}^{-1}(A)\cup\pi_{n+1,n}^{-1}(B)$
and $z\in I_{v,\\{v,u\\}}$. Since
$[z,\\{v,u\\}]\notin\pi_{n+1,n}^{-1}(A)\cup\pi_{n+1,n}^{-1}(B)$, we must have
$v\notin A_{n}\cup B_{n}$. This concludes showing that $\mathcal{F}_{n+1}$ is
a flow from $\pi_{n+1,n}^{-1}(A)$ to $\pi_{n+1,n}^{-1}(B)$.

Finally, if $v\in A_{n}$ and $z\in I_{v,\\{v,u\\}}$ then
$[z,\\{v,u\\}]\in\pi_{n+1,n}^{-1}(A)$. Hence

$\displaystyle I(\mathcal{F}_{n+1})$
$\displaystyle=\sum_{[z,\\{v,u\\}]\in\pi_{n+1,n}^{-1}(A)}{\rm
div}(\mathcal{F}_{n+1})([z,\\{v,u\\}])$ $\displaystyle=\quad\sum_{a\in
I}\mathcal{F}_{a}\cdot\sum_{v\in A_{n}}{\rm div}(\mathcal{F}_{n})(v)$
$\displaystyle=\quad\sum_{a\in I}\mathcal{F}_{a}\cdot I(\mathcal{F}_{n})$
$\displaystyle=\quad 1.$

Thus $\mathcal{F}$ is a unit flow, and the claim follows. ∎

######  Corollary 4.34.

If $e=\\{v,u\\}\in E_{n}$ is any edge then

(4.35) ${\rm
Mod}_{p}\left(\Theta_{v,e}^{(m)},G_{m}\right)=(\mathcal{M}_{p})^{m}$

###### Proof.

By applying Propositions 4.30 and 4.32 to the $\Theta_{v,e}$-admissible
density $\rho_{e}$ and the unit flow $\mathcal{F}_{v,e}$ from $I_{v,e}$ to
$I_{u,e}$ respectively, we obtain the
$\Theta\left(I_{v,e}^{(m)},I_{u,e}^{(m)}\right)$-admissible density $\rho_{m}$
and the unit flow $\mathcal{F}_{m}$ from $I_{v,e}^{(m)}$ to $I_{u,e}^{(m)}$.
By (4.31) and (4.33), they satisfy

$\mathcal{M}_{p}(\rho_{m})=(\mathcal{M}_{p})^{m}\text{ and
}\mathcal{E}_{q}(\mathcal{F}_{m})=(\mathcal{R}_{p})^{m}.$

By (4.28) we have
$(\mathcal{M}_{p})^{\frac{1}{p}}(\mathcal{R}_{p})^{\frac{1}{q}}=1$, and thus
$\mathcal{M}_{p}(\rho_{m})^{\frac{1}{p}}\mathcal{E}_{q}(\mathcal{F}_{m})^{\frac{1}{q}}=1$.
Thus Proposition 4.7 yields (4.35). ∎

######  Corollary 4.36.

The density $\widetilde{\rho}_{m}$ is $\Theta_{m}$-admissible and there is a
constant $C=C(p,C_{\deg},\lvert E_{1}\rvert)$ so that

(4.37) $\mathcal{M}_{p}(\widetilde{\rho}_{m})\leq
C\cdot(\mathcal{M}_{p})^{m}.$

Here $\widetilde{\rho}_{m}$ is as in (4.15).

###### Proof.

Admissibility is clear and the $p$-mass estimate follows by

$\mathcal{M}_{p}(\widetilde{\rho}_{m})\leq\lvert E_{1}\rvert\cdot\max_{e\in
E_{1}}\mathcal{M}_{p}(\rho_{e,m})\stackrel{{\scriptstyle(\ref{eq: Edge- and
Vertex modulus})}}{{\leq}}C\cdot(\mathcal{M}_{p})^{m}.$

∎

######  Theorem 4.38.

Let $A,B\subseteq V_{n}$ be non-empty disjoint subsets. Then

(4.39) ${\rm Mod}_{p}(\pi_{n+m,n}^{-1}(A),\pi_{n+m,n}^{-1}(B))={\rm
Mod}_{p}(A,B)\cdot(\mathcal{M}_{p})^{m}.$

Moreover, if $\rho_{n}:V_{n}\to\mathbb{R}_{\geq 0}$ is the optimal
$\Theta(A,B)$-admissible density then the optimal
$\Theta(\pi_{n+m,n}^{-1}(A),\pi_{n+m,n}^{-1}(B))$-admissible density is

(4.40)
$\rho_{n+m}(e)=\rho_{n}(\pi_{n+m,n}(e))\cdot\rho_{\pi_{n+m,n}(e),m}\left(\sigma_{\pi_{n+m,n}(e),m}^{-1}(e)\right).$

In particular, ${\rm supp}\,\rho_{n+m}\subseteq\pi_{n+m,n}^{-1}({\rm
supp}\,\rho_{n}).$

###### Proof.

When we apply Proposition 4.30 to $\rho_{n}$, it follows from the proof of the
proposition, that the density constructed is exactly $\rho_{n+m}$ as defined
in (4.40). Furthermore, it follows from similar argument as in Corollary 4.34
that $\rho_{n+m}$ is indeed the optimal