The Assouad spectrum of Kleinian limit sets and Patterson–Sullivan measure

The Assouad dimension of the limit set of a geometrically finite Kleinian group with parabolics may exceed the Hausdorff and box dimensions. The Assouad spectrum is a continuously parametrised family of dimensions which ‘interpolates’ between the box and Assouad dimensions of a fractal set. It is designed to reveal more subtle geometric information than the box and Assouad dimensions considered in isolation. We conduct a detailed analysis of the Assouad spectrum of limit sets of geometrically finite Kleinian groups and the associated Patterson–Sullivan measure. Our analysis reveals several novel features, such as interplay between horoballs of different rank not seen by the box or Assouad dimensions.


Introduction
Non-elementary Kleinian groups generate important examples of dynamically invariant fractal sets living on the boundary of hyperbolic space.Seminal work of Patterson, Sullivan and others established that the Hausdorff, packing and box dimensions of the limit set coincide in the geometrically finite case and are given by the associated Poincaré exponent, denoted by δ.This is perhaps remarkable because dimension is a fine measure of how the limit set takes up space on small scales whereas the Poincaré exponent is a coarse measure of the rate of accumulation of orbits to the boundary.The Assouad dimension is another notion of fractal dimension, which first came to prominence due to the central role it played in, for example, embedding theory and conformal geometry, see [18,23].However, it is rapidly gaining prominence in the literature on fractal geometry and the dimension theory of dynamical systems, see [13].Fraser [12] proved that the Assouad dimension of the limit set of a geometrically finite Kleinian group is not necessarily given by the Poincaré exponent, but is instead given by max{δ, k max } where k max is the maximal rank of a parabolic fixed point.This result is the starting point of our work, which attempts to understand the gap in-between the box dimension and the Assouad dimension by considering the, more recently introduced, Assouad spectrum.This spectrum is a continuum of dimensions which interpolate between the box and Assouad dimensions in a meaningful sense, providing more nuanced information about the scaling structure of the fractal object at hand.This is part of a more general programme of 'dimension interpolation', which is proving to be a useful concept.For example, the Assouad spectrum has found relevance in surprising contexts such as the study of certain spherical maximal functions [2,24].We compute the Assouad spectrum and (its natural dual) the lower spectrum for limit sets of non-elementary geometrically finite Kleinian groups and the associated Patterson-Sullivan measure.These results shed some new light on the 'Sullivan dictionary' in the context of dimension theory, see [14].
Our proofs use a variety of techniques.We use the global measure formula which allows us to utilise the Patterson-Sullivan measure to estimate the cardinality of efficient covers.We take some inspiration from the paper [12] which dealt with the Assouad and lower dimensions of Kleinian limit sets.However, the Assouad and lower spectra require much finer control and therefore many of the techniques from [12] need refined and some need replaced.Since we consider several dual notions of dimension, some of the arguments are analogous and we do our best to suppress repetition.We stress, however, that calculating the lower spectrum (for example) is not usually a case of simply 'reversing' the Assouad spectrum arguments and subtle differences often emerge.For example, Bowditch's theorem describing the geometry near parabolic fixed points is only needed to study the lower spectrum.
For notational convenience, we write A B if there exists a constant C 1 such that A CB, and A B if B A. We write A ≈ B if A B and B A. The constant C is allowed to depend on parameters fixed in the hypotheses of the theorems presented, but not on parameters introduced in the proofs.
2 Definitions and Background

Dimensions of sets and measures
We recall the key notions from fractal geometry and dimension theory which we use throughout the paper.For a more in-depth treatment see the books [7,10] for background on Hausdorff and box dimensions, and [13] for Assouad type dimensions.Kleinian limit sets will be subsets of the d-dimensional sphere S d which we view as a subset of R d+1 .Therefore, it is convenient to recall dimension theory in Euclidean space only.
Let F ⊆ R d be non-empty.We write dim A F , dim L F and dim H F to denote the Assouad, lower, and Hausdorff dimension of F , respectively.We also write dim B F and dim B F for the upper and lower box dimensions of F and dim B F for the box dimension when it exists.We refer the reader to [7,10,13] for the precise definitions since we do not use them directly.It is useful to keep in mind that, for compact We write to denote the diameter of F .Given r > 0, we write N r (F ) to denote the smallest number of balls of radius r required to cover F .We write M r (F ) to denote the largest cardinality of a packing of F by balls of radius r centred in F .In what follows, it is easy to see that replacing N r (F ) by M r (F ) yields equivalent definitions and so we sometimes switch between minimal coverings and maximal packings in our arguments.This is standard in fractal geometry.The Assouad and lower spectra, introduced in [15], interpolate between the box dimensions and the Assouad and lower dimensions in a meaningful way.They provide a parametrised family of dimensions In particular, dim θ A F → dim B F as θ → 0. Whilst the analogous statement does not hold for the lower spectrum in general, it was proved in [13,Theorem 6.3.1] that dim θ L F → dim B F as θ → 0 provided F satisfies a strong form of dynamical invariance.Whilst the fractals we study are not quite covered by this result, we shall see that this 'interpolation' holds nevertheless.The limits lim θ→1 dim θ A F and lim θ→1 dim θ L F are known to exist in general, but are not necessarily equal to the Assouad and lower dimensions, respectively, although we shall see this will hold for the sets considered here.
There is an analogous dimension theory of measures, and the interplay between the dimension theory of fractals and the measures they support is fundamental to fractal geometry, especially in the dimension theory of dynamical systems.Let µ be a locally finite Borel measure on R d .Similar to above, we write dim A µ, dim L µ and dim H µ for the Assouad, lower and (lower) Hausdorff dimensions of µ, respectively.For θ ∈ (0, 1), the Assouad spectrum of µ with support F is given by s and otherwise it is 0. Once again, if one replaces r θ in the above definitions with a free scale R ∈ (r, 1), then one recovers the Assouad and lower dimensions of µ, respectively.The Assouad and lower dimensions of measures were introduced in [17], where they were referred to as the upper and lower regularity dimensions, respectively.It is known (see [11] for example) that and, if µ has support equal to a closed set F , then The upper box dimension of µ with support F is given by and the lower box dimension of µ is given by If dim B µ = dim B µ, then we refer to the common value as the box dimension of µ, denoted by dim B µ.These definitions of the box dimension of a measure were introduced only recently in [11] where it was also shown that, for θ ∈ (0, 1),

Kleinian groups and limit sets
For a more thorough study of hyperbolic geometry and Kleinian groups, we refer the reader to [1,3,19,21].For d 1, we model (d + 1)-dimensional hyperbolic space using the Poincaré ball model equipped with the hyperbolic metric d H and we call the boundary the boundary at infinity of the space (D d+1 , d H ). We denote by Con(d) the group of orientationpreserving isometries of (D d+1 , d H ). We will occasionally make use of the upper half space model H d+1 = R d × (0, ∞) equipped with the analogous metric.We say that a group is Kleinian if it is a discrete subgroup of Con(d), and given a Kleinian group Γ, the limit set of Γ is defined to be L(Γ) = Γ(0) \ Γ(0) where 0 = (0, . . ., 0) ∈ D d+1 .It is well known that L(Γ) is a compact Γ-invariant subset of S d .If L(Γ) contains zero, one or two points, it is said to be elementary, and otherwise it is non-elementary.In the non-elementary case, L(Γ) is a perfect set, and often has a complicated fractal structure.We consider geometrically finite Kleinian groups.Roughly speaking, this means that there is a fundamental domain with finitely many sides (we refer the reader to [9] for further details).We define the Poincaré exponent of a Kleinian group Γ to be Due to work of Patterson and Sullivan [22,27], it is known that for a non-elementary geometrically finite Kleinian group Γ, the Hausdorff dimension of the limit set is equal to δ.It was discovered independently by Bishop and Jones [6, Corollary 1.5] and Stratmann and Urbański [25,Theorem 3] that the box and packing dimensions of the limit set are also equal to δ.Even in the non-elementary geometrically infinite case, δ is still an important quantity.In fact it always gives the Hausdorff dimension of the radial limit set, and therefore also provides a lower bound for the Hausdorff dimension of the limit set, see [6].In general, δ can be difficult to compute or estimate, but there are various techniques available, see [16,20].
From now on we only discuss the case of non-elementary geometrically finite Γ.We write µ to denote the associated Patterson-Sullivan measure, which is a measure first constructed by Patterson in [22].Strictly speaking there is a family of (mutually equivalent) Patterson-Sullivan measures.However, we may fix one for simplicity (and hence talk about the Patterson-Sullivan measure since the dimension theory is the same for each measure).The geometry of Γ, L(Γ) and µ are heavily related.For example, µ is a δ-conformal Γ-ergodic Borel probability measure with support L(Γ).Moreover, µ has Hausdorff, packing and entropy dimension equal to δ, see [26].The limit set is Γ-invariant in the strong sense that g(L(Γ)) = L(Γ) for all g ∈ Γ.However, µ is only quasi-invariant and µ • g is related to µ by a geometric transition rule, see [8,Chapter 14] for a more detailed exposition of this.
The Assouad and lower dimensions of µ and L(Γ) were dealt with in [12].To state the results, we require some more notation.Suppose Γ contains at least one parabolic point, and denote by P ⊆ L(Γ) the countable set of parabolic fixed points.We may fix a standard set of horoballs {H p } p∈P (a horoball H p is a closed Euclidean ball whose interior lies in D d+1 and is tangent to the boundary S d at p) such that they are pairwise disjoint, do not contain the point 0, and have the property that for each g ∈ Γ and p ∈ P , we have g(H p ) = H g(p) , see [25,26].
We note that, for any p ∈ P , the stabiliser of p denoted by Stab(p) cannot contain any loxodromic elements, as this would violate the discreteness of Γ.We denote by k(p) the maximal rank of a free abelian subgroup of Stab(p), which must be generated by k(p) parabolic elements which all fix p, and call this the rank of p.We write It was proven in [27] that δ > k max /2.In [12], the following was proven: Theorem 2.1.Let Γ be a non-elementary geometrically finite Kleinian group.Then We will rely on Stratmann and Velani's global measure formula [26] which gives a formula for the measure of any ball centred in the limit set up to uniform constants.More precisely, given z ∈ L(Γ) and T > 0, we define z T ∈ D d+1 to be the point on the geodesic ray joining 0 and z which is hyperbolic distance T from 0. We write S(z, T ) ⊂ S d to denote the shadow at infinity of the d-dimensional hyperplane passing through z T which is normal to the geodesic joining 0 and z.The global measure formula states that µ(S(z, T )) ≈ e −T δ e −ρ(z,T )(δ−k(z,T )) (2.2) where k(z, T ) = k(p) if z T ∈ H p for some p ∈ P and 0 otherwise, and if z T ∈ H p for some p ∈ P and 0 otherwise.Basic hyperbolic geometry shows that S(z, T ) is a Euclidean ball centred at z with radius comparable to e −T , and so an immediate consequence of (2.2) is the following.
Theorem 2.2 (Global Measure Formula).Let z ∈ L(Γ), T > 0. Then we have An easy consequence of Theorem 2.2 is that if Γ contains no parabolic points, then . Therefore, we assume throughout that Γ contains at least one parabolic point.

Results
We assume throughout that Γ < Con(d) is a non-elementary geometrically finite Kleinian group containing at least one parabolic element, and write L(Γ) to denote the associated limit set and µ to denote the associated Patterson-Sullivan measure.Our first result gives formulae for the Assouad and lower spectra of µ, as well as the box dimension of µ.
We prove Theorem 3.1 in Sections 4.2 -4.4.It is perhaps noteworthy that k min and k max show up simultaneously in the formulae for dim θ A µ and dim θ L µ in the 'intermediate range' of δ.This does not happen for dim A µ and dim L µ (see Theorem 2.1) and is indicative of subtle interplay between horoballs of different rank detected by the Assouad and lower spectra.The next theorem provides formulae for the Assouad and lower spectra of L(Γ).
We prove Theorem 3.2 in Sections 4.5 and 4.6.

Preliminaries
As many of the proofs in the Kleinian setting are reliant on horoballs, we first establish various estimates describing the geometry of horoballs, including the 'escape functions' ρ(z, T ).We start with a simple lemma.One should think of the circle involved as a 2-dimensional slice of a horoball.
We also require the following lemma to easily estimate the 'escape function' at a parabolic fixed point.Proof.Let p S be the 'tip' of the horoball H p , in other words the point on the horoball H p which lies on the geodesic joining 0 and p.It is obvious that p T ∈ H p ⇐⇒ T S, so for sufficiently large T we have k(p, T ) = k(p).Also note that as T → ∞, as required.
The formulae for dim θ A µ and dim θ L µ sometimes involve both k min and k max .Consequently, we need to consider two standard horoballs, where we drag one towards the other.One could think of the following lemma as saying that the images of this horoball 'fill in the gaps' under the fixed horoball.
Lemma 4.3 (Horoball Radius Lemma).Let p, p ′ ∈ L(Γ) be two parabolic fixed points with associated standard horoballs H p and H p ′ respectively, and let f be a parabolic element which fixes p. Then for sufficiently large n, Proof.By considering an inverted copy of Z converging to p, one sees that |f n (p ′ ) − p| ≈ 1/n, so we need only prove that as otherwise the horoballs would eventually overlap with H p , using Lemma 4.1 and the fact that |f n (p ′ ) − p| ≈ 1 n , contradicting the fact that our standard set of horoballs is chosen to be pairwise disjoint.For the lower bound, consider a point u = p ′ on H p ′ , and its images under the action of f .Let v denote the 'shadow at infinity' of u (see Figure 1), and note that for sufficiently large n, |f n (v) − p| ≈ 1 n .As f n (u) lies on a horoball with base point p, we can use Lemma 4.1 to deduce that |f n (u) − f n (v)| ≈ 1/n 2 (see Figure 2), and therefore  The following lemma is also necessary for estimating hyperbolic distance, essentially saying that if z, u ∈ L(Γ) are sufficiently close, then d H (z T , u T ) can be easily estimated.Proof.This is immediate from the cross ratio formula for hyperbolic distance.Briefly, given points P and Q in the interior of the Poincaré disk, draw a geodesic between them which, upon extension, intersects the boundary of the disc at points A and B such that A is closer to P than Q (see Figure 3).Then we have Applying this formula with P = z T and Q = u T yields the result.
We also require the following lemma due to Fraser [12,Lemma 5.2], which allows us to count horoballs of certain sizes.Lemma 4.6.Let z ∈ L(Γ) and T > t > 0. For t sufficiently large, we have where P is the set of parabolic fixed points contained in L(Γ).

Lower bound
We show dim B µ max{δ, 2δ − k min }.Note that we have dim B µ dim B L(Γ) = δ, so it suffices to prove that dim B µ 2δ − k min , and therefore we may assume that δ k min .Let p be a parabolic fixed point such that k(p) = k min , and let ε ∈ (0, 1).By Lemma 4.2, it follows that for sufficiently large T > 0, we have Let θ ∈ (0, 1), let p ∈ L(Γ) be a parabolic fixed point such that k(p) = k max , f be a parabolic element fixing p, and n ∈ N be very large.Choose p = z 0 ∈ L(Γ), and let z = f n (z 0 ), noting that z → p as n → ∞.We assume n is large enough to ensure that the geodesic joining 0 and z intersects H p , and choose T > 0 to be the larger of two values such that z T lies on the boundary of H p .We now restrict our attention to the hyperplane H(p, z, z T ) restricted to D d+1 .Define v to be the point on H p ∩ H(p, z, z T ) such that v lies on the quarter circle with centre z and Euclidean radius e −T θ (see Figure 4 below).We also consider 2 additional points u and w, where u is the 'shadow at infinity' of v and w = u T θ .
Figure 4: An overview of the horoball H p along with our chosen points.We wish to find a lower bound for d H (z T θ , v).
Our goal is to bound d H (v, w) from below and d H (w, z T θ ) from above.Consider the Euclidean distance between z and p.Note that z T lies on the horoball H p , so using This gives us for some constants C 1 , C 2 .We can apply Lemma 4.4 to deduce that d H (z T θ , w) C 3 for some constant C 3 .Therefore, by the triangle inequality By Theorem 2.2, we have as required.
Upper bound: We show Let z ∈ L(Γ), T > 0. Note that we have so we assume that θ ∈ (0, 1/2).Then by Theorem 2.2, we have which gives as required.
Lower bound: We show Let θ ∈ (0, 1/2), let p, p ′ ∈ L(Γ) be parabolic fixed points such that k(p) = k max and k(p ′ ) = k min , and let f be a parabolic element fixing p.Let n be a large positive integer and let z = f n (p ′ ).By Lemma 4.3, we note that for sufficiently large n we may choose T such that k(z, T ) = k min , k(z, T θ) = k max , and |z − p| = e −T θ (see Figure 7).We can make use of Lemma 4.4 to deduce that for some constant C 1 .Also, by Lemma 4.2, given ε ∈ (0, 1) we have that ρ(p, T θ) (1 − ε)T θ for sufficiently large n.This gives Finally, we note that as |z − p| = e −T θ , by Lemma 4.1 we have |H z | ≈ e −2T θ (see Figure 8), which implies that for some constants C 2 , C 3 .Applying (2.2), we get and letting ε → 0 proves the desired lower bound.The case when θ 1/2 follows identically to the lower bound in Section 4.3.1,so we omit the details.
Let z ∈ L(Γ), T > 0, θ ∈ (0, 1/2) (the case when θ 1/2 follows similarly to the upper bound in Section 4.3.1).If z T and z T θ do not lie in a common standard horoball, we may use the fact that We have by Theorem 2.2 .
If z T and z T θ do lie in a common standard horoball H p and δ k(p), then we use the inequality Then we have if ρ(z, T ) − ρ(z, T θ) 0, and otherwise .
In all cases, we have as required.

The lower spectrum of µ
The proofs of the bounds for the lower spectrum follow similarly to the Assouad spectrum, and so we only sketch the arguments.

When δ > k max
Upper bound: We show Let θ ∈ (0, 1), p ∈ L(Γ) be a parabolic fixed point such that k(p) = k min , f be a parabolic element fixing p, and n ∈ N be very large.Choose p = z 0 ∈ L(Γ), and let z = f n (z 0 ).We choose T > 0 such that z T is the 'exit point' from H p .Identically to the lower bound in Section 4.3.1,we have that for some constant C. Applying Theorem 2.2, we have Let z ∈ L(Γ), T > 0. Note that dim θ L µ dim L µ = k min so we assume θ ∈ (0, 1/2).Then by Theorem 2.2, we have as required.
Similarly to the lower bound in Section 4.3.2,we only need to deal with the case when θ ∈ (0, 1/2).Let p, p ′ ∈ L(Γ) be parabolic fixed points such that k(p) = k min and k(p ′ ) = k max , and let f be a parabolic element fixing p.Let n be a large integer and let z = f n (p ′ ).Again, by Lemma 4.3, for sufficiently large n we may choose T such that k(z, T ) = k max and k(z, T θ) = k min .We may argue in the same manner as the lower bound in Section 4.3.2 to show that, for sufficiently large n, for some ε ∈ (0, 1) and some constants C 1 , C 2 .Applying Theorem 2.2 gives and letting ε → 0 proves the desired upper bound.
If z T and z T θ do lie in a common standard horoball H p and δ k(p), then we have if ρ(z, T ) − ρ(z, T θ) 0, and otherwise . .
In all cases, we have as required.

When δ
We show dim θ L µ = 2δ − k max .Let p, p ′ ∈ L(Γ) be two parabolic fixed points such that k(p ′ ) = k max , let f be a parabolic element fixing p and let n be a large positive integer.Let z = f n (p ′ ) and let ε ∈ (0, 1).Note that for sufficiently large n and Lemma 4.2, we may choose T such that which proves dim θ L µ 2δ − k max − ε(δ − k max ) and letting ε → 0 proves the upper bound.Recall that dim L µ = min{2δ − k max , k min }, so when δ (k min + k max )/2, we have 4.5 The Assouad spectrum of L(Γ)

When δ k min
We show As when δ k min , we need only prove the lower bound.To obtain this, we make use of the following result (see [13,Theorem 3.4.8]).
Proposition 4.7.Let F ⊆ R n and suppose that exists and ρ ∈ (0, 1) and dim L F = dim B F .Then for θ ∈ (0, ρ), We note that if δ k min , then certainly We now show that ρ = 1/2.By (4.9), we have ρ 1/2, so we need only show that ρ 1/2.To do this, we recall a result from [12], which states that using the orbit of a free abelian subgroup of the stabiliser of some parabolic fixed point p with k(p) = k max , we can find a bi-Lipschitz image of an inverted Z kmax lattice inside L(Γ) in S d , which gives us

.10)
The final equality giving the Assouad spectrum of 1/Z kmax is a straightforward calculation which we omit, but note that this can be derived along similar lines (and giving the same formula) as the treatment of the product set (1/N) kmax given in [15,Prop 4.5,Cor 6.4].This proves ρ 1/2, as required.Therefore, by Proposition 4.7, we have for θ ∈ (0, 1/2) as required.
The argument for the upper bound for the Assouad spectrum is nearly identical to the argument for the upper bound for the Assouad dimension given in [12], and so we omit any part of the argument which does not improve upon the bounds provided in the paper, in particular when the Assouad dimension is bounded above by δ.We also note that so we may assume that θ ∈ (0, 1/2).Let z ∈ L(Γ), ε > 0, and T be sufficiently large such that Our goal is to bound from above the cardinalities of X 0 , X 1 and X n separately.We start with X 0 , which we may assume is non-empty as we are trying to bound from above.We note that if there exists some p ∈ P with |H p | 10e −T θ and H p ∩ (∪ i∈X (x i ) T ) = ∅, then this p must be unique, i.e. if , then H p and H p ′ could not be disjoint.This means we can choose p ∈ P such that (x i ) T ∈ H p for all i ∈ X 0 , and also note that this forces z T θ ∈ H p .
If δ k(p), then by Theorem 2.2 where the second inequality follows from the fact that {x i } i∈X 0 is an e −T packing.Therefore Note that Fraser [12] estimates using ρ(x i , T ) − ρ(z, t) T − t + 10, but we can improve this for t = T θ with θ < 1/2 by using ρ(x i , T ) 0 and ρ(z, T θ) T θ.
If δ > k(p), then we refer the reader to [12,[4998][4999][5000], where it is shown that Therefore, we have regardless of the relationship between δ and k(p).
For X 1 , we have by the definition of X 1 and the fact that δ > k min .Therefore Finally, we consider the sets X n .If i ∈ X n for n 2, then ρ(x i , T ) > n − 1, and so (x i ) T ∈ H p for some p ∈ P with e −T |H p | < 10e −T θ .Furthermore, we may note that B(x i , e −T ) is contained in the shadow at infinity of the squeezed horoball 2e −(n−1) H p .Also note that as where the last three inequalities use Theorem 2.2, Lemma 4.6 and the fact that εT (1−θ) < ρ(x i , T ) n as i / ∈ X 1 .On the other hand, using the fact that which implies Combining (4.11), (4.12) and (4.13), we have and letting ε → 0 proves the desired upper bound.
Lower bound: We show Note that the case when θ 1/2 is an immediate consequence of (4.10), so we assume that θ ∈ (0, 1/2).In order to derive the lower bound, we let p ∈ L(Γ) be a parabolic fixed point with k(p) = k max , then we switch to the upper half-space model H d+1 and assume that p = ∞.Recalling the argument from [12, Section 5.1.1],we know that there exists some subset of L(Γ) which is bi-Lipschitz equivalent to Z kmax .Therefore, we may assume that we have a Z kmax lattice, denoted by Z, as a subset of L(Γ), and then use the fact that the Assouad spectrum is stable under bi-Lipschitz maps.We now partition this lattice into {Z k } k∈N , where We note that Let φ : H d+1 → D d+1 denote the Cayley transformation mapping the upper half space model to the Poincaré ball model, and consider the family of balls {B(z, 1/3)} z∈Z .Taking the φ-image of Z yields an inverted lattice and there are positive constants C 1 and C 2 such that if z ∈ Z k for some k, then where p 1 = φ(∞).
Let T > 0. We now choose a constant C 3 small enough which satisfies the following: • The set of balls k∈N z∈Z k (B(φ(z), C 3 /k 2 )) are pairwise disjoint.
This gives us We wish to estimate from below.Let k ∈ N satisfy the conditions given in the sum above, and write t = log(k 2 /C 3 ), and let ε ∈ (0, 1).We will restrict the above sum to ensure that T − t max{ε −1 , log10}, that is, we assume further that k a(ε)e T /2 √ C 3 where a(ε) = exp(− max{ε −1 , log10}).Let {y i } i∈Y be a centred e −T covering of B(φ(z), e −t ) ∩ L(Γ), and decompose Y as Y = Y 0 ∪ Y 1 where We wish to estimate the cardinalities of Y 0 and Y 1 , and we refer the reader to [12,Section 5.2] for the details regarding the estimates used.
If i ∈ Y 0 , then (y i ) T ∈ H p for some unique parabolic fixed point p with In order to estimate ρ(φ(z), t), we ensure that T is chosen large enough such that the line joining 0 and φ(z) intersects H p 1 , and let u > 0 be the larger of two values such that φ(z) u lies on the boundary of H p 1 (see Figure 10).Then note that and also that since |φ(z Therefore, we have for some constant C 4 .Therefore, by (4.16) Similarly, we can show that .
Therefore, we have and so by (4.15) e T (δ+ε(δ−kmax )) e T (δ+ε(δ−kmax )) where the second last inequality uses (4.14).We may assume ε is chosen small enough to ensure that k max − 2δ − 2ε(δ − k max ) < 0 and that T is large enough (depending on ε) such that a(ε) √ C 3 e T /2 > e T θ /C 1 .Then the smaller limit of integration provides the dominant term, so we have and letting ε → 0 proves the lower bound.

When δ k max
We show dim θ A L(Γ) = δ.This follows easily, since when δ k max , we have and so dim θ A L(Γ) = δ, as required.

4.6
The lower spectrum of L(Γ)

When δ k min
We show dim θ L L(Γ) = δ.When δ k min , we have and so dim θ L L(Γ) = δ, as required.

When
Similarly to the upper bound for the Assouad spectrum of L(Γ), the argument is essentially identical to the one given in [12] for the lower bound for the lower dimension, and so we only include parts of the argument which improve upon the bounds derived in that paper, and we may assume that θ ∈ (0, 1/2).Let z ∈ L(Γ), ε ∈ (0, 1), and so we have .
If δ < k(p), we refer the reader to [12,[5003][5004][5005] where it is shown that regardless of the relationship between δ and k(p).Now, suppose that Then ρ(y i , T ) > εT (1 − θ) and so similar to the X n case in the Assouad spectrum argument, we know that (y i ) T is contained in the squeezed horoball e −εT (1−θ) H p with e −T |H p | < 10e −T θ for some p ∈ P with p ∈ B(z, 10e −T θ ).
We also note that the Euclidean distance from (y i ) T to S d is comparable to e −T , and so by Pythagoras' Theorem, see Figure 11, we have As δ > k max /2, this means that balls with centres in Y 1 \ Y 0 1 cannot carry a fixed proportion of µ(B(z, e −T θ )) for sufficiently large T .It follows that by our assumption.This means that e −T θδ e −ρ(z,T θ)(δ−k(z,T θ)) µ(B(z, e −T θ )) µ(∪ i∈Y 0 1 B(y i , e −T )) i∈Y 0 1 (e −T ) δ e −ρ(y i ,T )(δ−k(y i ,T ))

.20)
At least one of (4.19) and (4.20) must hold, so we have In order to obtain the upper bound in this case, we require the following technical lemma due to Bowditch [9].Switch to the upper half space model H d+1 , and let p ∈ P be a parabolic fixed point with rank k min .We may assume by conjugation that p = ∞.A consequence of geometric finiteness is that the limit set can be decomposed into the disjoint union of a set of conical limit points and a set of bounded parabolic fixed points (this result was proven in dimension 3 partially by Beardon and Maskit [4] and later refined by Bishop [5], and then generalised to higher dimensions by Bowditch [9]).In particular, p = ∞ is a bounded parabolic point, and applying [9, Definition, Page 272] gives the following lemma.Note that by Lemma 4.8, we immediately get dim θ L L(Γ) k min for θ 1/2, so we may assume that θ ∈ (0, 1/2).
We proceed in a similar manner to the lower bound in Section 4.5.2.Let p ∈ L(Γ) be a parabolic fixed point with k(p) = k min , and then switch to the upper half-space model H d+1 , and assume that p = ∞.Similar to the lower bound in Section 4.5.2,we may assume that we have a Z k min lattice as a subset of L(Γ), which we denote by Z.We partition Z into {Z k } k∈N , where  Furthermore, using the fact that C 3 /k 2 > e −T and the fact that we are summing in k min directions, we have |K| e T k min 2 e T (1−θ)k min e T (1−θ)δ .
We now assume that T − t max{ε −1 , log10}.We let {x i } i∈X denote a centred e −T -packing of B(φ(z), e −t ) ∩ L(Γ) of maximal cardinality, and decompose X as For the details on estimating the cardinalities of X 0 , X 1 and X n , we refer the reader to [12,Section 5.1].
Suppose that i ∈ X 0 .Then (x i ) T ∈ H p for some unique parabolic fixed point p with |H p | 10e −t .If δ > k(p), then it can be shown that In a similar manner to the lower bound in Section 4.5.2,we can show that ρ(φ(z), t) C 4 for some constant C 4 , see (4.17).Therefore, we have

Lemma 4 . 2 (
Parabolic Centre Lemma).Let p ∈ L(Γ) be a parabolic fixed point with associated standard horoball H p .Then we have ρ(p, T ) ∼ T as T → ∞, and for sufficiently large T > 0 we have k(p, T ) = k(p).

Figure 1 :
Figure 1: An illustration showing the points u and v.The dashed arc shows the horoball along which the point u is pulled under the action of f .

Lemma 4 . 4 .
Let z, u ∈ L(Γ) and T > 0 be large.If |z − u| ≈ e −T , then d H (z T , u T ) ≈ 1. (This should be read as the implicit constants in the conclusion depend on the implicit constants in the assumption.)
and letting ε → 0 proves the lower bound.

Figure 7 :
Figure 7: Making use of Lemma 4.3 so we can choose our desired T .

Figure 9 :
Figure 9: Choosing the appropriate T .

Figure 11 :
Figure 11: Applying Pythagoras' Theorem using (y i ) T , p T and the centre of e −εT (1−θ) H p .