Equality of different definitions of conformal dimension for quasiself-similar and CLP spaces

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY-NC 4.0 https://creativecommons.org/licenses/by-nc/4.0/ Equality of different definitions of conformal dimension for quasiself-similar and CLP spaces © 2024 The Finnish Mathematical Society Published version Eriksson-Bique, Sylvester Eriksson-Bique, S. (2024). Equality of different definitions of conformal dimension for quasiselfsimilar and CLP spaces. Annales Fennici Mathematici, 49(2), 405-436. https://doi.org/10.54330/afm.146682 2024 Annales Fennici Mathematici Volumen 49, 2024, 405–436 Equality of different definitions of conformal dimension for quasiself-similar and CLP spaces Sylvester Eriksson-Bique Abstract. We prove that for a quasiself-similar and arcwise connected compact metric space all three known versions of the conformal dimension coincide: the conformal Hausdorff dimension, conformal Assouad dimension and Ahlfors regular conformal dimension. This answers a question posed by Murugan. Quasisimilar spaces include all approximately self-similar spaces. As an example, the standard Sierpiński carpet is quasiself-similar and thus the three notions of conformal dimension coincide for it. We also give the equality of the three dimensions for combinatorially p-Loewner (CLP) spaces. Both proofs involve using a new notion of combinatorial modulus, which lies between two notions of modulus that have appeared in the literature. The first of these is the modulus studied by Pansu and Tyson, which uses a Carathéodory construction. The second is the one used by Keith and Laakso (and later modified and used by Bourdon, Kleiner, Carrasco-Piaggio, Murugan and Shanmugalingam). By combining these approaches, we gain the flexibility of giving upper bounds for the new modulus from the Pansu–Tyson approach, and the ability of getting lower bounds using the Keith–Laakso approach. Additionally the new modulus can be iterated in self-similar spaces, which is a crucial, and novel, step in our argument. Konformisen Hausdorffin ulottuvuuden eri määritelmien yhtäsuuruus kvasi-itsesimilaarisille ja CLP-avaruuksille Tiivistelmä. Osoitamme, että kvasi-itsesimilaarisilla ja polkuyhtenäisillä kompakteilla metrisillä avaruuksille kaikki kolme tunnettua konformisen ulottuvuuden määritelmää ovat yhteneviä: konforminen Hausdorffin dimensio, konforminen Assouadin dimensio ja Ahlforsin-säännöllinen konforminen ulottuvuus. Tämä vastaa Muruganin esittämään avoimeen kysymykseen. Kvasi-itsesimilaariset avaruudet ovat approksimatiivisesti itsesimilaaristen avaruuksien yleistys. Tuloksiemme seurauksena esimerkiksi Sierpińskin matolla kaikki konformisen ulottuvuuden määritelmät antavat saman arvon. Tarkastelemme myös avaruuksia, jotka toteuttavat p-kombinatorisen Loewnerin ehdon (CLP). Osoitamme, että näilläkin avaruuksilla eri konformisen ulottuvuuden määritelmät antavat saman arvon. Todistuksissa hyödynnämme uutta kombinatorisen moduluksen määritelmää, joka saadaan yhdistelemällä yhtäältä Pansun ja Tysonin ja toisaalta Keithin ja Laakson moduluksien määritelmiä. Tämä synteesi omaa molempien määritelmien hyviä puolia: saamme Pansun ja Tysonin lähestymistavalla modulukselle ylärajoja, ja Keithin ja Laakson lähestymistavasta alarajoja. Olennaista todistuksellemme on myös, että uutta modulusta voi iteroida algoritmisesti. https://doi.org/10.54330/afm.146682 2020 Mathematics Subject Classification: Primary 30L10, 51F99. Key words: Conformal dimension, Assouad dimension, Ahlfors regular, self-similar, combinatorial Loewner property, modulus. The author was partially supported by Finnish Academy Grants n. 345005 and n. 356861. We thank Mathav Murugan for posing the question to us, for discussing the problem at the Okinawa Institute of Science and Technology (OIST) in June 2023 and for giving helpful comments on a preprint version of this paper. The work was started at the workshop “Random walks and analysis on metric spaces” held at OIST. We thank the institute for its hospitality and care—especially given the weak typhoon that overlapped the event. c 2024 The Finnish Mathematical Society 406 Sylvester Eriksson-Bique 1. Introduction In this paper we study the equivalence of three different versions of the definition of the conformal dimension. First, recall the definition of a quasisymmetry. Definition 1.1. We say that a homeomorphic map f:X→Yis a quasisymmetry, if there exists a homeomorphism η: [0,∞)→[0,∞)so that for all x, y, z ∈X with x6=z, we have (1.2) d(f(x), f(y)) d(f(x), f(z)) ≤ηd(x, y) d(x, z). We write X∼q.s. Y, if there exists a quasisymmetry f:X→Y. We say that fis an η-quasisymmetry, if it satisfies (1.2) with this specific function η. We consider the effect of quasisymmetries on the a) Hausdorff dimension, b) Assouad dimension and c) Ahlfors-regularity of the space. We briefly recall the definitions of these. If s∈[0,∞), we define the Hausdorff s-content (at scale δ∈ (0,∞]) of A⊂Xas (1.3) Hs δ(A) = inf (X i∈N diam(Ai)s:A⊂[ i∈N Ai,diam(Ai)≤δ). For future reference, also define Hausdorff measure by Hs(A) := lim δ→0Hs δ(A). The Hausdorff dimension of Xcan be defined as dimH(X) = inf{s > 0: Hs ∞(X)=0}. Hausdorff dimension is not stable when sequences of spaces “converge” (e.g. in the Gromov–Hausdorff sense), and this is one reason to introduce Assouad dimension. If A⊂Xis a subset, let (1.4) N(A, r) = inf {N:∃x1, . . . , xN, A ⊂SB(xi, r)}. Define the Assouad dimension in terms of the scale-invariant asymptotic behaviour of this quantity. dimA(X) = inf{s > 0: ∃C > 0,∀R > r > 0,∀z∈X, N(B(z, R), r)≤CRsr−s}. Finally, a special setting, where dimA(X) = dimH(X)is when Xis Ahlfors regular for some Q > 0. We say that Xis Q-Ahlfors regular, if there exists a Radon measure µon Xand some constant C≥1, with C−1rQ≤µ(B(z, r)) ≤CrQ. for every z∈Xand r∈(0,diam(X)). In this case, Q= dimA(X) = dimH(X). Even if the space is not Ahlfors regular, we always have the inequality dimH(X)≤ dimA(X), where the inequality may be strict. Further, while not every space is QAhlfors regular, spaces often can be deformed into such. A more detailed discussion on this and the three notions of dimension, as well as conformal dimension in general, is given in [18, Chapter 2]. An example of a quasisymmetric map is the identity map (X, d)→(X, dθ), for θ∈(0,1). Such a map increases all of the three notions of dimension. It is considerably harder to decrease dimension, and one is led to defining three quasisymmetric Equality of different definitions of conformal dimension for quasiself-similar and CLP spaces 407 invariants. Conformal Hausdorff dimension dimCH (X) = inf{dimH(Y): X∼q.s. Y} Conformal Assouad dimension dimCA(X) = inf{dimA(Y): X∼q.s. Y} Ahlfors regular conformal dimension dimCAR(X) = inf{Q= dimH(Y): X∼q.s. Y, Yis Q-Ahlfors regular}. The first of these was defined in [21], while the final one was used in [6]. The Assouad variant appeared already in [15]. In general, dimCH (X)≤dimCA(X)≤dimCAR(X). For uniformly perfect spaces dimCAR(X) = dimCA(X), see [18, Proposition 2.2.6.] and [12, Chapters 14 and 15]. The relationship between dimCH (X)and dimCA(X)has so far not been studied in detail, beyond giving simple examples such as the following, when they are not equal. Example 1.5. Let X=Z×R. The conformal Assouad dimension can only drop under blowing the space down, and thus dimCA(X)≥dimCA(R2) = 2. The latter follows since the topological dimension of the plane is 2, and the Hausdorff dimension is always greater than the topological dimension. However, dimCH(X) = dimH(X) = 1. If we set X=Z×R∪R×Z, we can even make Xconnected without altering the previous argument. It is possible to make the space compact and connected as well: Let X= ({1 n:n∈N} ∪ {0})×[0,1] ∪[0,1] ×({1 n:n∈N} ∪ {0}). In this case, a blow-up of the space is R2. Assouad dimension involves a scale-invariant quantitative condition, while Hausdorff dimension is merely a qualitative statement on the dimension of the space. Further, as the previous example indicates dimCA(X)has stability properties under limits, while dimCH (X)does not. This means, that one may only hope for their equality in the case where one assumes some form of self-similarity. Consequently Mathav Murugan asked if the different definitions of conformal dimension agree for self-similar spaces [19]. This question is quite natural, since many well-studied examples have self-similarity: iterated functions systems of finite type [20], Julia sets of rational maps equipped with a visual metric [4] and boundaries of Gromov-hyperbolic groups equipped with a visual metric [16]. Our main theorem answers this natural question in the affirmative. The notion of quasiself-similarity is given in Definition 2.4, and (to our knowledge) was introduced in [7]. Theorem 1.6. Let Xbe a compact quasiself-similar metric space, which is connected and locally connected. Then, dimCH (X) = dimCA(X) = dimCAR(X). As stated, the equality dimCA(X) = dimCAR(X)for uniformly perfect spaces was already known, and follows directly from [18, Proposition 2.2.6]. Our contribution is to prove dimCH (X) = dimCA(X). Indeed, this equality has many further consequences. One may define a zoo of other conformal dimensions, such as: conformal upper and lower Minkowski dimension, conformal packing dimension,. . . Since these dimensions lie between the Hausdorff dimension and the Assouad dimension, one gets equality for the corresponding notions of conformal dimension as well. Indeed, our results here clarify a central point of ambiguity in much of the literature on conformal dimension, where the equality of the different notions is not addressed, but rather avoided and bypassed. 408 Sylvester Eriksson-Bique The only other result, which states equality of dimCH (X)with dimCAR(X) = dimCA(X)is that of [24, Theorem 3.4] and [21, Proposition 2.9.], which apply when Xis Q-Ahlfors regular and possesses a curve family with positive continuous Qmodulus. We will discuss this further below. We are not aware of any other instances, where equality of all notions has been shown. A concrete corollary of Theorem 1.6 is the following new result. The n-dimensional Sierpi`nski sponge Mnis obtained by iteratively subdividing the side of an n-dimensional cube by three, and removing the central cube. Corollary 1.7. Let n≥2. If Mnis an n-dimensional Sierpsi`nski sponge, then dimCH (Mn) = dimCA(Mn) = dimCAR(Mn). We will also give a result for non-self-similar spaces, where self-similarity is replaced with the combinatorial Loewner property (CLP) from [5, 9]; see Section 4 for a definition. It is worth noting, that this assumption usually is verified in the selfsimilar setting, and thus is not so much more general than Theorem 1.6. We present this here, since the argument for it is a bit simpler than for the general self-similar case. Further, it is worth to record a proof for this result here, since the developed tools may be useful in tackling the question of Kleiner, which asks if self-similar combinatorially Loewner spaces are quasisymmetric to Loewner spaces; see [16] for further background and the question. Theorem 1.8. Let p∈(1,∞). Let Xbe a compact, doubling and LLC space, which is p-combinatorially Loewner metric space. We have dimCH (X) = dimCA(X) = dimCAR(X) = p. We note that Corollary 1.7 would also follow from this result, since Sierpi`nski sponges are p-combinatorially Lowener spaces; see the proofs in [5]. In the course of the proof of Theorem 1.8 we will present some stronger results for CLP spaces in Section 4. In fact, while the statement dimCH (X) = pis qualitative, we will give a quantitative statement, Proposition 4.9, which gives a lower bound for the Hausdorff measure of the images of balls under quasisymmetries. This inequality may be useful in other settings as well, and is a generalization of an inequality which appeared in the work of Heinonen and Koskela [13, Theorem 3.6]. We next describe the main idea of the proof. The key tool in a majority of the research on conformal dimension is a notion of modulus—in particular discretized versions of moduli of path families. These generalize the notion of continuous modulus (later, often, modulus), see e.g. [11, 12, 14] for background. Our proof is also based on defining a new type of discrete modulus—or rather, discrete admissibility—and relating it to conformal Hausdorff dimension. At this point, there are several variants of discrete modulus, each with its own setting and application see e.g. [21, 24, 15, 8, 23, 19, 17, 1]. (There are also other notions, such as trans-boundary modulus, see e.g. [22, 3], but these are not relevant for our discussion here.) We will not discuss all these moduli here, but will focus on those which motivate our approach. The motivation for our argument and notion of modulus comes from a result of Pansu [21, Proposition 2.9.], whose dual formulation1was given by Tyson in [24, Theorem 3.4]. Tyson shows that if Xis a Q-Ahlfors regular metric measure 1As a side note, we remark that Pansu considers measures on families of curves, while Tyson uses the notion of curve modulus in [24]. These two notions are roughly dual to each other, see e.g. [2, 10] for more precise statements. Equality of different definitions of conformal dimension for quasiself-similar and CLP spaces 409 space, and if it possesses a family of curves Γwith positive continuous modulus, then dimCH (X) = Q. The proof of Tyson uses the discrete Q-modulus as introduced by Pansu in [21]. To be very brief, this modulus is defined using a Carathèodory construction and involves discrete sums. One shows, both in [21] and [24], that the discrete Q-modulus is bounded from below by the continuous Q-modulus. Further, the discrete modulus, up to a variation of parameters, is invariant under quasisymmetries. The final nail in the coffin of the proof is that if dimH(Y)< Q, then the discrete Q-modulus vanishes on Y. Consequently, a family of curves with positive continuous Q-modulus obstructs lowering the dimension of Xby a quasisymmetry below Q. The previous proof relies heavily on the fact that we can use the notion of continuous modulus to give a lower bound for discrete modulus. In many settings, such as the Sierpiński sponges mentioned above, the continuous moduli of all curves vanishes. Thus, we lack this lower bound, and we need to find a way around this by giving a lower bound using a different quantity. In the quasiself-similar setting, and in the combinatorially Loewner setting, we can obtain this lower bound by slightly different mechanisms—and by employing a different modulus. In the work [15, 8, 19], the inability to lower the dimension can be converted to a lower bound on some moduli—see Theorem 3.5 for a precise statement. Thus, one can use their result to obtain a lower bound for a different discrete modulus, which we call the Keith–Laakso modulus and which is defined in Subsection 3.1. In the case of combinatorially Loewner spaces, the setting is a bit simpler and the lower bound is obtained directly by the assumption that the space is combinatorially Loewner [5, 9]: see Definition 4.1. At this juncture, we have two moduli: the Keith–Laakso modulus and that of Pansu and Tyson. For the first we can obtain lower bounds. For the second, one can show upper bounds. Indeed, for Pansu and Tyson, the notion of discrete modulus is such that it is very easy to prove that if dimH(Y)< Q, the discrete modulus vanishes. In the absence of Q-Ahlfors regularity, it is harder to give lower bounds for the modulus of Pansu and Tyson. On the other hand, for the Keith–Laakso modulus, one lacks the ability to give good upper bounds and thus to directly say that the discrete modulus vanishes if one has Hausdorff dimension lower than Q. The reason for this inability is the following technical, but crucial, point. The definition of Keith–Laakso modulus can be summarized as assigning a value ModKL p(Γ,U) for a specific curve family Γand a cover Uof X, which [5] calls a κ-approximation at some level r. The key feature of their κ-approximations is that all sets in Uhave roughly the same size. (See Subsection 2.4 for details.) This is also a key difference with the work in [21, 24], since there the Carathéodory construction involves arbitrary covers. Similarly, Assouad dimension involves covering the space by balls of the same size, whereas Hausdorff dimension involves coverings by sets of various sizes. To give estimates for Hausdorff dimension, we need to allow arbitrary covers in the definition of discrete modulus. We bridge this gap, by introducing a new notion of modulus Modp(Γ,U)which lies between those of Pansu and Tyson in [21, 24] and Keith, Laakso and others in [15, 8, 19]. First, we get more flexibility by allowing arbitrary covers Uthat consist of balls (or, in general, sufficiently round sets). This forces us to introduce a new admissibility condition, to address several key technical issues. Similar to Pansu’s discrete modulus, we can show that if dimH(Y)< Q, then this modulus is very small for a given cover. Further, in the self-similar and CLP 410 Sylvester Eriksson-Bique space settings, we can relate the Keith–Laakso modulus and the new modulus to each other. For combinatorially Loewner spaces, the story is easier to finish. One can bound Modp(Γ,U)from below using the Keith–Laakso modulus, which in term has a lower bound from the combinatorial Loewner assumption. This estimate is given in Proposition 4.4. This gives a contradiction to the previous paragraph’s conclusion of Modp(Γ,U)being small. In fact, this argument is somewhat easier to discover, and it served as a starting point for this paper and project. For this reason we also include the argument in this paper. Quickly, however, the author realized that a more technical version of the argument could be applied for general quasiself-similar spaces. For quasiself-similar spaces the argument is a bit different. Instead of directly using a lower bound, we use the fact that the ability to lower dimension gives an upper bound. Indeed, if there is a quasisymmetric map f:X→Yand if Yhas small Hausdorff measure, then we obtain a quantitative statement on moduli of annuli, see Lemma 5.2 and Proposition 5.23. Our quantitative statement can be converted algorithmically by using iteration to a statement on the smallness of the Keith–Laakso modulus. This allows us to prove the equality dimCA(X) = dimCH (X) for quasiself-similar spaces by using the result of Carrasco-Paggio, which we state below in Theorem 3.5. The iteration is algorithmic, but quite technical. The basic step of the iteration involves ideas from the proof of the result for CLP spaces. We will describe it in more detail in Subsection 5.2. 1.1. Outline. We will present some general terminology in Section 2. Then, in Section 3 we introduce the different notions of discrete modulus needed in this paper, and present some known results on their relationships with the conformal dimension. For technical reasons, we will use mostly a variant of this modulus, the Bourdon– Kleiner modulus defined in [5], instead of the Keith–Laakso modulus. However, we will relate the two moduli to each other. In Subsection 3.4, we give the new modulus that is key to the approach of this paper. In Section 4 we focus on CLP spaces. There, we prove Theorem 1.8, which is the equality of the definitions of conformal dimension for CLP spaces. In the process, we give some useful stronger results on discrete moduli, and precise quantitative estimates, which hold for CLP spaces. In Section 5 we focus on quasiself-similar spaces. There, we study moduli of annuli, and give a push-down algorithm to adjust the scale of covers. This is then used to give a relationship between the two moduli used. Finally, in Subsection 5.4 we collect the pieces and complete the proof of Theorem 1.6. 2. Notation and basic properties 2.1. Basic terminology. A compact metric space Xis equipped with a metric denoted by d, and open balls within it are B(z, r) := {w∈X:d(z, w)< r}for z∈X, r > 0. An inflation of a ball B=B(z, r)is denoted CB := B(z, Cr) for C > 0. Note that we consider each ball as having an associated center and radius—and it may happen that a different center and radius defines the same set. The radius of a ball is denoted rad(B). Diameters of sets A⊂Xwill be denoted diam(A) = supa,b∈Ad(a, b). A curve is a continuous map γ:I→X, where Xis a non-empty compact interval in R. We often conflate γand it’s image set Image(γ). Equality of different definitions of conformal dimension for quasiself-similar and CLP spaces 411 Recall the definition of N(A, r)from (1.4). We say that a metric space Xis metrically doubling, if there exists a constant D≥1, so that N(B(z, r), r/2) ≤D for every z∈Xand r > 0. We will need some connectivity properties. A space Xis called locally connected, if it has a neighborhood basis consising of connected open sets. A metric space is LLC, if for every x, y ∈X, there exists a curve γwith x, y ∈γand diam(γ)≤Cd(x, y). We will consider collections of balls, which are often denoted by a script letter B. For these, we define unions by setting SB:= SB∈B B, inflations by setting CB:= {CB :B∈ B} and radii rad(B) = supB∈B rad(B). If Ais any finite set, we denote by |A|its cardinality. 2.2. Relative distance and quasisymmetries. We need some standard results on quasisymmetries, which we prove here simply for the sake of completeness. See [12] for a general introduction, and in particular Poposition 10.6. therein for the following. Lemma 2.1. If f:X→Yis an η-quasisymmetry, then f−1is a ˜η-quasisymmetry with ˜η(t)=(η−1(t−1))−1. We note the convention that the value of ˜ηat zero is given by ˜η(0) = 0. Proof of Lemma 2.1. Let x, y, z ∈Yand let x0, y0, z0∈Xbe such that f(x0) = x, f(y0) = y, f(z0) = z. Since fis an η-quasisymmetry, we have d(f(x0), f(z0)) d(f(x0), f(y0)) ≤ηd(x0, z0) d(x0, y0). Taking reciprocals and an inverse function, we get d(x, y) d(x, z)≤ η−1 d(f(x0), f(y0)) d(f(x0), f(z0))−1!!−1 . Replacing f(x0), f(y0), f(z0)with x, y, z and x0, y0, z0with f−1(x), f−1(y), f−1(z)yields that f−1is an ˜η-quasisymmetry.  Let Xbe a complete metric space. A continuum E⊂Xis a compact connected set. A continuum is non-degenerate, if it contains more than one point. We define the relative distance between two non-degenerate continua E, F as ∆(E, F) := d(E, F) min{diam(E),diam(F)}. The following Lemma is also standard, see [12, proof of Proposition 10.8]. Lemma 2.2. Let f:X→Ybe an η-quasisymmetry and let E, F be two nondegenerate disjoint continua in X. Then, 1 2η(∆(E, F)−1)≤∆(f(E), f(F)) ≤η(2∆(E, F)). Proof. Assume by symmetry that diam(E)≤diam(F). Let x∈Eand y∈F be such that d(E, F) = d(x, y). Choose u∈E,v∈Fso that d(x, u), d(y, v)≥ diam(E)/2. This is possible by connectivity. Then, we have d(x, y) d(x, u)≤2∆(E, F)and d(y, x) d(y, v)≤2∆(E, F). 412 Sylvester Eriksson-Bique Let x0:= f(x),y0=f(y),u0=f(u),v0=f(v)be the image points in Y. We have, since ηis increasing and since fis an η-quasisymmetry: d(f(E), f(F)) ≤d(x0, y0)≤ηd(x, y) d(x, u)d(x0, u0)≤η(2∆(E, F)) diam(f(E)). Similarly, d(f(E), f(F)) ≤d(y0, x0)≤ηd(y, x) d(y, v)d(y0, v0)≤η(2∆(E, F)) diam(f(F)). The previous two inequalities combine to gives the inequality: ∆(f(E), f(F)) = d(f(E), f(F)) min{diam(f(E)),diam(f(F))}≤η(2∆(E, F)) . Applying this to the inverse f−1, which by Lemma 2.1 is an ˜η-quasisymmetric map, yields the other inequality of the claim.  The following lemma will also prove useful on a few occasions. Note that the additional assumption on the existence on y∈B(x, r)is automatically satisfied if X is connected and r < diam(X). Lemma 2.3. Let f:X→Ybe a quasisymmetric map and let B(x, r)be a ball in Xfor which there exists a y∈B(x, r)with d(x, y)≥r/2. Then, for every L≥1, we have f(B(x, Lr)) ⊂B(f(x), η(2L)d(f(x), f(y))). Proof. Let z∈B(x, Lr), and apply the η-quasisymmetry to the triple of points x, y, z. This gives d(f(x), f(z)) d(f(x), f(y)) ≤ηd(x, z) d(x, y)≤η(2L). Consequently, we get the claim from d(f(x), f(z)) ≤η(2L)d(f(x), f(y)).  2.3. Quasiself-similarity. We define a notion of quasiself-similarity. This is motivated by the notion of approximate self-similarity discussed in [5]. Definition 2.4. We say that a compact space Xis quasiself-similar, if there exists a homeomorphism η: [0,∞)→[0,∞)and a constant δ > 0so that for B(x, r)⊂Xthere is a η-quasisymmetry f:B(x, r)→Ux,r where Ux,r ⊂Xis an open set with diam(Ux,r)≥δdiam(X). We also say that Xis η-quasiself-similar, if this property holds for a given function η. The principal advantage of defining quasiself-similar spaces is that they are more general than approximately self-similar spaces. Further, quasiself-similarity is an invariant under quasisymmetries: if Xis quasiself-similar and Y∼q.s. X, then Yis also quasiself-similar. The same fails for approximate self-similarity. Quasiself-similar spaces are quite general. They include attractors of iterated functions systems of finite type [20], Julia sets of rational maps equipped with a visual metric [4] and boundaries of Gromov-hyperbolic groups equipped with a visual metric [16]. We recall the following result [7, Proposition 2.9.]. Lemma 2.5. If Xis a compact quasiself-similar space, which is connected and locally connected, then Xis LLC. Equality of different definitions of conformal dimension for quasiself-similar and CLP spaces 419 We claim that ρ∧UPΓ. Let γ∈Γbe arbitrary. Let Vγ⊂ V be the collection, where each set intersects γand such that the collection {B(zV, τrV) : V∈ Vγ}is disjoint with X V∈Vγ ρ(V)≥1. Let Uγ:= {U∈ U :U∩γ6=∅}. Since Uis a cover of X, for each V∈ Vγ, we may choose a UV∈ U so that U∩(V∩γ)6=∅. For each U∈ U, let Vγ,U ={V∈ Vγ: UV=U}. This means that (3.8) Vγ⊂[ U∈Uγ Vγ,U We also have by the first paragraph of the proof that |Vγ,U | ≤ Dfor every U∈ Uγ. Thus, for every U∈ Uγ, we have (3.9) ρ(U)≥X V∈Vγ,U ρ(V). By applying (3.8) and (3.9) we get: X U∩γ6=∅ ρ(U)≥X U∩γ6=∅X V∈Vγ,U ρ(V)≥X V∈Vγ ρ(V)≥1. Note that U=SV∈V UV. By using this, we estimate the p-energy of ρusing the bound |UV| ≤ Dfor every V∈ U. X U∈U ρ(U)p≤X V∈V X U∈UV ρ(U)p≤X V∈V Dp|UV|ρ(V)p≤X B∈B Dp+1ρ(V)p. Thus, the claim holds for C=Dp+1 after we take an infimum over ρ∧τ,VΓ. One of the benefits of this notion of modulus, is that we can give simple bounds for it in terms of the Hausdorff measure of the space. The following will be an example of such a bound that will be useful for us. Recall the definition of Hausdorff content Hp δfrom (1.3). Proposition 3.10. Let κ≥1, τ ≥4,R, r > 0. Let Xbe any connected compact metric space, and suppose that Γis a family of curves, where each curve in Γis contained in a ball B(x, R)⊂Xand has diameter at least r. Then, for every ∈(0,1), δ ∈(0,diam(X)/2) there exists a κ-round collection Vfor which Modp,τ (Γ,V)≤(20τ)pHp δ(B(x, R)) +  rp, and supV∈V rV≤δand each ball in Vintersects B(x, R)as well as some curve in Γ. Proof. Fix  > 0. From the definition of Hausdorff content in (1.3), and by replacing each set in the cover by an enclosing ball, we can find a covering Vof B(x, R)by balls V=B=B(zV, rV)with rV≤δso that X V∈V diam(V)p≤2pHp δ(B(x, R)) + (10τ)−p. Moreover, by possibly making the collection smaller, we assume that each ball V∈ V intersects B(x, R)and some curve in Γ. This modified collection still covers every 420 Sylvester Eriksson-Bique curve γ∈Γ. Now, Vis κ-round with κ= 1. Let ρ(V) = 10τdiam(V)/r. We have by the choice of Vand ρthat X V∈V ρ(V)p≤X V∈V (10τ)pdiam(V)pr−p≤(20τ)pHp δ(B(x, R)) +  rp. Therefore, the claim will follow once we show that ρ∧τ,VΓ. Let γ∈Γ. We need to find a collection Vγso that the properties i, ii and iii from Definition 3.6 hold. Since Vis a cover of γ, we have that {B(zV, τrV)}is a cover of γ. Applying the 5rcovering lemma, we get a finite subcollection Vγ⊂ V so that i) {B(zV, τrV): V∈ Vγ} is pairwise disjoint, ii) so that γ∩V6=∅for all V∈ Vγand so that we have that V0={B(zV,5τrV): V∈ Vγ}is a covering of γ. Note that diam(B(zV,5τrV)) ≤10τrV≤10τdiam(V) = ρ(V)r. Since the balls {B(zV,5τrV): V∈ Vγ}cover γ, we get X V∈V,V ∩γ6=∅ ρ(V)≥X V∈V0 diam(B(zV,5τrV))/r ≥diam(γ)/r ≥1. Thus, ρ∧τ,VΓand the claim follows.  The new notion of modulus is also invariant under quasisymmetries, except for adjusting the τparameter. In the following, if Γis a collection of curves in Xand f:X→Yis a homeomorphism, we write f(Γ) = {f◦γ:γ∈Γ}. The opposite inequality can be obtained by adjusting τ, and applying this lemma to the inverse mapping f−1. Lemma 3.11. Let τ≥4and let f:X→Ybe an η-quasisymmetry. If Vis a κ-round collection, and if Γis any collection of curves in X, then Modp,τ (Γ,V)≤Modp,max{4,η(τ)}(f(Γ), f(V)). Proof. Let τ0= max{4, η(2τ)}. By Lemma 2.9, we have that f(V)is a κ0-round collection for some κ0. Let ρ∧τ0,f(V)f(Γ). Define ρ(V) = ρ(f(V)) for V∈ V. We clearly have X V∈V ρ(V)p=X V∈f(V) ρ(V)p. Thus, the claim will follow, if we can show that ρ∧τ,VΓ. In the following, elements of f(V)will be written as f(V), where V∈ V. Let γ∈Γ. Then, f◦γ∈f(Γ) and, since ρ∧τ0,f(V)f(Γ), there exists a collection Uf(γ)⊂ f(V)so that i) {B(zf(V), τ0rf(V)): f(V)∈ Uf(γ)}is pairwise disjoint; ii) f(V)intersects γfor every f(V)∈ Uf(γ); iii) we have X U∈Uf(γ) ρ(U)≥1. Let Uγ={V∈ V :f(V)∈ Uf(γ)}. We need to check the three properties from Definition 3.6 of ρ∧τ,VΓ: a) {B(zV, τrV): V∈ Uγ}is pairwise disjoint; b) Vintersects γfor every V∈ Uγ; c) we have X U∈Uγ ρ(U)≥1. Equality of different definitions of conformal dimension for quasiself-similar and CLP spaces 421 From these, b) and c) follow immediately from the properties ii) and iii) above. By Lemma 2.3, we have f(B(zV, τrV)) ⊂f(B(zf(V), η(2τ)rf(V)), for every V∈ Uγ. Since τ0≥η(2τ), the disjointness in a) follows from that in i).  4. Combinatorially Loewner spaces 4.1. Definition and basic property. For two closed sets E, F, let Γ(E, F)be the collection of curves which join them. We adapt the definition of Bourdon and Kleiner of the combinatorial Loewner property slightly, as modified by Clais in [9, Definition 2.6]. Let Ukbe a sequence of κ-approximations at level 2−k. Definition 4.1. Fix p > 1. We say that a compact LLC space Xsatisfies the combinatorial p-Loewner property, if there exist some increasing continuous functions φ, ψ: (0,∞)→(0,∞)with limt→0ψ(t) = 0, with the following two properties. (1) For every pair of disjoint continua E, F ⊂Xand all k≥0with 2−k≤ min{diam(E),diam(F)}, we have φ(∆(E, F)−1)≤Modp,Uk(Γ(E, F)). (2) For every z∈Xand 0< r < R and all k≥0with 2−k≤r, we have Modp,Uk(Γ(B(z, r), X \B(z, R))) ≤ψr R−r. Spaces with the combinatorial p-Loewner property are also called CLP -spaces or p-CLP spaces, if we wish to explicate the exponent p > 1. We first note that a combinatorially p-Loewner space has conformal Assoad dimension, as well as Ahlfors regular conformal dimension, equal to p. This Lemma is quite well known and is a rather direct consequence from the known Theorem 3.5. However, we present a proof for the sake of clarity, and since its proof does not appear to have been published elsewhere. The proof is very similar, or rather a localized version, of the proof of [5, Corollary 3.7]. Later, we will prove Theorem 1.8, which is one of our main contributions, and which improves the following statement by showing that also dimCH (X) = p. Lemma 4.2. For a compact LLC space X, which is combinatorially p-Loewner, it holds that dimCA(X) = dimCAR(X) = p. Proof. Let ψand ψbe the functions appearing in Definition 4.1. Since Xis a compact LLC space, it is uniformly perfect, and by [18, Proposition 2.2.6] and [12, Chapters 14 and 15] we have dimCA(X) = dimCAR(X). Next, let Ukbe a sequence of κ-approximations at levels 2−kfor k∈N. Let z∈Xand 0< r ≤diam(X)/4. Then, by the LLC property, there exists a continuum E⊂B(z, r)with diam(E)≥r and another continuum F⊂B(z, 3r)\B(z, 2r)with diam(F)≥r. Since every curve connecting Eto Fcontains a sub-curve within Γ(B(z, r), X \B(z, 2r)), we have Modp,Uk(Γ(E, F)) ≤Modp,Uk(Γ(B(z, r), X \B(z, 2r))). Now, by the CLP property and since ∆(E, F)≤6, we get for all k≥0such that 2−k≤rthat φ(6−1)≤Modp,Uk(Γ(E, F)) ≤Modp,Uk(Γ(B(z, r), X \B(z, 2r))). 422 Sylvester Eriksson-Bique We thus get: lim inf m→∞ sup z∈X,k≥0 Modp,Um+k(ΓB(z,2−k),2)≥φ(6−1). Thus, p≤dimCAR(X)by Theorem 3.5. The inequality p≥dimCAR(X)follows by showing that for all  > 0, we have lim m→∞ sup z∈X,k≥0 Modp+,Um+k(ΓB(z,2−k),2)=0. The idea in showing this is to compare the discrete moduli with exponents p+and p. Indeed, we will show that for all m≥3we have (4.3) Modp+,Um+k(ΓB(z,2−k),2)≤ψ(22−m)Modp,Um+k(ΓB(z,2−k),2)≤ψ(22−m)ψ(1). Then, since limt→0ψ(t)=0, the claim follows. Let ρbe the optimal function for Modp,Um+k(ΓB(z,2−k),2), which exists by Lemma 3.2. We will show that ρ(U)≤ψ(21−m)for every U∈ Um+k. This uses a bound for modulus coming from [5, Lemma 2.3], which in turn relies on estimating the modulus of the curves which pass through the set U. Let U∈ Um+k. Let ΓUbe the collection of curves in ΓB(z,2−k),2which intersect U. Then any curve in ΓB(z,2−k),2which intersects Uwill contain a sub-curve connecting B(zU, rU)to X\B(zU,2m−1rU). Thus, Modp,Uk(ΓU)≤Modp,Uk(Γ(B(zU, rU), X \B(zU,2m−1rU)) ≤ψ(22−m). By [5, Lemma 2.3], we get for all U∈ Um+k ρ(U)≤Modp,Uk(ΓU)≤ψ(22−m). This, together with the optimality of ρyields X U∈U ρ(U)p+≤max U∈U ρ(U)X U∈U ρ(U)p≤ψ(22−m)Modp,Um+k(ΓB(z,2−k),2), which is the desired estimate (4.3).  4.2. Estimates for modulus. If the space is combinatorially Loewner, then we can give a lower bound of our modulus, which we introduced in Subsection 3.4, in terms of the Bourdon–Kleiner modulus. This is a strengthening of the Proposition 3.7. In a sense, the following Proposition is the starting point of our paper, since its argument was the first to be discovered. Proposition 4.4. Let k∈N,p > 1. Assume that Xis metrically doubling, LLC and combinatorially p-Loewner, and that κ≥1,τ≥4. There exists a constant C > 0so that the following holds for r > 0. Suppose that Uis a κ-approximation at level rand Vis a κ-round collection with inf{rV:V∈ V} ≥ 2r. If Γis a collection of curves in Xwith 2τsupV∈V rV≤diam(γ)for all γ∈Γ, then Modp,U(Γ) ≤CModp,τ (Γ,V). Proof. Assume that Modp,τ (Γ,V)<∞, and that ρ∧τ,VΓwith X V∈V ρp(V)<∞. For each V∈ V consider the collection of curves ΓV= Γ(B(zV, rV), X\B(zV,(τ− 1)rV)). By the p-combinatorial Loewner assumption and since r≤rV/2, we have (4.5) Modp,U(ΓV)≤C, Equality of different definitions of conformal dimension for quasiself-similar and CLP spaces 423 for C=ψ(1 τ−1)>0, where ψis from Definition 4.1. Notice, that while the combinatorial Loewner property only gives a bound for coverings at dyadic scales r= 2−k. Thus to obtain (4.5) requires also a the application of a simple result [5, Proposition 2.2.], which shows moduli with respect to κapproximations at comparable levels are comparable. Let ρV:U → [0,∞)be such that ρV∧UΓVand so that (4.6) X U∈U ρV(U)p≤2C. Let ρ(U) = max{ρV(U)ρ(V): V∈ V}. We claim that ρ∧UΓ. Let γ∈Γ. Since ρ∧τ,VΓ, there exists a collection Vγof V∈ V with (1) V∩γ6=∅for all V∈ Vγ; (2) {B(zV, τrV): V∈ Vγ}is a pairwise disjoint collection of balls; and (3) (4.7) X V∈Vγ ρ(V)≥1. For each V∈ Vγ, let γ|Vbe a minimal subcurve which connects B(zV, rV)to B(zV,(τ−1)rV). Such a subcurve exists since diam(γ)≥2τrVand γ∩B(zV, rV)6=∅. These subcurves are disjoint and d(γ|V, γ|V0)≥2 min{rV, rV0} ≥ 4r, for distinct V, V 0∈ Vγ. Therefore, if we let UV={U∈ U :U∩γ|V6=∅} for V∈ Vγ, then UV∩ UV0=∅for distinct V, V 0∈ Vγ. We also have, since ρV∧ΓVand ρ≥ρVρ(V) that (4.8) X U∈UV ρ(U)≥X U∈U,U∩γ|V6=∅ ρV(U)ρ(V)≥ρ(V). Now, let Uγ={U∈ U :U∩γ6=∅}. We also have [ V∈Vγ UV⊂ Uγ. By the disjointness of the collections UV, for distinct V∈ Vγ, and by applying (4.7), (4.8) and the choice of ρ, we get X U∈Uγ ρ(U)≥X V∈VγX U∈UV ρ(U)≥X V∈Vγ ρ(V)≥1. Thus, since γis arbitrary, ρ∧UΓ. Next, we show a mass-bound for ρ. For each U∈ U let VU∈ V be such that ρ(U) = ρVU(U)ρ(VU). This yields a partition of Uinto sets UV={U∈ U :VU=V}. Thus, we have, since UV⊂ U Modp,U(Γ) ≤X U∈U ρ(U)p=X V∈V X U∈UV ρV(U)pρ(V)p≤X V∈V ρ(V)pX U∈U ρV(U)p ≤X V∈V 2Cρ(V)p= 2CX V∈V ρ(V)p. By infimizing over ρsuch that ρ∧τ,VΓthe claim follows.  We obtain the following proposition, which gives a lower bound for the Hausdorff measure of a combinatorially Loewner space. In this way, this generalizes to combinatorially Loewner spaces the classical estimate of Heinonen and Koskela, [13, 424 Sylvester Eriksson-Bique Theorem 3.6]. That result is much easier to show using continuous modulus. For discrete modulus one needs to do some extra work. Proposition 4.9. Let Xbe a p-combinatorially Loewner LLC and metrically doubling space. Then, there exists a constant C≥1so that for every r∈(0,diam(X)) and any x∈Xwe have Hp(B(x, r)) ≥Crp. Proof. Let x∈X. It is sufficient to prove (4.10) Hp(B(x, 2L0r)) ≥Crp. for some uniform constants L0≥1, C > 0for all r∈(0,diam(X)/8). Since Xis LLC, we can find a continuum E⊂B(x, r)with r≥diam(E)≥r/2and x∈E. Further, there exists a continuum F⊂B(x, 4r)\B(x, 3r)with 8r≥diam(F)≥r. We have 1≤∆(E, F)≤16. Let Γbe the collection of continuous curves connecting Eto F. Next, our strategy in proving (4.10) is to show three estimates. We will show that. A) There is a collection ΓBof curves so that for any κ-approximation Uat a small enough level the quantity Modp,U(Γ \ΓB)can be bounded from below by using the CLP property, and each curve in Γ\ΓBis contained in a ball of definite size. B) Proposition 3.10 gives a lower bound for the Hausdorff measure in terms of the discrete modulus Modp,τ (Γ \ΓB,V). C) Finally, Proposition 4.4 is used to find a small enough level so that Modp,τ (Γ\ ΓB,V)is bounded from below by Modp,U(Γ \ΓB)for some κ-approximation Uat a small enough level. These estimates together yield the desired bound. We focus on A) first and determine ΓB. Let Ube a κ-approximation at level 2−k for some k∈Ns.t. 2−k≤min{diam(E),diam(F)}. We have Modp,U(Γ) ≥φ(16−1). Let L≥2be such that ψ(2L−1)≤2−1φ(16−1). Let ΓBbe the collection of curves γ∈ΓXwith a subcurve in Γ(B(x, r), X \B(x, Lr)). We have, since Xis CLP, that Modp,U(ΓB)≤Modp,U(Γ(B(x, r), X \B(x, Lr))) ≤ψ(2L−1)≤φ(16−1) 2. Thus, by subadditivity of modulus, we get for ΓG:= ΓX\ΓBthe estimate (4.11) Modp,U(ΓG)≥φ(16−1) 2. Next, we deduce B) in our strategy. Let τ≥4. Choose δ∈(0,4−1τ−1r). Each of the curves in ΓGhas diameter at least rand is contained in B(x, Lr). So, we can apply Proposition 3.10 to find for any  > 0a1-round collection Vof balls which intersect B(x, Lr)and some curve in ΓGwith rad(V)≤δand with (4.12) Modp,τ (ΓG,V)≤(20τ)p(Hp δ(B(x, Lr)) + )r−p≤(20τ)p(Hp(B(x, Lr)) + )r−p. Finally, we deduce C). Each curve in ΓGconnects Eto F, and thus we have diam(γ)≥rfor all γ∈ΓG. This means that 2τsupV∈V rV≤infγ∈ΓGdiam(γ). Thus, Equality of different definitions of conformal dimension for quasiself-similar and CLP spaces 425 by Proposition 4.4, there exists a constant Cso that (4.13) Modp,U(ΓG)≤CModp,τ (ΓG,V), if kis so large so that inf{rV:V∈ V} ≥ 2−k−1. By combining estimates A–C), we get the following once kis large enough φ(η(16)−1) 2 (4.11) ≤Modp,U(ΓG)(4.13) ≤CModp,τ (ΓG,V) (4.12) ≤(20τ)pC(Hp(B(x, Lr)) + )r−p. Consequently, since this holds for all  > 0, we get φ(η(16)−1) 2(20τ)pCrp≤ Hp(B(x, L0r)). This yields the desired estimate (4.10).  4.3. Proof of Theorem 1.8. Using the previous properties, we are able to prove the equality of different forms of conformal dimension for CLP spaces. Proof of Theorem 1.8. Assume that Xis combinatorially p-Loewner. By Lemma 4.2, we have dimCA(X) = dimCAR(X) = p. We also have dimCH (X)≤dimCAR(X) = p. Thus, we only need to show that dimCH (X)≥p. Let f:X→Ybe a quasisymmetry. The space Yis p-combinatorially Loewner, since the combinatorial Loewner property is invariant under quasisymmetries, see [5, Theorem 2.6 (2)]. It is also easy to see, that the LLC and metric doubling properties are invariant under quasisymmetries, and thus Yis LLC and metric doubling. Then, by Proposition 4.9 there exists a constant Cso that we have for every y∈Yand any r∈(0,diam(Y)) that Hp(B(y, r)) ≥Crp>0. From the definition of Hausdorff dimension, and since Hp(B(y, r)) >0if and only if Hp ∞(B(y, r)) >0, we have dimH(Y)≥p. Consequently, by taking an infimum over all Ywhich are quasisymmetric to X, we get dimCH (X)≥p. 5. Quasiself-similar spaces 5.1. Uniform bound for annuli. As discussed in the introduction, the case of quasiself-similar spaces requires some more care. We do this by considering first moduli of annuli. Define an annulus as A(x, r, R) := B(x, R)\B(x, r). Definition 5.1. Let p∈(1,∞). Let τ≥4. We say that a metric space Xhas uniformly small p-moduli of annuli, if there exists ∈(0,1) and constants 0< δ−< δ+< τ−1, so that the following holds. For every annulus A(x, r, (τ−2)r)in X, with x∈X, r ∈(0,2−1τ−1diam(X)), there exists a finite collection of balls Vx,r contained in B(x, τr)and which intersect B(x, (τ−2)r), with rV∈[δ−r, δ+r]for each V∈ Vx,r, and there exists a function ρx,r :Vx,r →[0,∞)with ρx,r∧τ,Vx,r Γ(B(x, r), X \B(x, (τ−2)r)) and with X B∈VB ρx,r(B)p≤. 426 Sylvester Eriksson-Bique The following lemma is a refinement of Proposition 3.10 to the quasiself-similar setting. Lemma 5.2. Suppose that dimCH (X)< p,p∈(1,∞), and Xis an arcwise connected quasiself-similar compact metric space. Then Xhas uniformly small pmoduli of annuli. Proof. Assume that Xis η-quasiself-similar and let τ≥4. Fix any δ+∈(0, τ−1). Since dimCH (X)< p, there exists a compact space Ywith dimH(Y)< p and a quasisymmetry g:X→Y. Fix C≥1, σ ∈(0,2−1)to be determined. By adjusting η, we may assume that gis an η-quasisymmetry. Let  > 0, and choose a covering of Yby a collection of balls BYwith X B∈BY diam(B)p≤C−pdiam(Y)p, and for which rad(B)≤σdiam(Y)for every B∈ BY. Let A(x, r, (τ−2)r)be an annulus in Xwith x∈Xand r∈(0,2−1τ−1diam(X)). There is a homeomorphism f:B(x, 2τr)→U, for some open set U⊂X, which is an η-quasisymmetry, where diam(U)≥δdiam(X). We first define the collection Vx,r used in Definition 5.1. For each B=B(y, s)∈ BYwith B∩g(f(B(x, (τ−2)r))) 6=∅, choose xVB∈(g◦f)−1(B)∩B(x, (τ−2)r), and let rVB= sup{d(z, xB): z∈(g◦ f)−1(2B)∩B(x, τr)}. Define VB:= B(xVB, rVB). Let Vx,r := {VB:B∈ BY, B ∩g(f(B(x, (τ−2)r))) 6=∅} be the collection of balls we seek. Next, we give bounds for rVBby using the fact that Xis connected and that g◦fis a ˜η-quasisymmetry with ˜η=η◦η. Since diam(U)≥δdiam(X), we can choose a, b ∈Uwith d(a, b)≥2−1δdiam(X). Choose a point c∈Xso that d(g(c), g(a)) ≥diam(Y)2−1. Since gis an ηquasisymmetry, we have d(g(a), g(c)) d(g(a), g(b)) ≤ηd(c, a) d(b, a)≤η(2δ−1). Thus, (5.3) d(g(a), g(b)) ≥η(2δ−1)−12−1diam(Y). We will use (5.3) to give an upper bound for rVBfor each VB∈ Vx,r, where B∈ BY. Let u, v ∈B(x, 2τr)be such that f(u) = a, f(v) = b. Choose s, t ∈ (g◦f)−1(2B)∩B(x, τr)so that d(s, t)≥rVB/2. Up to possibly switching uand v, and a, b, we can assume by (5.3) that (5.4) d(g(f(s)), g(a)) ≥d(g(a), g(b)) 2≥η(2δ−1)−1diam(Y)2−2. We have (5.5) d(g(f(s)), g(f(u))) d(g(f(s)), g(f(t))) ≤˜ηd(s, u) d(s, t). Since g(f(s)), g(f(t)) ∈2B, we get d(g(f(s)), g(f(t))) ≤4rad(B)≤4σdiam(Y). Thus, from (5.4), we get 1 24η(2δ−1)σ=diam(Y) 24η(2δ−1)σdiam(Y)≤d(g(f(s)), g(a)) d(g(f(s)), g(f(t))). Equality of different definitions of conformal dimension for quasiself-similar and CLP spaces 427 By combining this with (5.5), we deduce ˜η−11 24ση(2δ−1)≤d(s, u) d(s, t)≤6τr rVB . Thus, rVB≤6τ ˜η−11 24ση(2δ−1)r. Choose now σ≤˜η(6τ δ+)−1η(2δ−1)−12−4. We then have, rVB≤δ+r. Since δ+<1, we also have rVB≤rand since xVB∈B(x, (τ−2)r)we clearly have VB⊂B(x, τr). Next, we give a uniform lower bound for the radii rVBfor VB∈ Vx,r . Since BY is finite, there exists a constant β > 0so that rad(B)≥βdiam(Y)for all B∈ BY. Choose δ−= ˜η−1(β)/2. Let c∈B(xVB, δ−r)be an arbitrary point. Also, choose b∈B(x, 2τr)with d(b, xVB)≥r, which is possible by connectivity. Then, by the quasisymmetry condition, we get d(g(f(c)), g(f(xVB))) d(g(f(b)), g(f(xVB))) ≤˜ηd(c, xVB) d(b, xVB)≤˜η(δ−). The choice of δ−guarantees ˜η(δ−)≤β, and thus d(g(f(c)), g(f(xVB))) ≤˜η(δ−) diam(Y)≤rB. Therefore, since g(f(xVB)) ∈B, we get g(f(c)) ∈2B. This holds for all c∈ B(xVB, δ−r), and thus g(f(B(xVB, δ−r))) ⊂2B. This yields, by connectivity and the definition of rVBthat rVB≥δ−r. Finally, we define the admissible function ρ. Define ρ(V) = max{CdiamY(B) diamY(Y)−1:VB=V, B ∈ BY} for V∈ Vx,r. We have (5.6) X V∈Vx,r ρ(V)p≤CpX B∈BY diamY(B)pdiamY(Y)−p≤, since for every V∈ Vx,r there exists at least one B∈ BYso that VB=V, and for every B∈ BYthere is only one V∈ Vx,r for which VB=V. Next, we show that ρ∧τ,Vx,r Γ(B(x, r), X \B(x, (τ−2)r)). Let γ∈Γ(B(x, r), X \ B(x, (τ−2)r)) be arbitrary. Let σbe a sub-curve of γso that σ⊂(τ−2)Band σ∈Γ(B(x, r), X \B(x, (τ−2)r)). To show admissibility, we will combine the fact that BYcovers g◦f◦σwith a lower bound for the diameter of g◦f◦σ. Since σconnects B(x, r)to X\B(x, (τ−2)r)there exist j, k ∈σwith d(j, k)≥ (τ−3)r. Let a, b and u, v be as before. By possibly switching jand k, we can assume that d(j, u)≥d(j, k)/2≥2−1(τ−3)r. We get d(g(f(j)), g(f(u))) d(g(f(j)), g(f(k)) ≤˜ηd(j, u) d(j, k)≤˜η4τr (τ−3)r≤˜η(16). Thus, (5.7) diam(g◦f◦σ)≥d(g(f(j)), g(f(k))) ≥d(g(f(j)), g(f(u))) ˜η(16)−1. 428 Sylvester Eriksson-Bique Next, d(j, u)≥2−1(τ−3)r≥d(u, v)/8. Thus, by a similar reasoning that uses the quasisymmetry of g◦fand by employing (5.3), we get d(g(f(j)), g(f(u))) ≥d(g(f(u), g(f(v)))˜η(8)−1 ≥˜η(8)−1η(2δ−1)−12−1diam(Y). (5.8) By combining (5.7) and (5.8), we obtain (5.9) diam(g◦f◦σ)≥d(g(f(j)), g(f(k)) ≥˜η(16)−1˜η(8)−1η(2δ−1)−12−1diam(Y). Recall that Vx,r consists of balls. The open sets Vx,r cover the ball B(x, 2(τ−2)), and thus the curve σ. Therefore, by the Vitali covering theorem, there exists a finite collection of balls Vγwith σ⊂S5τVγ, and for which τVγare disjoint, and so that each ball in Vγintersects γ. For each V∈ Vγ, choose a ball B(V)∈ BYso that V=VB(V)and ρ(V) = CdiamY(B(V)) diamY(Y)−1. First, we note that the quasisymmetry condition and Lemma 2.3, we have g(f(5τV )) ⊂˜η(10τ)B(V). Therefore, we get that the balls ˜η(10τ)B(V)for V∈ Vγcover g(f(σ)). Thus, X V∈Vγ ρ(V) = X V∈Vγ CdiamY(B(V)) diamY(Y)−1 ≥X V∈Vγ C(2˜η(10τ))−1diamY(Y)−1diamY(˜η(10τ)B(V)) (5.9) ≥diam(g◦f◦σ)C(2˜η(10τ))−1diamY(Y)−1 ≥C2−2˜η(16)−1˜η(8)−1η(2δ−1)−1˜η(10τ)−1. If C≥4˜η(16)˜η(8)η(2δ−1)˜η(10τ), then ρ∧τ,Vx,r Γ(B(x, r), X \B(x, (τ−2)r)) is admissible and the claim follows.  5.2. Algorithm for pushing down a cover. The following lemma describes a “push down” algorithm. It uses admissible functions for annuli in order to push down a collection of balls Band a strongly discretely τ-admissible function ρ. This is done by replacing a ball B∈ B by a collection BBand an associated function ρB. A new admissible function ρis defined by taking a maximum over B∈ B, and a new collection by taking a union of all the new balls. This arguments for admissibility and the construction of ρare similar to Proposition 4.4. To distinguish the “parent” balls from the “descendant balls”, we will bold the parent balls. This replacement algorithm is depicted and explained more in Figure 1. As seen in this figure, we permit all sorts of overlaps, and balls of different sizes. This is one of the technical reasons for using the new modulus from Subsection 3.4. Recall that ΓB,L denotes the collection of curves γconnecting Bto X\LB. Lemma 5.10. Let , η ∈(0,1). Assume that Bis a finite collection of balls, Γis a collection of curves, 2(τ−2)rad(B)≤infγ∈Γdiam(γ)and ρ∧BΓ. Suppose further that C ⊂ B is any finite collection of balls, and for every B∈ C, there exists a finite collection of balls BBand a function ρB:BB→[0,∞)with (1) rad(BB)≤τ−1rad(B), (2) ρB∧τ,BBΓB,(τ−2), Equality of different definitions of conformal dimension for quasiself-similar and CLP spaces 435 We are left to prove the converse inequality. Since Xis connected, compact, locally connected and quasiself-similar, by Lemma 2.5 Xis LLC. Suppose that pis arbitrary and dimCH(X)< p. Fix any sequence of κ-approximations {Uk}k∈N, where Ukis at scale 2−k. By Lemma 5.2, we have that Xhas uniformly small moduli of annuli. Then, by Proposition 5.23, we have that lim inf m→∞ sup x∈X,k∈N {Modp(ΓB(x,2−k),2,U): Uis a κ-approximation at level 2−k−m}= 0. Then, Theorem 3.5 implies that dimCAR(X)≤p. Since p > dimCH (X)is arbitrary, this completes the proof.  References [1] Albin, N., and M. Brunner,R. Perez,P. Poggi-Corradini, and N. Wiens: Modulus on graphs as a generalization of standard graph theoretic quantities. - Conform. Geom. Dyn. 19, 2015, 298–317. [2] Ambrosio, L.,S. Di Marino, and G. Savaré: On the duality between p-modulus and probability measures. - J. Eur. Math. Soc. (JEMS) 17:8, 2015, 1817–1853. [3] Bonk, M., and S. Merenkov: Quasisymmetric rigidity of square Sierpiński carpets. - Ann. of Math. (2) 177:2, 2013, 591–643. [4] Bonk, M., and D. Meyer: Expanding Thurston maps. - Math. Surveys Monogr. 225, Amer. Math. Soc., Providence, RI, 2017. [5] Bourdon, M., and B. Kleiner: Combinatorial modulus, the combinatorial Loewner property, and Coxeter groups. - Groups Geom. Dyn. 7:1, 2013, 39–107. [6] Bourdon, M., and H. Pajot: Cohomologie lpet espaces de Besov. - J. Reine Angew. Math. 558, 2003, 85–108. [7] Carrasco-Piaggio, M.: Jauge conforme des espaces métriques compacts. - PhD thesis, Université de Provence-Aix-Marseille I, 2011. [8] Carrasco Piaggio, M.: On the conformal gauge of a compact metric space. - Ann. Sci. Éc. Norm. Supér. (4), 46:3, 2013, 495–548. [9] Clais, A.: Combinatorial modulus on boundary of right-angled hyperbolic buildings. - Anal. Geom. Metr. Spaces 4, 2016, 1–53. [10] David, G. C., and S. Eriksson-Bique: Infinitesimal splitting for spaces with thick curve families and Euclidean embeddings. - Ann. Inst. Fourier (Grenoble) (to appear). [11] Fuglede, B.: Extremal length and functional completion. - Acta Math. 98, 1957, 171–219. [12] Heinonen, J.: Lectures on analysis on metric spaces. - Universitext, Springer-Verlag, New York, 2001. [13] Heinonen, J., and P. Koskela: Quasiconformal maps in metric spaces with controlled geometry. - Acta Math. 181:1, 1998, 1–61. [14] Heinonen, J.,P. Koskela,N. Shanmugalingam, and J. T. Tyson: Sobolev spaces on metric measure spaces. - New Math. Monogr. 27, Cambridge Univ. Press, Cambridge, 2015. [15] Keith, S., and T. Laakso: Conformal Assouad dimension and modulus. - Geom. Funct. Anal. 14:6, 2004, 1278–1321. [16] Kleiner, B.: The asymptotic geometry of negatively curved spaces: uniformization, geometrization and rigidity. - In: Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006, vol. 2, 2007, 743–768. [17] Lindquist, J.: Weak capacity and modulus comparability in Ahlfors regular metric spaces. - Anal. Geom. Metr. Spaces 4:1, 2016, 399–424. [18] Mackay, J. M., and J. T. Tyson: Conformal dimension: theory and application. - Univ. Lecture Ser. 54, Amer. Math. Soc., Providence, RI, 2010. 436 Sylvester Eriksson-Bique [19] Murugan, M.: Conformal Assoaud dimension as the critical exponent for combinatorial modulus. - Ann. Fenn. Math. 48:2, 2023, 453–491. [20] Ngai, S.-M., and Y. Wang: Hausdorff dimension of self-similar sets with overlaps. - J. London Math. Soc. (2) 63:3, 2001, 655–672. [21] Pansu, P.: Dimension conforme et sphère à l’infini des variétés à courbure négative. - Ann. Acad. Sci. Fenn. Ser. A I Math. 14:2, 1989, 177–212. [22] Schramm, O.: Transboundary extremal length. - J. Anal. Math. 66, 1995, 307–329. [23] Shanmugalingam, N.: On Carrasco Piaggio’s theorem characterizing quasisymmetric maps from compact doubling spaces to Ahlfors regular spaces. - In: Potentials and Partial Differential Equations, The Legacy of David R. Adams (series: Advances in Analysis and Geometry), 2023, 1–23. [24] Tyson, J. T.: Sets of minimal Hausdorff dimension for quasiconformal maps. - Proc. Amer. Math. Soc. 128:11, 2000, 3361–3367. Received 9 November 2023 •Revision received 2 May 2024 •Accepted 5 June 2024 Published online 24 June 2024 Sylvester Eriksson-Bique University of Jyväskylä Department of Mathematics and Statistics Seminaarinkatu 15, PO Box 35 FI-40014 University of Jyväskylä, Finland sylv[email protected]