The Assouad dimension of self-affine measures on sponges

Abstract We derive upper and lower bounds for the Assouad and lower dimensions of self-affine measures in 
$\mathbb {R}^d$
 generated by diagonal matrices and satisfying suitable separation conditions. The upper and lower bounds always coincide for 
$d=2,3$
 , yielding precise explicit formulae for those dimensions. Moreover, there are easy-to-check conditions guaranteeing that the bounds coincide for 
$d \geqslant 4$
 . An interesting consequence of our results is that there can be a ‘dimension gap’ for such self-affine constructions, even in the plane. That is, we show that for some self-affine carpets of ‘Barański type’ the Assouad dimension of all associated self-affine measures strictly exceeds the Assouad dimension of the carpet by some fixed 
$\delta>0$
 depending only on the carpet. We also provide examples of self-affine carpets of ‘Barański type’ where there is no dimension gap and in fact the Assouad dimension of the carpet is equal to the Assouad dimension of a carefully chosen self-affine measure.


Introduction: dimensions of self-affine measures
Let ν be a compactly supported Borel probability measure in R d .The Assouad and lower dimensions of ν quantify the extremal local fluctuations of the measure by considering the relative measure of concentric balls.In particular, a measure is doubling if and only if it has finite Assouad dimension, e.g.[11,Lemma 4.1.1].Write supp(ν) to denote the support of ν and |F | to denote the diameter of a non-empty set F .The Assouad dimension of ν is defined by dim A ν = inf s 0 : there exists C > 0 such that, for all x ∈ supp(ν) and for all 0 < r < R < |supp If |supp(ν)| = 0, then dim L ν = 0.The Assouad and lower dimensions of measures were introduced by Käenmäki, Lehrbäck and Vuorinen [16], where they were originally 1 referred to as the upper and lower regularity dimensions, respectively.We are interested in the Assouad and lower dimensions of self-affine measures.Given a finite index set I = {1, . . ., N }, an affine iterated function system (IFS) on R d is a finite family F = {f i } i∈I of affine contracting maps f i (x) = A i x + t i .The IFS determines a unique, non-empty compact set F , called the attractor, that satisfies the relation Given a probability vector p = (p(i)) i∈I with strictly positive entries, the self-affine measure ν p fully supported on F is the unique Borel probability measure The measure ν p has an equivalent characterisation as the push-forward of the Bernoulli measure generated by p under the natural projection from the symbolic space to the attractor.More precisely, given p, the Bernoulli measure on the symbolic space Σ = I N is the product measure µ p = p N .The natural projection π : Σ → F is given by where Computing (or estimating) the dimensions of self-affine measures in general is a hard problem.Moreover, many self-affine measures fail to be doubling (and so have infinite Assouad dimension) and so some conditions are needed in order to obtain sensible results.The specific self-affine measures we are able to handle are those supported on 'Barański type sponges'.That is, the A i are diagonal matrices and we assume a separation condition (the 'very strong SPPC', see Definition 2.1) which, roughly speaking, says that all relevant projections of the measure satisfy the, more familiar, strong separation condition (SSC).For such measures we derive upper and lower bounds for the Assouad and lower dimensions, see Theorem 2.5.Moreover, the upper and lower bounds agree when d = 2, 3 (see Lemma 3.2) and also in many other cases in higher dimensions.It remains an interesting open problem whether our bounds are sharp in full generality, see Question 2.6.One of the main technical challenges in considering 'Barański type sponges' instead of, for example, those of 'Bedford-McMullen' or 'Lalley-Gatzouras' type is that we have to control the ratio of the measure of approximate cubes with 'different orderings'.As such we develop a number of technical tools which may have further application, e.g. the subdivision argument used in proving Proposition 5.4.An interesting consequence of our results is that there can be a 'dimension gap' for such self-affine constructions, even in the plane, see Corollary 2.7 and Proposition 4.1.

Main results: dimension bounds and dimension gaps
2.1.Our model and assumptions.We call a self-affine set F a (self-affine) sponge if the linear part A i of each f i is a diagonal matrix with entries λ .When d = 2 sponges are more commonly referred to as self-affine carpets and when d = 1 they are self-similar sets.The original model for a self-affine carpet was introduced independently by Bedford [3] and McMullen [20] and later generalised by Lalley and Gatzouras [14], Barański [1] and many others.The dimension theory of self-affine carpets is well-developed, although several interesting questions remain such as the question of whether self-affine carpets necessarily support an invariant measure of maximal Hausdorff dimension, see [22].A recent breakthrough established that this was false for sponges with d = 3 [6], that is, the existence of a 'dimension gap' was established for certain examples.This dimension gap result resolved a long standing open problem in dynamical systems.
Generally, much less is known about sponges in dimensions d 3. The objective of this paper is to contribute to this line of research.A number of results concern the higher dimensional Bedford-McMullen sponges, see Example 2.4 for the formal definition.Their Hausdorff and box dimensions were determined by Kenyon and Peres [17] while their Assouad and lower dimensions were calculated by Fraser and Howroyd [12].Olsen [21] studied multifractal properties of self-affine measures supported by these sponges and Fraser and Howroyd [13] derived a formula for the Assouad dimension of such measures.The lower and Assouad dimensions of Lalley-Gatzouras sponges, see Example 2.3, are also known [5,15].
Without loss of generality we assume that f i ([0, 1] d ) ⊂ [0, 1] d and that there is no i = j such that f i (x) = f j (x) for every x ∈ [0, 1] d .To avoid unwanted complication with notation, we also assume that λ (n) i ∈ (0, 1) for every i ∈ I and 1 n d.
We make one further simplification by assuming that all pairs of coordinates are distinguishable, i.e.
for any m = n ∈ {1, . . ., d} there exists i ∈ I such that λ (2.1) Otherwise, the sponge is not 'genuinely self-affine' in all coordinates.The case when not all pairs of coordinates are distinguishable can be handled by 'gluing' together nondistinguishable coordinates as was done by Howroyd [15] but we omit further discussion of such examples.The orthogonal projections of F onto the principal n-dimensional subspaces play a vital role in the arguments.Let S d be the symmetric group on the set {1, . . ., d}.For a permutation σ = (σ 1 , . . ., σ d ) ∈ S d of the coordinates, let E σ n denote the n-dimensional subspace spanned by the coordinate axes indexed by σ 1 , . . ., σ n .Notice that E σ n = E ω n as long as {σ 1 , . . ., σ n } and {ω 1 , . . ., ω n } are the same sets.The permutation appears in the notation rather than just the set of indices because the ordering of coordinates will play a role in how the subspace is 'built up' from its lower dimensional subspaces.Let Π σ n : [0, 1] d → E σ n be the orthogonal projection onto E σ n .For n = d, Π σ d is simply the identity map.We say that f i and f j overlap exactly on Observe that if f i and f j overlap exactly on E σ n then they also overlap exactly on E σ m for all 1 m n but may not overlap exactly on any E σ ′ n for some other σ ′ ∈ S d .Recall Σ = I N is the space of all one-sided infinite words i = i 1 , i 2 , . ... Slightly abusing notation, we also write i = i 1 , . . ., i k ∈ I k for a finite length word or i|k = i 1 , . . ., i k for the truncation of i ∈ Σ.For r > 0, the r-stopping of i ∈ Σ in the n-th coordinate (for n = 1, . . ., d) is the unique integer L i (r, n) for which (2.2) We distinguish between two different kinds of orderings.We say that i ∈ Σ determines a σ-ordered cylinder at scale r if σ d = σ d (i, r) is the largest index that satisfies and then where to make the ordering unique, we use the convention that if It is a strictly σ-ordered cylinder if all inequalities in (2.3) are strict.This corresponds to the ordering of the length of the sides of the cylinder set f i|L i (r,σ d ) ([0, 1] d ) with σ d corresponding to the shortest side and σ 1 the longest.Moreover, we say that i ∈ Σ determines a σ-ordered cube at scale r if Here the ordering is made unique with the following rule: if coordinates k < m satisfy L i (r, k) = L i (r, m), then k precedes m in σ if and only if i ℓ .This corresponds to the ordering of the sides of a symbolic approximate cube to be formally introduced in Section 5. Note that the ordering of i as a cylinder or as a cube at a scale r need not be the same.Of importance are the different orderings that are 'witnessed' by an i ∈ Σ at some scale r: A := {σ ∈ S d : there exist i ∈ Σ and r > 0 such that i determines a σ-ordered cube at scale r} (2.5) and B := {σ ∈ S d : there exist i ∈ Σ and r > 0 such that i determines a strictly σ-ordered cylinder at scale r}.(2.6) Clearly B ⊆ A because if σ ∈ B is witnessed by j at scale r, then by defining i := j|L j (r, σ d ), i.e. repeating the word j|L j (r, σ d ) infinitely often, there is r ′ small enough such that (2.4) holds.We give a more detailed account of the relationship between A and B in Section 3, where we show that A = B for d = 2 and 3, however, also present a four dimensional example for which B ⊂ A. A simple example to determine A and B is when the sponge F satisfies the coordinate ordering condition, i.e. there exists a permutation σ ∈ S d such that 0 < λ (2.7) In this case, L r (i, σ d ) L r (i, σ d−1 ) . . .L r (i, σ 1 ) for every i ∈ Σ and r > 0, hence, A = B = {σ} and only the projections Π σ n F play a role in the study of F .For each permutation σ ∈ A we define index sets I σ d ⊇ I σ d−1 ⊇ . . .⊇ I σ 1 with I σ d := I as follows.Initially set I σ d = I σ d−1 = . . .= I σ 1 and then repeat the following procedure for all pairs i < j (i, j ∈ I).Starting from n = d − 1 and decreasing n, check whether f i and f j overlap exactly on E σ n .If they do not overlap exactly for any n, then move onto the next pair (i, j), otherwise, take the largest n ′ for which f i and f j overlap exactly and remove j from I σ n ′ , I σ n ′ −1 , . . ., I σ 1 and then move onto the next pair (i, j).The sets 8) The sponge satisfies the very strong SPPC if (0, 1) d can be replaced with [0, 1] d .
If (2.8) is only assumed for n = d, the rather weaker condition is known as the rectangular open set condition, e.g.[9].The following are the natural generalisations of Barański [1], Lalley-Gatzouras [14] and 20] carpets to higher dimensions.
Example 2.2.A Barański sponge F ⊂ [0, 1] d satisfies that for all σ ∈ S d and i, j ∈ I, Observe that a carpet on the plane satisfies the SPPC if and only if it is either Barański (when #A = 2) or Lalley-Gatzouras (when #A = 1).Therefore, this definition combines these two classes in a natural way.Moreover, for dimensions d 3 it is a wider class of sponges than simply the union of the Barański and Lalley-Gatzouras class.For d = 3, we give a complete characterisation of the new classes that emerge in Section 4.2.
The very strong SPPC is a natural extension of the very strong separation condition first introduced by King [18] to study the fine multifractal spectrum of self-affine measures on Bedford-McMullen carpets.It was later adapted to higher dimensional Bedford-McMullen sponges by Olsen [21].It is also assumed by Fraser and Howroyd [12,13] when calculating the Assouad dimension of self-afffine measures on these sponges.In fact, in this case the very strong separation condition is a necessary assumption.Without it, one can construct a carpet which does not carry any doubling self-affine measure, see [12,Section 4.2] for an example.

Main result.
In order to state our main result we need to introduce additional probability vectors derived from p = (p(i)) i∈I by 'projecting' it onto subsets I σ n ⊆ I.For σ ∈ A and 1 n d − 1 let , where p σ n (i) : Observe that due to the SPPC, p σ n (i) can also be calculated by This gives rise to the conditional measure , where in case n = 1, we define Π σ 0 i = ∅, I σ 0 = {∅} and p σ 0 (∅) = 1.This is a natural extension of the conditional probabilities introduced by Olsen [21] for Bedford-McMullen sponges.For m n and i ∈ I σ m , we slightly simplify notation by writing A specific choice of p has particular importance.For i ∈ I σ n (0 n d − 1), define s σ n (i) to be the unique number which satisfies the equation This is the similarity dimension of the IFS given by the "fibre above" i.The SPPC implies that s σ n (i) ∈ [0, 1].We define the σ-ordered coordinate-wise natural measure as For Bedford-McMullen sponges, Fraser and Howroyd [12] used the terminology coordinate uniform measure since in that case the natural measure along a fibre simplifies to the uniform measure.This measure has the special property that We are now ready to state our main result.In particular, for the σ-ordered coordinate-wise natural measure Symbolic arguments used in our proof are collected in Section 5 while the theorem itself is proved in Section 6.The result generalises the formula in [13, Theorem 2.6] for dim A ν p in case of Bedford-McMullen sponges.A sufficient condition for the lower and upper bounds to coincide is if A = B.This occurs when F is a Lalley-Gatzouras sponge in any dimensions, moreover, we prove in Section 3 that A = B for all F satisfying the SPPC in dimensions d = 2 and 3.However, A = B is not a necessary condition.We give an example in four dimensions for which the lower and upper bounds coincide even though B ⊂ A, see Proposition 3.4.Finding a potential example for max σ∈B S(p, σ) < max σ∈A S(p, σ) seems to be a more delicate matter and is a natural direction for further research.
Question 2.6.Is it true that max σ∈B S(p, σ) = max σ∈A S(p, σ) even if B ⊂ A? If not then what is the correct value of dim A ν p ?

A dimension gap: examples and non-examples.
Very often it is the case that one of the bounds to determine some dimension of a set is obtained by calculating the respective dimension of measures supported by the set.For example, for the Assouad dimension Luukkainen and Saksman [19] and for the lower dimension Bylund and Gudayol [4] The well-known mass distribution principle and Frostman's lemma combine to provide a similar result for the Hausdorff dimension, for example, see [7].There is also a relatively new notion of box or 'Minkowski' dimension for measures and again there is a similar result, see [8,Theorem 2.1].Therefore, it is interesting to see whether the dimension of a set is still attained by restricting to a certain class of measures (e.g.dynamically invariant measures) or if there is a strictly positive 'dimension gap'.
Self-affine measures supported on carpets and sponges have been used to showcase both kinds of behaviour.Here we just give a few highlights and direct the interested reader to the book [11,Chapter 8.5] for a more in-depth discussion.The Hausdorff dimension of a Lalley-Gatzouras carpet is attained by a self-affine measure [14], however, this is not the case in higher dimensions by the counterexample of Das and Simmons [6].The box dimension of a Bedford-McMullen carpet is attained by a self-affine measure if and only if the carpet has uniform fibres, see [2].The Assouad and lower dimensions of a Lalley-Gatzouras sponge are simultaneously realised by the same self-affine measure, namely the coordinate-wise natural measure [15].
Going beyond the Lalley-Gatzouras class, one might expect that if A = B then one of the coordinate-wise natural measures could still realise the Assouad dimension and potentially another the lower dimension.An interesting corollary of Theorem 2.5 is that this is not the case in general.A strictly positive dimension gap can occur on the plane, noting that dim A F and dim L F were calculated by Fraser [10] using covering arguments.
Corollary 2.7.There exists a Barański carpet F such that for some δ F > 0 depending only on F .Moreover, there also exists a Barański carpet These families of examples are presented in Section 4.1.Finding conditions under which there is a dimension gap also seems a delicate issue.Question 2.8.Is it possible to give simple necessary and/or sufficient conditions for general self-affine carpets satisfying the very strong SPPC for there to be a dimension gap in the sense of (2.12)?
An unfortunate consequence of Corollary 2.7 is that in general the class of self-affine measures is insufficient to use in order to determine dim A F .Question 2.9.What class P of measures should be used on the plane to ensure inf ν∈P dim A ν = dim A F ?For example, can P be taken to be the set of invariant measures?

Comparing orderings of cubes and cylinders
In this section we establish some further relationships between A and B, recall (2.5) and (2.6).We say that coordinate x dominates coordinate y, denoted y ≺ x, if for every i ∈ I. (3.1) Since any two coordinates x = y are distinguishable (2.1), there actually exists an i for which the inequality is strict.A consequence of (3.1) is that L i (r, y) L i (r, x) for all i ∈ Σ and r > 0, therefore, x must precede y in any σ ∈ A. As a result, if there is a chain of coordinates x n ≺ x n−1 ≺ . . .≺ x 1 , then #A d!/n!.Moreover, if y ≺ x, then the orthogonal projection onto the xy-plane must be a Lalley-Gatzouras carpet with coordinate x the dominant, while if neither dominates the other, then the projection is a Barański carpet.In general, we say F is a genuine Barański sponge if there do not exist coordinates x, y with x ≺ y.An example with two maps is if λ We start with a useful equivalent characterisation of B by a condition on the maps of the IFS.Let P I denote the set of all probability vectors on I.For a coordinate x and p ∈ P I , we define the Lyapunov exponent to be χ x (p) := − i∈I p(i) log λ (x) i .Observe that if y ≺ x, then χ x (p) < χ y (p) for every p ∈ P I .The following lemma shows that to determine B it is enough to see how P I gets partitioned by the different orderings of Lyapunov exponents.Proof.By introducing the empirical probability vector t K i = (t K i (i)) i∈I with coordinate for i ∈ Σ, K ∈ N and i ∈ I, we can express for any coordinate n, By definition, if σ ∈ B then there exist i ∈ Σ and r > 0 such that This clearly implies then there also exists q ∈ P I arbitrarily close to p with the property that each element has the form q(i) = a i /K for some a i , K ∈ N and still χ σ 1 (q) < χ σ 2 (q) < . . .< χ σ d (q).Then any i ∈ Σ such that t K i = q and r = K ℓ=1 λ Consider the set Q := {p ∈ P I : there exist x = y such that χ x (p) = χ y (p)}.It is the union of lower dimensional slices of P I .Since all pairs of coordinates are distinguishable (2.1), for every q ∈ Q with χ σ 1 (q) χ σ 2 (q) . . .χ σ d (q) there exists p ∈ P I \Q with χ σ 1 (p) < χ σ 2 (p) < . . .< χ σ d (p).Therefore, dropping the word 'strictly' from the definition of B in (2.6) gives the same set of orderings.
The relationship B ⊆ A always holds.It is interesting to see whether the inclusion is strict or not.Proof.For d = 2 the claim is automatic.For d = 3, choose σ ∈ A. Then there exists i ∈ Σ and r > 0 such that L i (r, σ 3 ) L i (r, σ 2 ) L i (r, σ 1 ).We claim that the cylinder Indeed, the way we have made the σ-ordering unique implies that . If the cylinder is not strictly σ-ordered, then based on the discussion before Lemma 3.2 one can construct a strictly σ-ordered cylinder from a small perturbation of p.
However, in four dimensions the inclusion B ⊆ A can be strict.Our example relies on the following lemma.Lemma 3.3.Assume the sponge F satisfying the SPPC is the attractor of an IFS consisting of two maps f 1 , f 2 ordered (2, 1, 3, 4) and (1, 2, 4, 3), respectively.Then (2) .

Examples
4.1.Planar Barański carpets with different behaviour.The Assouad dimension of planar Barański carpets F was determined by Fraser [10].Using our Theorem 2.5, we can check whether dim A F = dim A ν p for some self-affine measure ν p or if there is a dimension gap in the sense of (2.12).Surprisingly, both behaviours are witnessed by simple families of examples.Recall, in the Lalley-Gatzouras class dim A F is always achieved by the (only) coordinate-wise natural measure.
Defining maps for a Barański carpet with strictly positive dimension gap (left), and where the Assouad dimension of F is attained for correctly chosen parameters (right).
Our first example shows a positive dimension gap.Let F be a Barański carpet which is not in the Lalley-Gatzouras class that satisfies the very strong SPPC with its first level cylinders arranged in a way that there is no exact overlap when projecting to either coordinate axis, see left hand side of Figure 1 for an example.In particular, this contains all genuine Barański carpets defined by two maps.Let a i = λ  Without loss of generality we assume that t s.The very strong SPPC implies that s < 1.The formula from [10] shows that dim L F = t s = dim A F .Proposition 4.1.For a Barański carpet F described above there is a strictly positive dimension gap, i.e. there exists δ F > 0 such that Proof.The condition that there is no exact overlap when projecting to either coordinate axis implies that p σ 1 (i) = p(i) and so P σ 1 (i) = 1 for all i ∈ I = {1, . . ., N }.Applying Theorem 2.5, we immediately obtain Since F is not in the Lalley-Gatzouras class and s t, there exists ℓ ∈ I such that b s ℓ > a s ℓ .Fix 0 < ε < b s ℓ − a s ℓ and first consider any p that satisfies p(i) a s i + ε for every by the choice of ε.Now assume that p is such that there exists j ∈ I satisfying p(j) > a s j + ε.Since i∈I a s i = 1, the pigeon hole principle implies that there exists k ∈ I such that 0 < p(k) a s k − ε/(N − 1).Using this particular index, Therefore, choosing Proposition 4.1 shows that if a genuine Barański carpet whose Assouad dimension is realised by a self-affine measure exists, then its defining IFS must have at least three maps.Our second example shows that such a carpet does exist using only three maps.Giving a complete characterisation for Barański carpets with three maps seems possible but perhaps tedious.However, it is straightforward to give an easy to check sufficient condition (valid for all Barański carpets) ensuring that the Assouad dimension of the carpet is attained by a Bernoulli measure.Comparing the formula from [10] with Theorem 2.5 shows that dim A F = max σ∈A S q σ , σ .Therefore, if σ satisfies that S q σ , σ max S q ω , ω , S q σ , ω , then dim A F = dim A ν q σ .To demonstrate this, consider the Barański carpet whose first level cylinders are depicted with the three shaded rectangles on the right hand side of Figure 1.To ensure the attractor is a genuine Barański carpet, assume a 1 < b 1 and a 2 > min{b 2 , b 3 }.Define r, s, t as follows: b r 2 + b r 3 = 1, a s 1 + a s 2 = 1 and b t 1 + b t 2 + b t 3 = 1.We assume max{s, t} < 1 so that the maps can be arranged in a way that satisfies the very strong SPPC.It follows from the formulas in [10] that dim A F = max{s + r, t}.
Assuming that y ≺ x, we have A ⊆ {(x, y, z), (x, z, y), (z, x, y)}.Hence, projection onto yz-plane never plays a role.If #A = 1, then F is a Lalley-Gatzouras sponge.There are potentially three possibilities for #A = 2: (1) A = {(x, y, z), (x, z, y)}, i.e. max{y, z} ≺ x.In this case, the projection onto both the xy and xz-planes are Lalley-Gatzouras carpets with x being the dominant side.(2) A = {(x, z, y), (z, x, y)}, i.e. y ≺ min{x, z}.In this case, projection onto the xz-plane is a Barański carpet and y ≺ min{x, z} implies that projection onto either the xy-plane or yz-plane can be arbitrary.
The third option is not possible due to the following.
Proposition 4.3.Let F be a three dimensional sponge that satisfies the SPPC, y ≺ x and (x, y, z), (z, x, y) ∈ A. Then also (x, z, y) ∈ A.
Proof.All maps of the IFS defining F can not be ordered the same way, therefore, without loss of generality we assume that corresponding to ordering (x, y, z) and (z, x, y), respectively.We also assume that f 1 and f 2 do not overlap exactly on neither the xy nor on the xz-plane.
According to Lemma 3.1 it is enough to show that there exists p ∈ P I such that χ x (p) < χ z (p) < χ y (p).Consider p = (p, 1 − p, 0, . . ., 0) noting that the calculations that follow can also be adapted to small enough perturbations of p. Straightforward algebraic manipulations yield where and .
, which is always true because of (4.1).This completes the proof.

Symbolic arguments
In this section we work on the symbolic space Σ = I N of all one-sided infinite words i = i 1 , i 2 , . . .with the Bernoulli measure µ p = p N .Recall all notation from Section 2. Throughout the section a σ-order always refers to a cube as defined in (2.4).We define symbolic cubes whose image under the natural projection (1.1) well-approximate Euclidean balls on the sponge F .Let Σ σ r := {i ∈ Σ : i is σ-ordered at scale r}.We define the σ-ordered symbolic r-approximate cube containing i ∈ Σ σ r to be where i ∧ j denotes the longest common prefix of i and j.This is the natural extension of the notion of approximate squares used extensively in the study of planar carpets.Due to (2.2), the image π(B i (r)) is contained within a hypercuboid of [0, 1] d aligned with the coordinate axes with side lengths at most r.Observe that if i ∈ Σ σ r , then for all j ∈ B i (r) also j ∈ Σ σ r .Thus, we identify the σ-ordering of B i (r) with the σ-ordering of i at scale r.If i ∈ Σ σ r , then the surjectivity of the maps Π σ n implies that B i (r) can be identified with a sequence of symbols of length L i (r, σ 1 ) of the form where we set L i (r, σ d+1 ) := 0.
The following lemmas collect important properties about the µ p measure of a symbolic r-approximate cube.The first one is the extension of [21, eq. ( 6.2)].We use the convention that any empty product is equal to one.
Motivated by the definition of dim A ν, the goal is to bound the ratio µ p (B i (R))/µ p (B i (r)) for approximate cubes with different orderings.The first step is to consider when B i (R) and B i (r) have the same ordering.
Let λ min := min n,i λ n , then choose one arbitrarily.Lemma 5.3.Fix σ ∈ A and assume that both B i (R) and B i (r) are σ-ordered, where 0 < R 1 and r < λ min R. Then there exists a constant C > 1 depending only on the sponge F such that .
Proof.It follows from Lemma 5.1 that The requirement that r < λ min R ensures that L i (R, σ n ) < L i (r, σ n ) for all n.We bound each exponent individually to obtain From definition (2.2) of L i (r, n) it follows that there exists C > 1 such that which together with (5.4) concludes the proof.Now we extend Lemma 5.3 so that B i (R) and B i (r) can have different orderings.This step is not necessary if F is a Lalley-Gatzouras sponge and represents one of the key technical challenges in the paper. .
5.1.Proof of Proposition 5.4.Let σ i (r) denote the ordering of B i (r) and assume σ i (R) = σ i (r).Trying to estimate the ratio µ p (B i (R))/µ p (B i (r)) directly using Lemma 5.1 did not lead us to a proof.Instead, the rough idea is to divide the interval [r, R] of scales into a uniformly bounded number of subintervals so that the ordering at roughly the two endpoints of a subinterval are the same.Then we repeatedly apply Lemma 5.3 to each subinterval.The next lemma allows us to make a subdivision.
Lemma 5.5.Fix ε > 0 such that 1 − ε > max n,i λ i .There exists a constant C 1 = C 1 (F, p, ε) < ∞ such that for all i ∈ Σ and 0 < R 1, Proof.First assume that σ i (R) = σ i ((1 − ε)R) = σ and consider the symbolic representation of B i (R) and B i ((1 − ε)R).They could be different at indices ) to a ratio p/q, where p q (p is a sum containing q by (2.9)) and both p, q are uniformly bounded away from 0 (simply because p and q are sums of different terms of p which all are strictly positive to begin with).Therefore, there exists a uniform upper bound C for p/q.As a result, completing the proof in this case by setting We claim that even if σ i (R) = σ i ((1 − ε)R), there still exists an ordering ω such that the value of µ p (B i (R)) is the same when calculating it with σ i (R) or ω and likewise, the value of µ p (B i ((1 − ε)R)) is the same when calculating it with σ i ((1 − ε)R) or ω.Hence, we may apply the previous argument to ω.
To see the claim, first observe that the ordering σ The assumption on ε implies that L We fix an (arbitrary) ordering of all Y ℓ , Z ℓ and define By Remark 5.2, µ p (B i (R)) can be calculated by another ordering that only permutes elements within any of the blocks X ℓ .The ordering ω clearly satisfies this.It remains to argue that ω also works for σ ).These imply that σ i ((1 − ε)R) can be obtained from ω by only permuting elements within a block Y ℓ or Z ℓ .This exactly means that µ p B i ((1 − ε)R) can be calculated using ω.
We next define the scales where we subdivide We are ready to conclude the proof.We suppress multiplicative constants c depending only on F by writing X Y if X cY .First using Lemma 5.5 and then Lemma 5.3, we get the upper bound .
The lower bound is very similar.Lemma 5.5 is not necessary because µ p (B i (R)) µ p B i ((1 − ε)R) holds for any R > 0 and one uses min σ∈A S(p, σ) instead in the last step.The proof of Proposition 5.4 is complete.
6. Proof of Theorem 2.5 6.1.Transferring symbolic estimates to geometric estimates.The very strong SPPC implies that there exists δ 0 > 0 depending only on the sponge F such that for every σ ∈ A, 1 n d and i, j ∈ I for which f i and f j do not overlap exactly on The next lemma allows us to replace a Euclidean ball B(x, r) with the image of an approximate cube of roughly the same diameter under the natural projection π, recall (1.1).It is an adaptation of [21, Proposition 6.2.1].The short proof is included for completeness. .
We use this j together with k σ n (n = 1, . . ., d), see (5.2), to build the i ∈ Σ from the statement of the proposition.We now construct the sequence of triples (R, r, i).This is done by first choosing a decreasing sequence of R tending to 0 with the first term sufficiently small.For a particular R the associated r and i are built as follows.Choosing r in the range (6.3) is possible for sufficiently small R since the bound on the left is o(R) as R → 0 due to (6.2).Moreover, this allows the choice to be made whilst also ensuring R/r → ∞.Let i = i 1 , i 2 , • • • ∈ Σ be such that for ℓ = L i (R, σ n ) + 1, . . ., L i (r, σ n ) and all other entries are j.Note that the upper bound from (6.3) immediately guarantees for all n = 1, . . ., d.In order to show that i is indeed well-defined, we claim that the lower bound from (6.3) guarantees L i (r, σ n ) < L i (R, σ n−1 ) (6.4) for n = 2, . . ., d + 1, where we adopt the convention that L i (r, σ d+1 ) = 0.This takes more work.Proceeding by (backwards) induction let n ∈ {2, . . ., d} and assume that (6.4) holds for n + 1, . . ., d + 1.The goal is to establish (6.4) for n.By definition of L i (R, σ n−1 ), see (2.2), λ min R < as required.
6.3.Final claim for σ-ordered coordinate-wise natural measures.The claim for the σ-ordered coordinate-wise natural measure q σ follows from the simple observation that |supp(ν)| > 0, the lower dimension of ν is dim L ν = sup s 0 : there exists C > 0 such that, for all x ∈ supp(ν) and for all 0 < r < R < |supp(ν)|, 1) d ) = ∅.In other words, the IFSs generated on the coordinate axes by indices I σ 1 satisfy the open set condition.This clearly implies the SPPC.Example 2.3.A Lalley-Gatzouras sponge F ⊂ [0, 1] d satisfies the SPPC and the coordinate ordering condition (2.7) for some σ ∈ S d .Example 2.4.A Bedford-McMullen sponge F ⊂ [0, 1] d is a Barański sponge which satisfies the coordinate ordering condition (hence, is also a Lalley-Gatzouras sponge) and λ

Theorem 2 . 5 .
Let ν p be a self-affine measure fully supported on a self-affine sponge satisfying the very strong SPPC.Then max σ∈B S(p, σ) dim A ν p max σ∈A S(p, σ) and min σ∈A S(p, σ) dim L ν p min σ∈B

Lemma 3 . 2 .
If d = 2 or 3, then A = B for every sponge F satisfying the SPPC.

( 2 )
i , moreover, define s and t to be the unique solutions to the equations

Proposition 4 . 2 .
Consider a Barański carpet as on the right hand side of Figure 1.Assume s + r > t.If a 2 max{b 2 , b 3 } and b 1+r/s 1

3 )
With this notation S(p, σ) = d n=1 s σ n and S(p, σ) = d n=1 s σ n .If there are multiple choices for either k σ n or k σ

Proposition 5 . 4 .
Assume 0 < R 1 and r < λ min R. Then there exists a constant C > 1 depending only on the sponge F such thatC −1 R r min σ∈A S(p,σ) µ p (B i (R)) µ p (B i (r)) C R r max σ∈A S(p,σ)

3 )
i (R,σ ℓ )−L i (r,σ ℓ+1 ) λ (σ n−1 ) k σ ℓ L i (r,σ ℓ )−L i (R,σ ℓ ), where we have changed the use of the index ℓ slightly.Invoking (6.1) and using (2, which proves (6.4).With the sequence now in place, the result follows easily.For all triples (R, r, i) in the sequence, Lemma 5.1 gives µ p (B i (R)) µ p (B i (r ∈ B if and only if there exists p = (p, 1 − p) such that χ 1 (p) < χ 2 (p) < χ 3 (p) < χ 4 (p).Notice that χ 2 (p) < χ 3 (p) for any p because coordinate 2 dominates coordinate 3. From the other two inequalities χ 1 (p) < χ 2 (p) and χ 3 (p) < χ 4 (p), we can express p to obtain Characterisation of SPPC in dimension three.Recall from Lemma 3.2 that A = B in d = 3 and notation y ≺ x from (3.1).If no coordinate dominates any of the other, then (recall) F is a genuine Barański sponge and projection to any of the three principal planes is a Barański carpet.For genuine Barański sponges with d = 3, it is not necessarily true that B = S 3 .For example, take an IFS consisting of two maps with λ The last two terms are at most s + r if and only if a 2 max{b 2 , b 3 } and the first term is at most s + r if and only if b for each 1 n d, but necessarily agree at all other indices due to the choice of ε.Where they agree, the corresponding terms simply cancel out in µ p (B i (R))/µ p B i ((1 − ε)R) .Hence, there are at most 1 + 2 + . . .+ d < d 2 different indices of interest.An index where they differ corresponds in µ p