A new perspective on the Sullivan dictionary via Assouad type dimensions and spectra
AA new perspective on the Sullivan dictionaryvia Assouad type dimensions and spectra
Jonathan M. Fraser and Liam StuartThe University of St Andrews, ScotlandE-mails: [email protected] and [email protected] 31, 2020
Abstract
We conduct a detailed analysis of the Assouad type dimensions and spectra in the contextof limit sets of geometrically finite Kleinian groups and Julia sets of parabolic rational maps.Our analysis includes the Patterson-Sullivan measure in the Kleinian case and the analogousconformal measure in the Julia set case. Our results constitute a new perspective on theSullivan dictionary between Kleinian groups and rational maps. We show that there existboth strong correspondences and strong differences between the two settings. The differenceswe observe are particularly interesting since they come from dimension theory, a subjectwhere the correspondence described by the Sullivan dictionary is especially strong.
Mathematics Subject Classification
Key words and phrases : Kleinian group, rational map, Julia set, limit set,Patterson-Sullivan measure, conformal measure, parabolicity, Assouad dimension, Assouadspectrum, Sullivan dictionary.JMF was financially supported by an
EPSRC Standard Grant (EP/R015104/1) and a
Leverhulme TrustResearch Project Grant (RPG-2019-034). LS was financially supported by the University of St Andrews. a r X i v : . [ m a t h . D S ] J u l ontents µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.3 The Assouad spectrum of µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.4 The lower spectrum of µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.5 The Assouad spectrum of L (Γ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.6 The lower spectrum of L (Γ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.3 The Assouad dimension of m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.4 The lower dimension of m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.5 The Assouad spectrum of m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.6 The lower spectrum of m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.7 The Assouad dimension of J ( T ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.8 The lower dimension of J ( T ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.9 The Assouad spectrum of J ( T ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.10 The lower spectrum of J ( T ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 References 64 Introduction
The Sullivan dictionary provides a correspondence between Kleinian groups and rational maps.Seminal work of Sullivan in the 1980s [49] resolved a long-standing problem in complex dy-namics by proving that the Fatou set of a rational map has no wandering domains. This workserved to establish remarkable connections between the dynamics of rational maps and the ac-tions of Kleinian groups. This connection subsequently stimulated activity in both the complexdynamics and hyperbolic geometry communities and led to what is now known as the
Sullivandictionary ; see, for example, [37]. The Sullivan dictionary provides a framework to study therelationships between Kleinian groups and rational maps. In many cases there are analogousresults, even with similar proofs, albeit expressed in a different language.Both (geometrically finite) Kleinian groups and rational maps generate important examples ofdynamically invariant fractal sets: limit sets in the Kleinian case, and
Julia sets in the ra-tional map case. The Sullivan dictionary is very well-suited to understanding the connectionsbetween these two families of fractal and the correspondence is especially strong in the contextof dimension theory: in both settings there is a ‘critical exponent’ which describes all of themost commonly used notions of fractal dimension. In the Kleinian case the critical exponent isthe Poincar´e exponent, denoted by δ , and in the rational map case the critical exponent is thesmallest zero of the topological pressure, denoted by h , see [16]. In both settings the criticalexponent coincides with the Hausdorff, packing and box dimensions of the fractal as well as theHausdorff, packing, and entropy dimensions of the associated ergodic conformal measure.This paper is dedicated to dimension theory in the context of the Sullivan dictionary and we findthat, by slightly expanding the family of dimensions considered, a much richer and more variedtapestry of results emerges. The dimensions we consider are of ‘Assouad type’ and include theAssouad dimension, the lower dimension, and the Assouad and lower spectrum. The Assouaddimension is perhaps the most widely used of these notions and stems from work in embeddingtheory and conformal geometry, see [33, 42]. It has recently been gaining substantial attentionin the fractal geometry literature, however, see [22]. The lower dimension is the natural ‘dual’to the Assouad dimension and it is particularly useful to consider these notions together. TheAssouad and lower spectrum were introduced much more recently in [25] and provide an ‘in-terpolation’ between the box dimension and the Assouad and lower dimensions, respectively.The motivation for the introduction of these ‘dimension spectra’ was to gain a more nuancedunderstanding of fractal sets than that provided by the dimensions considered in isolation. Thisis already proving a fruitful programme with applications emerging in a variety of settings in-cluding to problems in harmonic analysis, see work of Anderson, Hughes, Roos and Seeger [3]and [43].The Assouad and lower dimensions of limit sets of geometrically finite Kleinian groups andassociated Patterson-Sullivan measures were found in [21]. These results already highlight thesignificance of the Assouad dimension in this setting since it is not generally given by the Poincar´e3xponent δ . We find that this phenomenon also occurs in the rational map setting, with theAssouad dimension of the Julia set of a parabolic rational map being given by max { , h } , seeTheorem 3.4. Our main results consist of precise formulae for the Assouad and lower spectra oflimit sets of geometrically finite Kleinian groups (Theorem 3.2) and the associated Patterson-Sullivan measures (Theorem 3.1), as well as the Assouad and lower dimensions and spectra ofJulia sets of parabolic rational maps (Theorem 3.4) and their associated h -conformal measures(Theorem 3.3). Many of these results may be of independent interest. That said, our mainmotivation is to use this rich family of dimensions and spectra to compare the Kleinian andrational map settings, especially to identify differences between the theories. We refer to thesedifferences as new ‘non-entries’ in the Sullivan dictionary.Our proofs use a variety of techniques. A central concept is that of global measure formulae which allow us to use conformal measures to estimate the size of efficient covers. We take someinspiration from the paper [21] which dealt with the Assouad and lower dimensions of Kleinianlimit sets. However, the Assouad and lower spectra require much finer control and thereforemany of the techniques from [21] need refined and some need replaced. We adapt this broadapproach to the Julia setting which, for example, requires replacing horoballs with canonicalballs . This is often more awkward and connected to the lack of understanding of the ‘hidden3-dimensional geometry’ of Julia sets, see [37, 38]. We also take inspiration from the papers[18, 44, 45, 46, 47] where ideas from Diophantine approximation are applied in the contextof conformal dynamics. In order to adequately describe the local behaviour around parabolicpoints, we rely on structural results such as Bowditch’s theorem in the Kleinian setting and the(quantitative) Leau-Fatou flower theorem in the Julia setting. Since we consider several dualnotions of dimension, some of the arguments are analogous and we do our best to suppress rep-etition. We stress, however, that calculating the lower dimension (for example) is not usually acase of simply ‘reversing’ the Assouad dimension arguments and subtle differences often emerge.For example, Bowditch’s theorem and the Leau-Fatou flower theorem are only needed to studythe lower dimension and spectrum.The rest of the paper is structured as follows. In Section 2, we provide all the necessary back-ground on dimension theory, Kleinian groups, limit sets, rational maps and Julia sets. In Section3, we state the main results of our paper, including a detailed discussion of our new perspectiveon the Sullivan dictionary in Section 3.3. Sections 4 and 5 contain the proofs of our results.For notational convenience throughout, we write A (cid:46) B if there exists a constant C (cid:62) A (cid:54) CB , and A (cid:38) B if B (cid:46) A . We write A ≈ B if A (cid:46) B and B (cid:46) A . The constant C isallowed to depend on parameters fixed in the hypotheses of the theorems presented, but not onparameters introduced in the proofs. 4 Definitions and Background
We recall the key notions from fractal geometry and dimension theory which we will use through-out the paper. For a more in-depth treatment see the books [9, 19, 36] for background on Haus-dorff and box dimensions, and [22] for Assouad type dimensions. We will work with fractals intwo distinct settings. Kleinian limit sets will be subsets of the d -dimensional sphere S d which weview as a subset of R d +1 . On the other hand, Julia sets will be subsets of the Riemann sphereˆ C = C ∪ {∞} . However, by a standard reduction we will assume that the Julia sets are boundedsubsets of the complex plane C , which we identify with R . Therefore, it is convenient to recalldimension theory in Euclidean space only.Let F ⊆ R d . Perhaps the most commonly used notion of fractal dimension is the Hausdorff di-mension. We write dim H F for the Hausdorff dimension of F , but refer the reader to [9, 19, 36]for the precise definition since we do not use it directly. We write | F | = sup x,y ∈ F | x − y | ∈ [0 , ∞ ]to denote the diameter of F . Given r >
0, we write N r ( F ) for the smallest number of balls ofradius 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 an equivalent definition and so we sometimes switch between minimal coveringsand maximal packings in our arguments. This is standard in fractal geometry.For a non-empty bounded set F ⊆ R d , the upper and lower box dimensions of F are defined bydim B F = lim sup r → log N r ( F ) − log r and dim B F = lim inf r → log N r ( F ) − log r , respectively. If dim B F = dim B F , we call the common value the box dimension of F and denoteit by dim B F . While we also will not work with the definition of box dimension directly, it isinstructive to consider it before the related Assouad and lower dimensions. The key differenceis that the box dimension seeks global covers, whereas the Assouad and lower dimensions seeklocal covers. The Assouad dimension of F ⊆ R d is defined bydim A F = inf (cid:40) s (cid:62) | ∃ C > ∀ < r < R : ∀ x ∈ F : N r ( B ( x, R ) ∩ F ) (cid:54) C (cid:18) Rr (cid:19) s (cid:41) . Similarly, the lower dimension of F is defined bydim L F = sup (cid:40) s (cid:62) | ∃ C > ∀ < r < R (cid:54) | F | : ∀ x ∈ F : N r ( B ( x, R ) ∩ F ) (cid:62) C (cid:18) Rr (cid:19) s (cid:41) . Importantly, for compact F we have dim L F (cid:54) dim H F (cid:54) dim B F (cid:54) dim B F (cid:54) dim A F. The Assouad and lower spectrum, introduced in [25], interpolate between the box dimensionsand the Assouad and lower dimensions in a meaningful way. They provide a parametrised family5f dimensions by fixing the relationship between the two scales r < R used to define Assouadand lower dimension. Studying the dependence on the parameter within this family thus yieldsfiner and more nuanced information about the local structure of the set. For example, one mayunderstand which scales ‘witness’ the behaviour described by the Assouad and lower dimensions.For θ ∈ (0 , Assouad spectrum of F is given bydim θ A F = inf (cid:40) s (cid:62) | ∃ C > ∀ < r < ∀ x ∈ F : N r ( B ( x, r θ ) ∩ F ) (cid:54) C (cid:18) r θ r (cid:19) s (cid:41) and the lower spectrum of F bydim θ L F = sup (cid:40) s (cid:62) | ∃ C > ∀ < r < ∀ x ∈ F : N r ( B ( x, r θ ) ∩ F ) (cid:62) C (cid:18) r θ r (cid:19) s (cid:41) . It was shown in [25] that for a bounded set F ⊆ R d , we havedim B F (cid:54) dim θ A F (cid:54) min (cid:26) dim A F, dim B F − θ (cid:27) (2.1)dim L F (cid:54) dim θ L F (cid:54) dim B F. In particular, dim θ A F → dim B F as θ →
0. Whilst the analogous statement does not hold forthe lower spectrum in general, it was proved in [22, Theorem 6.3.1] that dim θ L F → dim B F as θ → F satisfies a strong form of dynamical invariance. Whilst the fractals we studyare not quite covered by this result, we shall see that this interpolation holds nevertheless. Thelimit lim θ → dim θ L F is known to exist in general (see [13, Theorem 1.1]) and moreover can take anyvalue in the range [dim qL F, dim B F ].We write dim qA F to denote the quasi-Assouad dimension of F and dim qL F to denote the quasi-lower dimension of F . These were introduced in [14] and [31] and, due to work in [13, 23, 27],it is known that lim θ → dim θ A F = dim qA F and lim θ → dim θ L F = dim qL F. In many cases of interest, including the ones we discuss in this paper, we have dim qA F = dim A F and dim qL F = dim L F , so we make no further mention of the quasi-dimensions.There is an analogous dimension theory of measures, and the interplay between the dimensiontheory of fractals and the measures they support is fundamental to fractal geometry, especiallyin the dimension theory of dynamical systems. Let µ be a locally finite Borel measure on R d ,i.e. µ ( B ( x, r )) < ∞ for all x ∈ R d and r >
0. We write supp( µ ) = { x ∈ R d | µ ( B ( x, r )) > r > } for the support of µ . We say that µ is fully supported on a set F ⊆ R d ifsupp( µ ) = F . Similar to above, we write dim H µ for the (lower) Hausdorff dimension of µ andnote that dim H µ (cid:54) supp( µ ) and, for compact F ,dim H F = sup { dim H µ | supp( µ ) ⊆ F } , Assouad dimension of µ with supp( µ ) = F is defined bydim A µ = inf (cid:40) s (cid:62) | ∃ C > ∀ < r < R < | F | : ∀ x ∈ F : µ ( B ( x, R )) µ ( B ( x, r )) (cid:54) C (cid:18) Rr (cid:19) s (cid:41) and, provided | supp( µ ) | = | F | >
0, the lower dimension of µ is given bydim L µ = sup (cid:40) s (cid:62) | ∃ C > ∀ < r < R < | F | : ∀ x ∈ F : µ ( B ( x, R )) µ ( B ( x, r )) (cid:62) C (cid:18) Rr (cid:19) s (cid:41) and otherwise it is 0. By convention we assume that inf ∅ = ∞ .These were introduced in [28, 29], where they were referred to as the upper and lower regularitydimensions of µ . It is well known (see [22, Lemma 4.1.2]) that for a Borel probability measure µ supported on a closed set F ⊆ R d , we havedim L µ (cid:54) dim L F (cid:54) dim A F (cid:54) dim A µ and furthermore, we have the stronger fact thatdim A F = inf { dim A µ | µ is a Borel probability measure fully supported on F } and dim L F = sup { dim L µ | µ is a Borel probability measure fully supported on F } . For θ ∈ (0 , Assouad spectrum of µ with supp( µ ) = F is given bydim θ A µ = inf (cid:40) s (cid:62) | ∃ C > ∀ < r < | F | : ∀ x ∈ F : µ ( B ( x, r θ )) µ ( B ( x, r )) (cid:54) C (cid:18) r θ r (cid:19) s (cid:41) and provided | supp( µ ) | = | F | >
0, the lower spectrum of µ is given bydim θ L µ = sup (cid:40) s (cid:62) | ∃ C > ∀ < r < | F | : ∀ x ∈ F : µ ( B ( x, r θ )) µ ( B ( x, r )) (cid:62) C (cid:18) r θ r (cid:19) s (cid:41) and otherwise it is 0. It is known (see [20] for example) that for any measure µ ,dim L µ (cid:54) dim θ L µ (cid:54) dim θ A µ (cid:54) dim A µ and if µ is fully supported on a closed set F , thendim θ L µ (cid:54) dim θ L F (cid:54) dim θ A F (cid:54) dim θ A µ. The upper box dimension of µ with supp( µ ) = F is given bydim B µ = inf (cid:110) s | ∃ C > ∀ < r < | F | : ∀ x ∈ F : µ ( B ( x, r )) (cid:62) Cr s (cid:111) lower box dimension of µ is given bydim B µ = inf (cid:110) s | ∃ C > ∀ r > ∃ < r < r : ∀ x ∈ F : µ ( B ( x, r )) (cid:62) Cr s (cid:111) . If dim B µ = dim B µ , then we refer to the common value as the box dimension of µ , denoted bydim B µ . These definitions of the box dimension of a measure were introduced only recently in[20]. Similar to the case for Assouad dimension, we can express the box dimensions of a set F in terms of the box dimensions of measures supported on F . More precisely, for a non-emptycompact set F ⊆ R d , it was shown in [20] thatdim B F = inf (cid:8) dim B µ | µ is a finite Borel measure fully supported on F (cid:9) and dim B F = inf { dim B µ | µ is a finite Borel measure fully supported on F } . Furthermore, it was shown that the upper box dimension of µ can be related to the Assouadspectrum of µ in a similar manner to sets. In particular, for a measure µ supported on a compactset F and for θ ∈ (0 , B µ (cid:54) dim θ A µ (cid:54) min (cid:26) dim A µ, dim B µ − θ (cid:27) . For a more thorough study of hyperbolic geometry and Kleinian groups, we refer the reader to[2, 4, 35]. For d (cid:62)
1, we model ( d + 1)-dimensional hyperbolic space using the Poincar´e ballmodel D d +1 = { z ∈ R d +1 | | z | < } equipped with the hyperbolic metric d H defined by ds = 2 | dz | − | z | and we call the boundary S d = { z ∈ R d +1 | | z | = 1 } the boundary at infinity of the space( D d +1 , d H ). We denote by Con( d ) the group of orientation-preserving 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 withthe analogous metric.We say that a group is Kleinian if it is a discrete subgroup of Con( d ), and given a Kleiniangroup Γ, the limit set of Γ is defined to be L (Γ) = Γ( ) \ Γ( )where = (0 , . . . , ∈ D d +1 . It is well known that L (Γ) is a compact Γ-invariant subset of S d , see Figure 1. If L (Γ) contains zero, one or two points, it is said to be elementary , andotherwise it is non-elementary . In the non-elementary case, L (Γ) is a perfect set, and often8as a complicated fractal structure. We consider geometrically finite Kleinian groups. Roughlyspeaking, this means that there is a fundamental domain with finitely many sides (we refer thereader to [10] for further details). We define the
Poincar´e exponent of a Kleinian group Γ to be δ = inf s > | (cid:88) g ∈ Γ e − sd H ( ,g ( )) < ∞ . Due to work of Patterson and Sullivan [41, 48], it is known that for a non-elementary geomet-rically finite Kleinian group Γ, the Hausdorff dimension of the limit set is equal to δ . It wasdiscovered independently by Bishop and Jones [8, Corollary 1.5] and Stratmann and Urba´nski[44, Theorem 3] that the box and packing dimensions of the limit set are also equal to δ .From now on we only discuss the non-elementary geometrically finite case. We denote by µ the Patterson-Sullivan measure , which is a measure first constructed by Patterson in [41]. Thegeometry of Γ, L (Γ) and µ are heavily related. For example, µ is a conformal Γ-ergodic Borelprobability measure which is fully supported on L (Γ). Moreover, µ has Hausdorff, packing andentropy dimension equal to δ , see [8, 41, 47, 48].The Assouad and lower dimensions of µ and limit sets of non-elementary geometrically finiteKleinian groups were dealt with in [21]. To state the results, we require some more notation. LetΓ be a non-elementary Kleinian group which contains at least one parabolic point, and denoteby P ⊆ L (Γ) the countable set of parabolic 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 tothe boundary S d at p ) such that they are pairwise disjoint, do not contain the point , and havethe property that for each g ∈ Γ and p ∈ P , we have g ( H p ) = H g ( p ) , see [44, 47].We note that for any p ∈ P , the stabiliser of p denoted by Stab( p ) cannot contain any loxodromicelements, as this would violate the discreteness of Γ. We denote by k ( p ) the maximal rank of afree abelian subgroup of Stab( p ), which must be generated by k ( p ) parabolic elements which allfix p , and call this the rank of p . We write k min = min { k ( p ) | p ∈ P } k max = max { k ( p ) | p ∈ P } . It was proven in [48] that δ > k max /
2. In [21], the following was proven:
Theorem 2.1.
Let Γ be a non-elementary geometrically finite Kleinian group. Then dim A L (Γ) = max { δ, k max } dim L L (Γ) = min { δ, k min } dim A µ = max { δ − k min , k max } dim L µ = min { δ − k max , k min } . We resolve the Assouad and lower spectra of µ and L (Γ) in Theorems 3.1 and 3.2. An importantresult used in the proofs of the box and Assouad dimensions of limit sets is Stratmann and9elani’s global measure formula , derived in [47], which gives a formula for the measure of anyball centred in the limit set up to uniform constants. More precisely, given z ∈ L (Γ) and T > z T ∈ D d +1 to be the point on the geodesic ray joining and z which is hyperbolicdistance T from . We write S ( z, T ) ⊂ S d to denote the shadow at infinity of the d -dimensionalhyperplane passing through z T which is normal to the geodesic joining and z . The globalmeasure 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 ρ ( z, T ) = inf { d H ( z T , y ) | y / ∈ H p } if z T ∈ H p for some p ∈ P and 0 otherwise. Basic hyperbolic geometry shows that S ( z, T ) is aEuclidean ball centred at z with radius comparable to e − T , and so an immediate consequenceof (2.2) is the following. Theorem 2.2 (Global Measure Formula I) . Let z ∈ L (Γ) , T > . Then we have µ ( B ( z, e − T )) ≈ e − T δ e − ρ ( z,T )( δ − k ( z,T )) . We will repeatedly make use of this fact throughout. An easy consequence of Theorem 2.2 isthat if Γ contains no parabolic points, thendim A L (Γ) = dim L L (Γ) = dim A µ = dim L µ = dim B µ = δ, and dim θ A L (Γ) = dim θ A µ = dim θ L L (Γ) = dim θ L µ = δ for all θ ∈ (0 , d = 2 and the boundary S has been identifiedwith R ∪ {∞} . Parabolic points with rank 1 are easily identified.10 .3 Rational maps and Julia sets For a more detailed discussion on the dynamics of rational maps, see [5, 12, 40]. Let T : ˆ C → ˆ C denote a rational map, and denote by J ( T ) the Julia set of T , which is equal to the closure ofthe repelling periodic points of T , see Figure 2. The Julia set is closed and T -invariant. We mayassume that J ( T ) is a compact subset of C by a standard reduction. If this is not the case, thenwe can conjugate a point z / ∈ J ( T ) to ∞ via a M¨obius inversion and then the closedness of theresulting Julia set ensures it lies in a bounded region of C . This is essentially just choosing adifferent point on the Riemann sphere to represent the point at infinity. A periodic point ξ ∈ ˆ C with period p is said to be rationally indifferent if( T p ) (cid:48) ( ξ ) = e πiq for some q ∈ Q . We say that T is parabolic if J ( T ) contains no critical points of T , but containsat least one rationally indifferent point. We will assume throughout that T is parabolic. Wedenote by Ω the finite set of parabolic points of T , and let Ω = { ξ ∈ Ω | T ( ξ ) = ξ, T (cid:48) ( ξ ) = 1 } . As J ( T n ) = J ( T ) for every n ∈ N , we may assume without loss of generality that Ω = Ω . We write h = dim H J ( T ). It was proven in [16] that h can also be characterised by the smallestzero of the function P ( t ) = P ( T, − t log | T (cid:48) | )where the P on the right denotes the topological pressure . A similar result for hyperbolic Juliasets with ‘smallest’ replaced with ‘only’ is often referred to as the Bowen-Manning-McCluskeyformula , see [11, 34].We recall, see [18, 45], that for each ω ∈ Ω , we can find a ball U ω = B ( ω, r ω ) with sufficientlysmall radius such that on B ( ω, r ω ), there exists a unique holomorphic inverse branch T − ω of T such that T − ω ( ω ) = ω . For a parabolic point ω ∈ Ω , the Taylor series of T about ω is of theform z + a ( z − ω ) p ( ω )+1 + · · · . We call p ( ω ) the petal number of ω , and we write p min = min { p ( ω ) | ω ∈ Ω } p max = max { p ( ω ) | ω ∈ Ω } . It was proven in [1] that h > p max / (1 + p max ). We define the set of pre-parabolic points J p ( T )by J p ( T ) = ∞ (cid:83) k =0 T − k ( Ω ). It was proven in [16] that there exists a constant C > ξ ∈ J ( T ) \ J p ( T ), we can associate a unique maximal sequence of integers n j ( ξ ) suchthat for each j ∈ N , the inverse branches T − n j ( ξ ) ξ are well defined on B ( T n j ( ξ ) ( ξ ) , C ). We call J r ( T ) = J ( T ) \ J p ( T ) the radial Julia set . Following [45, 46], we define r j ( ξ ) = | ( T n j ( ξ ) ) (cid:48) ( ξ ) | − r j ( ξ )) j ∈ N the hyperbolic zoom at ξ . Similarly, for each ξ ∈ J p ( T ), wecan associate its terminating hyperbolic zoom ( r j ( ξ )) j ∈{ ,...,l } . We also require the concept of a canonical ball , see [46]. Let ω ∈ Ω , and let I ( ω ) = T − ( ω ) \ { ω } . Then for each integer n (cid:62) canonical radius r ξ at ξ ∈ T − n ( I ( ω )) by r ξ = | ( T n ) (cid:48) ( ξ ) | − and we call B ( ξ, r ξ ) the canonical ball . We will use the fact that r ξ ≈ r l , where r l is the lastelement in the terminating hyperbolic zoom at ξ .Due to work from Aaronson, Denker and Urba´nski [1, 15, 16] it is known that given a parabolicrational map T , there exists a unique h -conformal measure m supported on J ( T ), i.e. m is aprobability measure such that for each Borel set F ⊂ J ( T ) on which T is injective, m ( T ( F )) = (cid:90) F | T (cid:48) ( ξ ) | h dm ( ξ ) . In [18], it was shown that m has Hausdorff dimension h , and also that the box and packingdimensions of J ( T ) are equal to h . It also follows from, for example, [46] that m is exactdimensional and therefore the packing and entropy dimensions are also given by h . We resolvethe Assouad and lower dimensions and spectra of J ( T ) and m in Theorems 3.3 and 3.4. Similarto the Kleinian setting, it was shown in [45] that m also has an associated global measure formulawhich we will make use of throughout. Theorem 2.3 (Global Measure Formula II) . Let T be a parabolic rational map with Julia set J ( T ) of Hausdorff dimension h . Let m denote the associated h-conformal measure supportedon J ( T ) . Then there exists a function φ : J ( T ) × R + → R + such that for all ξ ∈ J ( T ) and < r < | J ( T ) | , we have m ( B ( ξ, r )) ≈ r h φ ( ξ, r ) . The values of φ are determined as follows: i) Suppose ξ ∈ J r ( T ) has associated optimal sequence ( n j ( ξ )) j ∈ N and hyperbolic zooms ( r j ( ξ )) j ∈ N and r is such that r j +1 ( ξ ) (cid:54) r < r j ( ξ ) for some j ∈ N and T k ( ξ ) ∈ U ω for all n j ( ξ ) < k < n j +1 ( ξ ) and for some ω ∈ Ω . Then φ ( ξ, r ) ≈ (cid:16) rr j ( ξ ) (cid:17) ( h − p ( ω ) r > r j ( ξ ) (cid:16) r j +1 ( ξ ) r j ( ξ ) (cid:17) p ( ω ) (cid:16) r j +1 ( ξ ) r (cid:17) h − r (cid:54) r j ( ξ ) (cid:16) r j +1 ( ξ ) r j ( ξ ) (cid:17) p ( ω ) . ii) Suppose ξ ∈ J p ( T ) has associated terminating optimal sequence ( n j ( ξ )) j =1 ,...,l and hyperboliczooms ( r j ( ξ )) j =1 ,...,l . Suppose T n l ( ξ ) ( ξ ) = ω for some ω ∈ Ω . If r > r l ( ξ ) , the values of φ aredetermined as in the radial case, and if r (cid:54) r l ( ξ ) , then φ ( ξ, r ) ≈ (cid:18) rr l ( ξ ) (cid:19) ( h − p ( ω ) . J ( T ) contains no parabolic points, then a simpleconsequence of Theorem 2.3 is thatdim A J ( T ) = dim L J ( T ) = dim A m = dim L m = dim B m = h and dim θ A J ( T ) = dim θ A m = dim θ L J ( T ) = dim θ L m = h for all θ ∈ (0 , J ( T ) contains at least one parabolicpoint.Figure 2: An example of a parabolic Julia set. Parabolic points with petal number 4 are easilyspotted. Given the theory established above, many direct correspondences between the Kleinian groupand rational map settings are already evident. We discuss some of these in detail here to putour results in a wider context. For us, the most important entry in the Sullivan dictionary(mentioned above) is that, in both cases, the Hausdorff, box and packing dimensions of theassociated invariant fractal and the Hausdorff, packing and entropy dimensions of the associatedconformal measure are all equal and given by a natural critical exponent. The relationshipbetween Kleinian groups and rational maps goes much deeper than this, however, and nowserves to connect many problems and communities across conformal geometry. See work ofSullivan, McMullen, and many others [37, 39, 49]. We can draw some comparisons between thetheory we have discussed above as follows: 13leinian JuliaKleinian group Γ rational map T Kleinian limit set L (Γ) Julia set J ( T )Poincar´e exponent δ critical exponent h Patterson-Sullivan measure µ h -conformal measure m global measure formula for µ global measure formula for m set of parabolic points P set of pre-parabolic points J p ( T )rank of parabolic point k ( p ) petal number of parabolic point p ( ω )dimension bound δ > k max / h > p max / (1 + p max )horoballs canonical ballsTable 1: Some ‘entries’ in the Sullivan dictionary.With the results we prove in this paper, we are able to provide several new entries in the Sullivandictionary. Perhaps more strikingly, our work also provides several new ‘non-entries’, that is,where the correspondence between the two settings breaks down. This is novel in the setting ofdimension theory, where the Sullivan dictionary provides a particularly strong correspondence.Our new entries and non-entries are discussed in Section 3.3. We assume throughout that Γ < Con( d ) is a non-elementary geometrically finite Kleinian groupcontaining at least one parabolic element, and write L (Γ) to denote the associated limit setand µ to denote the associated Patterson-Sullivan measure. Recall that the Assouad and lowerdimensions of L (Γ) and µ were found in [21], see Theorem 2.1. Our first result gives formulaefor the Assouad and lower spectrum of µ , as well as the box dimension of µ . Theorem 3.1.
Let θ ∈ (0 , . Then i) dim B µ = max { δ, δ − k min } . ii) If δ < k min , then dim θ A µ = δ + min (cid:26) , θ − θ (cid:27) ( k max − δ ) , if k min (cid:54) δ < ( k min + k max ) / , then dim θ A µ = 2 δ − k min + min (cid:26) , θ − θ (cid:27) ( k min + k max − δ ) and if δ (cid:62) ( k min + k max ) / , then dim θ A µ = 2 δ − k min . iii) If δ > k max , then dim θ L µ = δ − min (cid:26) , θ − θ (cid:27) ( δ − k min ) , f ( k min + k max ) / < δ (cid:54) k max , then dim θ L µ = 2 δ − k max − min (cid:26) , θ − θ (cid:27) (2 δ − k min − k max ) and if δ (cid:54) ( k min + k max ) / , then dim θ L µ = 2 δ − k max . We prove Theorem 3.1 in Sections 4.2 - 4.4. The next theorem provides formulae for the Assouadand lower spectra of L (Γ). Theorem 3.2.
Let θ ∈ (0 , . i) If δ < k max , then dim θ A L (Γ) = δ + min (cid:26) , θ − θ (cid:27) ( k max − δ ) and if δ (cid:62) k max , then dim θ A L (Γ) = δ. ii) If δ (cid:54) k min , then dim θ L L (Γ) = δ , and if δ > k min , then dim θ L L (Γ) = δ − min (cid:26) , θ − θ (cid:27) ( δ − k min ) . We prove Theorem 3.2 in Sections 4.5 and 4.6. We defer a detailed discussion of these resultsuntil Section 3.3, where they will be compared with the analogous results in the rational mapsetting.
We assume throughout that T is a parabolic rational map, and write J ( T ) to denote the as-sociated Julia set and m to denote the associated h -conformal measure. We start with thedimensions and dimension spectra of m . Theorem 3.3.
Let θ ∈ (0 , . Then i) dim B m = max { h, h + ( h − p max } . ii) dim A m = max { , h + ( h − p max } . iii) dim L m = min { , h + ( h − p max } . iv) If h < , then dim θ A m = h + min (cid:26) , θ p max − θ (cid:27) (1 − h ) and if h (cid:62) , then dim θ A m = h + ( h − p max . v) If h < , then dim θ L m = h + ( h − p max and if h (cid:62) , then dim θ L m = h + min (cid:26) , θ p max − θ (cid:27) (1 − h ) . We prove Theorem 3.3 in Sections 5.2 - 5.6. Turning our attention to J ( T ), we have the following.15 heorem 3.4. Let θ ∈ (0 , . Then i) dim A J ( T ) = max { , h } . ii) dim L J ( T ) = min { , h } . iii) If h < , then dim θ A J ( T ) = h + min (cid:26) , θ p max − θ (cid:27) (1 − h ) and if h (cid:62) , then dim θ A J ( T ) = h. iv) If h < , then dim θ L J ( T ) = h and if h (cid:62) , then dim θ L J ( T ) = h + min (cid:26) , θ p max − θ (cid:27) (1 − h ) . We prove Theorem 3.4 in Sections 5.7 - 5.10. Interpolation between dimensions
In both settings, the Assouad spectrum always interpolates between the upper box and As-souad dimensions of the respective sets and measures regardless of what form it takes, that is,lim θ → dim θ A F = dim A F where F can be replaced by µ, L (Γ) , m or J ( T ). Recall that this interpo-lation does not hold in general. Similar interpolation holds as θ → Failure to witness the box dimension of measures
For the measures µ and m , the lower spectrum does not generally tend to the box dimensionas θ →
0. In fact, if the lower spectrum does tend to the box dimension as θ →
0, then it isconstant and δ = k min = k max (in the Kleinian setting) and h = 1 (in the Julia setting).3) Formulae for the spectra
In fact, the formulae for the Assouad spectra provide an even stronger correspondence than thatdescribed above. For F a given set or measure, consider ρ = inf { θ ∈ (0 , | dim θ A F = dim A F } . It turns out that ρ provides the unique phase transition in the spectra and, moreover,dim θ A F = min (cid:26) dim B F + (1 − ρ ) θ (1 − θ ) ρ (dim A F − dim B F ) , dim A F (cid:27) where F can be replaced by µ, L (Γ) , m or J ( T ). This formula, and the fact that the Assouadspectrum can be expressed purely in terms of the phase transition and the box and Assouad di-mensions, has appeared in a variety of settings, see [22, Section 17.7] and the discussion therein.For example, this formula also holds for self-affine Bedford-McMullen carpets [24]. The phase16ransition ρ often has a natural ‘geometric significance’ for the objects involved and has led toa new ‘dictionary’ extending beyond the setting discussed here.4) The phase transition and the Hausdorff dimension bound
There is a correspondence between the phase transition ρ and the general lower bounds forthe Hausdorff dimension. Applying (2.1) shows that, for any non-empty bounded set F , ρ (cid:62) − dim B F/ dim A F. When the spectra are non-constant, in the Kleinian setting we always have ρ = 1 /
2, and in the Julia setting we always have ρ = 1 / (1 + p max ). Combining this with thegeneral Hausdorff dimension bounds δ > k max / k max ρ and h > p max / (1 + p max ) = p max ρ inboth settings yields ρ > − dim B F/ dim A F , showing that the upper bound from (2.1) is neverachieved in either setting.5) The realisation problem
Given the interplay between dimensions of sets and dimensions of measures seen in Section 2.1,one may ask if it is possible to construct an (invariant) measure ν which realises the dimensionsof an (invariant) set F , that is, dim ν = dim F . One can ask this about a particular choice ofdimension dim or if a single measure can be constructed to solve the problem for several notionsof dimension simultaneously. We note that the measures µ and m always realise the Hausdorffdimensions of L (Γ) and J ( T ) respectively. As for the Assouad and lower dimensions, we notethat µ realises the Assouad dimension of L (Γ) when δ (cid:54) ( k min + k max ) / δ (cid:62) ( k min + k max ) /
2. Similarly, for m to realise the Assouad dimension of J ( T )we require h (cid:54)
1, and for m to realise the lower dimension of J ( T ) we require h (cid:62)
1. Notethat a similar relationship holds for the box dimension too: in the Kleinian setting we require δ (cid:54) k min and in the Julia setting we require h (cid:54) A special case
Finally, we observe that in the (very) special case k min = k max = p max = 1, the formulae forthe Assouad type dimensions and spectra are identical in the Kleinian and Julia settings. Doesthis suggest that this special case is one where we can expect the Sullivan dictionary to yield aparticularly strong correspondence in other settings? Assouad dimension
Our results show that Julia sets of parabolic rational maps can never have full Assouad dimen-sion, that is, we always have dim A J ( T ) <
2. This uses our result together with [1, Theorem8.8] which proves that h <
2. This is in stark contrast to the situation for Kleinian limit setswhere it is perfectly possible for the Assouad dimension to be full, that is, Γ ∈ Con( d ) withdim A L (Γ) = d = dim S d for any integer d (cid:62)
1, even when the limit set is nowhere dense in S d which implies dim H L (Γ) < d (see [50, Theorem D]). We note that dim A J ( T ) < R d must have Assouad dimensionstrictly less than d . Our results can thus be viewed as a refinement of the observation that Julia17ets are porous.2) Lower dimension
Our results, together with the standard bound h > p max / (1 + p max ), show that dim L J ( T ) =min { , h } > p max / (1 + p max ), that is, the lower dimension respects the general lower boundsatisfied by the Hausdorff dimension. Again, this is in stark contrast to the situation forKleinian limit sets where the standard bound for Hausdorff dimension is δ > k max / L L (Γ) = min { k min , δ } < k max / Relationships between dimensions
An interesting aspect of dimension theory is to consider what configurations are possible betweenthe different notions of dimension in a particular setting. We refer the reader to [22, Section17.5] for a more general discussion of this. Our results show thatdim L J ( T ) < dim H J ( T ) < dim A J ( T )is impossible in the Julia setting but the analogous configuration is possible in the Kleiniansetting. Configuration Fuchsian Kleinian JuliaL=H=A (cid:88) (cid:88) (cid:88) L=H < A (cid:88) (cid:88) (cid:88) L < H=A × (cid:88) (cid:88) L < H < A × (cid:88) × Table 2: Summarising the possible relationships between the lower, Hausdorff, and Assouaddimensions of Fuchsian limit sets, Kleinian limits sets and parabolic Julia sets with the obviouslabelling. The label ‘Fuchsian’ refers to the Kleinian setting when d = 1. The symbol (cid:88) means that the configuration is possible, and × means the configuration is impossible. Inother situations it is interesting to add box dimension into this discussion, but here this alwayscoincides with Hausdorff dimension and so we omit it.4) The forms of the spectra
Turning our attention to the measures, we note that the Assouad and lower spectra of µ in theKleinian setting can take 3 different forms, in comparison to the Julia setting where we onlyhave 2 possibilities for m . We also note that in the Kleinian setting, both k min and k max appearin the formulae for the Assouad and lower spectra, sometimes simultaneously, but in the Juliasetting only p max appears.5) The realisation problem for dimension spectra
One can also extend the realisation problem to the Assouad and lower spectra: when does an (in-variant) set support an (invariant) measure with equal Assouad or lower spectra? In the Kleiniansetting, we have dim θ A µ = dim θ A L (Γ) when δ (cid:54) k min and dim θ L µ = dim θ L L (Γ) when δ (cid:62) k max .This can leave a gap when k min < δ < k max where neither of the spectra are realised by the18atterson-Sullivan measure. This is in contrast to the Julia setting where dim θ A m = dim θ A J ( T )when h (cid:54) θ L m = dim θ L J ( T ) when h (cid:62)
1, and so at least one of the spectra is alwaysrealised by m .6) Dimension spectra as a fingerprint
Suppose it is not true that k min = k max = p max = 1. Then simply by looking at plots of theAssouad and lower spectra, one can determine whether the set in question is a Kleinian limit setor a Julia set. Whenever the Assouad spectrum is non-constant in either the Kleinian or Juliasetting, there is a unique phase transition at ρ = inf { θ ∈ (0 , | dim θ A F = dim A F } . Moreover ρ = 1 / ρ = 1 / (1 + p max ) in the Julia setting. Note that in theKleinian setting the phase transition is constant across all Kleinian limit sets, whereas in theJulia setting the phase transition depends on the rational map T . This allows one to distinguishbetween the Assouad spectrum of a Kleinian limit set and a Julia set just by looking at thephase transition, provided p max (cid:54) = 1. However, even if p max = 1, the spectra will still distinguishbetween the two settings provided we do not also have k min = k max = 1. In order to give a visual idea of the results of Theorems 3.1 - 3.4, we plot the Assouad and lowerspectra for some examples. In the Kleinian setting, we assume that d = 2 throughout for a moredirect comparison with the Julia setting, and plot the following cases: when δ (cid:54) k min , when k min < δ < k max and when δ (cid:62) k max . In the Julia setting, we consider when h < h (cid:62)
1. The following are plots of the Assouad and lower spectra as functions of θ ∈ (0 , µ and m are plotted with dashed lines, and the spectra of L (Γ) and J ( T ) by solidlines. The Assouad spectra are plotted in black and the lower spectra are plotted in grey. . . . . . . . . θ . . . . . . . . θ Figure 3: On the left, we have a Kleinian limit set with δ = 0 . k min = k max = 1, and onthe right we have a Julia set with h = 0 . p max = 2.19 . . . . θ . . . . θ Figure 4: On the left, we have a Kleinian limit set with δ = 1 . k min = k max = 1, and onthe right we have a Julia set with h = 1 . p max = 4. . . . . θ Figure 5: In this case, we have aKleinian limit set with δ = 1 . k min =1 and k max = 2. Recall that in theJulia setting, we always have eitherdim θ A m = dim θ A J ( T ) or dim θ L m =dim θ L J ( T ), and so these plots are im-possible in the Julia setting. As many of the proofs in the Kleinian setting are reliant on horoballs, many of the results in thissection involve establishing estimates for quantities regarding horoballs, including the ‘escapefunctions’ ρ ( z, T ). We start with the following lemma: one should think of the circle involvedas a 2-dimensional slice of a horoball. Lemma 4.1 (Circle Lemma) . Let
R > , and consider a circle centred at (0 , R ) with radius R ,parametrised by x ( θ ) = R sin θ and y ( θ ) = R (1 − cos θ ) (0 (cid:54) θ < π ) . For sufficiently small θ , wehave (cid:112) Ry ( θ )2 (cid:54) x ( θ ) (cid:54) (cid:112) Ry ( θ ) (4.1)( x ( θ )) R (cid:54) y ( θ ) (cid:54) x ( θ )) R . (4.2)
Proof.
By Taylor’s Theorem, we have x ( θ ) = R (cid:18) θ − θ
3! + θ − . . . (cid:19) y ( θ ) = R (cid:18) θ − θ
4! + θ − . . . (cid:19) .
20e note that x ( θ ) (cid:54) Rθ for all θ and y ( θ ) (cid:62) Rθ / θ . So we get x ( θ ) (cid:54) (cid:112) Ry ( θ )for sufficiently small θ . Similarly, we have that x ( θ ) (cid:62) Rθ/ y ( θ ) (cid:54) Rθ for sufficientlysmall θ , which gives y ( θ ) (cid:54) x ( θ )) R as required.We also require the following lemma to easily estimate the ‘escape function’ at a parabolic fixedpoint. Lemma 4.2 (Parabolic Centre Lemma) . Let p ∈ L (Γ) be a parabolic fixed point with associatedstandard horoball H p . Then we have ρ ( p, T ) ∼ T as T → ∞ , and for sufficiently large T > wehave k ( p, T ) = k ( p ) .Proof. Let p S be the ‘tip’ of the horoball H p , in other words the point on the horoball H p whichlies on the geodesic joining and p . It is obvious that p T ∈ H p ⇐⇒ T (cid:62) S , so for sufficientlylarge T we have k ( p, T ) = k ( p ). Also note that1 (cid:62) ρ ( p, T ) T = d H ( p T , p S ) T = T − ST → T → ∞ as required.One may note that some of the formulae for dim θ A µ and dim θ L µ involve both k min and k max .Consequently, in these cases we will need to consider two standard horoballs, where we drag onecloser to the other. One could think of the following lemma as saying that the images of thishoroball ‘fill in the gaps’ under the fixed horoball. Lemma 4.3 (Horoball Radius Lemma) . Let p, p (cid:48) ∈ L (Γ) be two parabolic fixed points withassociated standard horoballs H p and H p (cid:48) respectively, and let f be a parabolic element whichfixes p . Then for sufficiently large n , | f n ( p (cid:48) ) − p | ≈ n and | f n ( H p (cid:48) ) | = | H f n ( p (cid:48) ) | ≈ n . Proof.
By considering an inverted copy of Z converging to p , one sees that | f n ( p (cid:48) ) − p | ≈ /n , sowe need only prove that | H f n ( p (cid:48) ) | ≈ /n . Note that clearly | H f n ( p (cid:48) ) | (cid:46) /n , as otherwise thehoroballs would eventually overlap with H p , using Lemma 4.1 and the fact that | f n ( p (cid:48) ) − p | ≈ n ,contradicting the fact that our standard set of horoballs is chosen to be pairwise disjoint. Forthe lower bound, consider a point u (cid:54) = p (cid:48) on H p (cid:48) , and its images under the action of f . Let v denote the ‘shadow at infinity’ of u (see Figure 6), and note that for sufficiently large n , | f n ( v ) − p | ≈ 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 ) | ≈ /n (see Figure 7), and therefore | H f n ( p (cid:48) ) | (cid:38) /n , as required.21 p (cid:48) uv Figure 6: An illustration showing the points u and v . The dashed arc shows the horoball alongwhich the point u is pulled under the action of f . f n ( v ) f n ( u ) f n ( p (cid:48) ) ≈ n Figure 7: Applying Lemma 4.1, considering the dashed horoball.The following lemma is also necessary for estimating hyperbolic distance, essentially saying thatif z, u ∈ L (Γ) are sufficiently close, then d H ( z T , u T ) can be easily estimated. Lemma 4.4.
Let z, u ∈ L (Γ) and T > be large. If | z − u | ≈ e − T , then d H ( z T , u T ) ≈ . Proof.
In order to bound d H ( z T , u T ), we make use of a formula for hyperbolic distance, sometimesreferred to as the cross ratio. Given points P and Q , draw a geodesic between them whichintersects the boundary of the disc at points A and B such that A is closer to P than Q (seeFigure 8). Then we have d H ( P, Q ) = log | AQ || BP || AP || BQ | . (4.3)Using the formula above with P = z T and Q = u T , letting A = z (cid:48) and B = u (cid:48) , and also notingthat | AQ | = | BP | and | AP | = | BQ | , we have d H ( z T , u T ) ≈ log | z (cid:48) − u T | | z (cid:48) − z T | . (cid:48) u T u (cid:48) z uz (cid:48) z T Figure 8: The cross ratio visualised.Now, note that as | z − u | ≈ e − T , we also have | z (cid:48) − u | (cid:46) e − T , and so by the triangle inequalitywe have | z (cid:48) − u T | (cid:54) | z (cid:48) − u | + | u − u T | (cid:46) e − T . We also note that clearly | z (cid:48) − z T | (cid:38) e − T , so by applying (4.3) we get d H ( z T , u T ) (cid:46) log | z (cid:48) − u T | | z (cid:48) − z T | (cid:46) . As d H ( z T , u T ) >
0, we have d H ( z T , u T ) ≈
1, as required.Given a standard horoball H p and λ ∈ (0 , λH p the squeezed horoball which still hasbase point p , but has Euclidean diameter scaled by a factor of λ , i.e. | λH p | = λ | H p | . We alsowrite Π : D d +1 \ { } → S d to denote the projection defined by choosing Π( z ) ∈ S d such that , z , and Π( z ) are collinear. Given A ⊂ D d +1 \ { } , we call Π( A ) the shadow at infinity of A ,and note that for a horoball H p , Π( H p ) is a Euclidean ball with | Π( H p ) | ≈ | H p | . We require thefollowing lemma due to Stratmann and Velani [47, Corollary 3.5] regarding squeezed horoballs. Lemma 4.5.
Let H p be a standard horoball for some p ∈ P , and let λ ∈ (0 , . Then µ (Π( λH p )) ≈ λ δ − k ( p ) | H p | δ . We also require the following lemma due to Fraser [21, Lemma 5.2], which allows us to counthoroballs of certain sizes.
Lemma 4.6.
Let z ∈ L (Γ) and T > t > . For t sufficiently large, we have (cid:88) p ∈ P ∩ B ( z,e − t ) e − t > | H p | (cid:62) e − T | H p | δ (cid:46) ( T − t ) µ ( B ( z, e − t )) where P is the set of parabolic fixed points contained in L (Γ) . µ • We show dim B µ (cid:54) max { δ, δ − k min } .Let z ∈ L (Γ) and T >
0. We have µ ( B ( z, e − T )) (cid:38) e − T δ e − ρ ( z,T )( δ − k ( z,T )) (cid:62) e − T δ (cid:0) e − T (cid:1) max { ,δ − k min } = (cid:0) e − T (cid:1) max { δ, δ − k min } B µ (cid:54) max { δ, δ − k min } . • We show dim B µ (cid:62) max { δ, δ − k min } .Note that we have dim B µ (cid:62) dim B L (Γ) = δ , so it suffices to prove that dim B µ (cid:62) δ − k min , andtherefore we may assume that δ (cid:62) k min .Let p be a parabolic fixed point such that k ( p ) = k min , and let ε ∈ (0 , T >
0, we have µ ( B ( p, e − T )) (cid:46) e − T δ e − ρ ( p,T )( δ − k ( p )) (cid:54) e − T δ e − (1 − ε ) T ( δ − k min ) = (cid:0) e − T (cid:1) δ +(1 − ε )( δ − k min ) which proves dim B µ (cid:62) δ − k min − ε ( δ − k min ), and letting ε → µ δ < k min • Lower bound : We show dim θ A µ (cid:62) δ + min (cid:110) , θ − θ (cid:111) ( k max − δ ).Let θ ∈ (0 , p ∈ L (Γ) be a parabolic fixed point such that k ( p ) = k max , f be a parabolicelement fixing p , and n ∈ N be very large. Choose p (cid:54) = z ∈ L (Γ), and let z = f n ( z ), notingthat z → p as n → ∞ . We assume n is large enough to ensure that the geodesic joining and z intersects H p , and choose T > z T lies on the boundaryof H p .We now restrict our attention to the hyperplane H ( p, z, z T ) restricted to D d +1 . Define v to bethe point on H p ∩ H ( p, z, z T ) such that v lies on the quarter circle with centre z and Euclideanradius e − T θ (see Figure 9 below). We also consider 2 additional points u and w , where u is the‘shadow at infinity’ of v and w = u T θ . S d p zz T z T θ vwu ≈ e − Tθ Figure 9: An overview of the horoball H p along with our chosen points. We wish to find a lowerbound 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 using244.1) we have, for sufficiently large n , | z − p | (cid:46) (cid:112) | z − z T | (cid:46) e − T .p zz T (cid:46) e − T (cid:46) e − T Figure 10: Bounding | z − p | from above using Lemma 4.1.Also note that clearly | z − u | (cid:46) e − T θ . We can apply (4.2), this time considering the point v ,and so for sufficiently large n , we obtain | u − v | (cid:46) (cid:16) e − T + e − T θ (cid:17) (cid:46) e − min { T θ,T } .p uvzz T (cid:46) e − T + e − Tθ (cid:46) e − min { Tθ,T } Figure 11: Bounding | u − v | from above, again using Lemma 4.1.This gives us d H ( v, w ) = log | z − z T θ || u − v | (cid:62) log e − T θ /C C e − min { T θ,T } = min { T (1 − θ ) , T θ } − log( C C )for some constants C , C . We can apply Lemma 4.4 to deduce that d H ( z T θ , w ) (cid:54) C for someconstant C . Therefore, by the triangle inequality ρ ( z, T θ ) = d H ( z T θ , v ) (cid:62) d H ( w, v ) − d H ( z T θ , w ) (cid:62) min { T (1 − θ ) , T θ } − log( C C ) − C . (4.4)25y Theorem 2.2, we have µ ( B ( z, e − T θ )) µ ( B ( z, e − T )) ≈ e − δT θ e − δT e ρ ( z,T θ )( k max − δ ) (cid:38) (cid:16) e T (1 − θ ) (cid:17) δ e min { T (1 − θ ) ,T θ } ( k max − δ ) by (4 . (cid:16) e T (1 − θ ) (cid:17) δ (cid:16) e T (1 − θ ) (cid:17) min { , θ − θ } ( k max − δ ) = (cid:16) e T (1 − θ ) (cid:17) δ +min { , θ − θ } ( k max − δ ) which gives dim θ A µ (cid:62) δ + min (cid:26) , θ − θ (cid:27) ( k max − δ )as required. • Upper bound : We show dim θ A µ (cid:54) δ + min (cid:110) , θ − θ (cid:111) ( k max − δ ).Let z ∈ L (Γ), T >
0. Note that we havedim θ A µ (cid:54) dim A µ = k max so we assume that θ ∈ (0 , / µ ( B ( z, e − T θ )) µ ( B ( z, e − T )) ≈ e − δT θ e − δT e − ρ ( z,T θ )( δ − k ( z,T θ )) e − ρ ( z,T )( δ − k ( z,T )) (cid:46) (cid:16) e T (1 − θ ) (cid:17) δ e ρ ( z,T θ )( k max − δ ) (cid:54) (cid:16) e T (1 − θ ) (cid:17) δ e T θ ( k max − δ ) = (cid:16) e T (1 − θ ) (cid:17) δ + θ − θ ( k max − δ ) which gives dim θ A µ (cid:54) δ + θ − θ ( k max − δ )as required. k min (cid:54) δ < ( k min + k max ) / • Lower bound : We show dim θ A µ (cid:62) δ − k min + min (cid:110) , θ − θ (cid:111) ( k min + k max − δ ).Let θ ∈ (0 , / p, p (cid:48) ∈ L (Γ) be parabolic fixed points such that k ( p ) = k max and k ( p (cid:48) ) = k min ,and let f be a parabolic element fixing p . Let n be a large positive integer and let z = f n ( p (cid:48) ).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 12).We can make use of Lemma 4.4 to deduce that d H ( z T θ , p
T θ ) (cid:54) C C . Also, by Lemma 4.2, given ε ∈ (0 ,
1) we have that ρ ( p, T θ ) (cid:62) (1 − ε ) T θ for sufficiently large n . This gives ρ ( z, T θ ) (cid:62) ρ ( p, T θ ) − C (cid:62) (1 − ε ) T θ − C . (4.5)Finally, we note that as | z − p | = e − T θ , by Lemma 4.1 we have | H z | ≈ e − T θ (see Figure 13),which implies that ρ ( z, T ) (cid:62) log e − T θ /C C e − T = T (1 − θ ) − log( C C ) (4.6)for some constants C , C . pzz T z T θ p T θ e − Tθ Figure 12: Making use of Lemma 4.3 so we can choose our desired T . z z T ≈ e − T ≈ e − Tθ Figure 13: Calculating ρ ( z, T ) using Lemma 4.1.27pplying (2.2), we get µ ( B ( z, e − T θ )) µ ( B ( z, e − T )) ≈ e − δT θ e − δT e ρ ( z,T θ )( k max − δ ) e − ρ ( z,T )( δ − k min ) (cid:38) (cid:16) e T (1 − θ ) (cid:17) δ e (1 − ε ) T θ ( k max − δ )+(1 − θ ) T ( δ − k min ) by (4 .
5) and (4 . (cid:16) e T (1 − θ ) (cid:17) δ e T (1 − θ ) δ + T (1 − θ ) k min + T θ ( k min + k max − δ ) − εT θ ( k max − δ ) (cid:62) (cid:16) e T (1 − θ ) (cid:17) δ − k min + θ − θ ( k min + k max − δ ) − ε θ − θ ( k max − δ ) which proves dim θ A µ (cid:62) δ − k min + θ − θ ( k min + k max − δ ) − ε θ − θ ( k max − δ )and letting ε → θ (cid:62) / • Upper bound : We show dim θ A µ (cid:54) δ − k min + min (cid:110) , θ − θ (cid:111) ( k min + k max − δ ).Let z ∈ L (Γ), T > θ ∈ (0 , /
2) (the case when θ (cid:62) / z T and z T θ do not lie in a common standard horoball, we may use the factthat ρ ( z, T ) (cid:54) T (1 − θ ) − ρ ( z, T θ ) . (4.7)We have by Theorem 2.2 µ ( B ( z, e − T θ )) µ ( B ( z, e − T )) ≈ e − δT θ e − δT e − ρ ( z,T θ )( δ − k ( z,T θ )) e − ρ ( z,T )( δ − k ( z,T )) (cid:54) e − δT θ e − δT e − ρ ( z,T θ )( δ − k max ) e − ρ ( z,T )( δ − k min ) = (cid:16) e T (1 − θ ) (cid:17) δ e ρ ( z,T θ )( k max − δ )+ ρ ( z,T )( δ − k min ) (cid:54) (cid:16) e T (1 − θ ) (cid:17) δ e T θ ( k min + k max − δ )+ T (1 − θ )( δ − k min ) by (4 . (cid:54) (cid:16) e T (1 − θ ) (cid:17) δ − k min + θ − θ ( k min + k max − δ ) If z T and z T θ do lie in a common standard horoball H p and δ (cid:62) k ( p ), then we use the inequality | ρ ( z, T ) − ρ ( z, T θ ) | (cid:54) T (1 − θ ) . (4.8)Then we have µ ( B ( z, e − T θ )) µ ( B ( z, e − T )) ≈ e − δT θ e − δT e − ρ ( z,T θ )( δ − k ( z,T θ )) e − ρ ( z,T )( δ − k ( z,T )) = (cid:16) e T (1 − θ ) (cid:17) δ e ( ρ ( z,T ) − ρ ( z,T θ ))( δ − k ( p )) (cid:54) (cid:16) e T (1 − θ ) (cid:17) δ e T (1 − θ )( δ − k min ) by (4 . (cid:16) e T (1 − θ ) (cid:17) δ − k min (cid:54) (cid:16) e T (1 − θ ) (cid:17) δ − k min + θ − θ ( k min + k max − δ ) ρ ( z, T ) − ρ ( z, T θ ) (cid:62)
0, and otherwise µ ( B ( z, e − T θ )) µ ( B ( z, e − T )) ≈ (cid:16) e T (1 − θ ) (cid:17) δ e ( ρ ( z,T ) − ρ ( z,T θ ))( δ − k ( p )) (cid:54) (cid:16) e T (1 − θ ) (cid:17) δ (cid:54) (cid:16) e T (1 − θ ) (cid:17) δ − k min + θ − θ ( k min + k max − δ ) If δ < k ( p ), then we use ρ ( z, T θ ) − ρ ( z, T ) (cid:54) T θ. (4.9)By Theorem 2.2, µ ( B ( z, e − T θ )) µ ( B ( z, e − T )) ≈ (cid:16) e T (1 − θ ) (cid:17) δ e ( ρ ( z,T ) − ρ ( z,T θ ))( δ − k ( p )) (cid:54) (cid:16) e T (1 − θ ) (cid:17) δ e T θ ( k ( p ) − δ ) by (4 . (cid:54) (cid:16) e T (1 − θ ) (cid:17) δ + θ − θ ( k max − δ ) (cid:54) (cid:16) e T (1 − θ ) (cid:17) δ − k min + θ − θ ( k min + k max − δ ) In all cases, we have dim θ A µ (cid:54) δ − k min + θ − θ ( k min + k max − δ )as required. δ (cid:62) ( k min + k max ) / • We show dim θ A µ = 2 δ − k min .This follows easily, since 2 δ − k min = dim B µ (cid:54) dim θ A µ (cid:54) dim A µ = 2 δ − k min and so dim θ A µ =2 δ − k min , as required. µ The proofs of the bounds for the lower spectrum follow similarly to the Assouad spectrum, andso we only sketch the arguments. δ > k max • Upper bound : We show dim θ L µ (cid:54) δ − min (cid:110) , θ − θ (cid:111) ( δ − k min ).Let θ ∈ (0 , p ∈ L (Γ) be a parabolic fixed point such that k ( p ) = k min , f be a parabolicelement fixing p , and n ∈ N be very large. Choose p (cid:54) = z ∈ L (Γ), and let z = f n ( z ). We choose T > z T is the ’exit point’ from H p . Identically to the lower bound in Section 4.3.1,we have that ρ ( z, T θ ) (cid:62) min { T (1 − θ ) , T θ } − C C . Applying Theorem 2.2, we have µ ( B ( z, e − T θ )) µ ( B ( z, e − T )) ≈ e − δT θ e − δT e − ρ ( z,T θ )( δ − k min ) (cid:46) (cid:16) e T (1 − θ ) (cid:17) δ e − min { T (1 − θ ) ,T θ } ( δ − k min ) = (cid:16) e T (1 − θ ) (cid:17) δ (cid:16) e T (1 − θ ) (cid:17) − min { , θ − θ } ( δ − k min ) = (cid:16) e T (1 − θ ) (cid:17) δ − min { , θ − θ } ( δ − k min ) which gives dim θ L µ (cid:54) δ − min { , θ − θ } ( δ − k min )as required. • Lower bound : We show dim θ L µ (cid:62) δ − min (cid:110) , θ − θ (cid:111) ( δ − k min ).Let z ∈ L (Γ), T >
0. Note that dim θ L µ (cid:62) dim L µ = k min so we assume θ ∈ (0 , / µ ( B ( z, e − T θ )) µ ( B ( z, e − T )) ≈ e − δT θ e − δT e − ρ ( z,T θ )( δ − k ( z,T θ )) e − ρ ( z,T )( δ − k ( z,T )) (cid:38) (cid:16) e T (1 − θ ) (cid:17) δ e − ρ ( z,T θ )( δ − k min ) (cid:62) (cid:16) e T (1 − θ ) (cid:17) δ e − T θ ( δ − k min ) = (cid:16) e T (1 − θ ) (cid:17) δ − θ − θ ( δ − k min ) which gives dim θ L µ (cid:62) δ − θ − θ ( δ − k min )as required. ( k min + k max ) / < δ (cid:54) k max • Upper bound : We show dim θ L µ (cid:54) δ − k max − min (cid:110) , θ − θ (cid:111) (2 δ − k min − k max ).Similarly to the lower bound in Section 4.3.2, we only need to deal with the case when θ ∈ (0 , / p, p (cid:48) ∈ L (Γ) be parabolic fixed points such that k ( p ) = k min and k ( p (cid:48) ) = k max , andlet f be a parabolic element fixing p . Let n be a large integer and let z = f n ( p (cid:48) ). Again, byLemma 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, forsufficiently large n , ρ ( z, T θ ) (cid:62) (1 − ε ) T θ − C and ρ ( z, T ) (cid:62) T (1 − θ ) − C for some ε ∈ (0 ,
1) and some constants C , C . Applying Theorem 2.2 gives µ ( B ( z, e − T θ )) µ ( B ( z, e − T )) ≈ e − δT θ e − δT e ρ ( z,T θ )( k min − δ ) e − ρ ( z,T )( δ − k max ) (cid:46) (cid:16) e T (1 − θ ) (cid:17) δ e (1 − ε ) T θ ( k min − δ )+(1 − θ ) T ( δ − k max ) = (cid:16) e T (1 − θ ) (cid:17) δ e T (1 − θ ) δ − T (1 − θ ) k max − T θ (2 δ − k min − k max )+ εT θ ( δ − k min ) = (cid:16) e T (1 − θ ) (cid:17) δ − k max − θ − θ (2 δ − k min − k max )+ ε θ − θ ( δ − k min ) θ L µ (cid:54) δ − k max − θ − θ (2 δ − k min − k max ) + ε θ − θ ( δ − k min )and letting ε → • Lower bound : We show dim θ L µ (cid:62) δ − k max − min (cid:110) , θ − θ (cid:111) (2 δ − k min − k max ).Let z ∈ L (Γ), T > θ ∈ (0 , /
2) (the case when θ (cid:62) / z T and z T θ do not lie in a common standard horoball. ApplyingTheorem 2.2 gives µ ( B ( z, e − T θ )) µ ( B ( z, e − T )) ≈ e − δT θ e − δT e − ρ ( z,T θ )( δ − k ( z,T θ )) e − ρ ( z,T )( δ − k ( z,T )) (cid:62) e − δT θ e − δT e − ρ ( z,T θ )( δ − k min ) e − ρ ( z,T )( δ − k max ) = (cid:16) e T (1 − θ ) (cid:17) δ e ρ ( z,T θ )( k min − δ )+ ρ ( z,T )( δ − k max ) (cid:62) (cid:16) e T (1 − θ ) (cid:17) δ e T θ ( k max + k min − δ )+ T (1 − θ )( δ − k min ) by (4 . (cid:62) (cid:16) e T (1 − θ ) (cid:17) δ − k min − θ − θ (2 δ − k min − k max ) . If z T and z T θ do lie in a common standard horoball H p and δ (cid:54) k ( p ), then we have µ ( B ( z, e − T θ )) µ ( B ( z, e − T )) ≈ e − δT θ e − δT e − ρ ( z,T θ )( δ − k ( z,T θ )) e − ρ ( z,T )( δ − k ( z,T )) = (cid:16) e T (1 − θ ) (cid:17) δ e ( ρ ( z,T ) − ρ ( z,T θ ))( δ − k ( p )) (cid:62) (cid:16) e T (1 − θ ) (cid:17) δ e T (1 − θ )( δ − k max ) by (4 . (cid:16) e T (1 − θ ) (cid:17) δ − k max (cid:62) (cid:16) e T (1 − θ ) (cid:17) δ − k max − θ − θ (2 δ − k min − k max ) if ρ ( z, T ) − ρ ( z, T θ ) (cid:62)
0, and otherwise µ ( B ( z, e − T θ )) µ ( B ( z, e − T )) ≈ (cid:16) e T (1 − θ ) (cid:17) δ e ( ρ ( z,T ) − ρ ( z,T θ ))( δ − k ( p )) (cid:62) (cid:16) e T (1 − θ ) (cid:17) δ (cid:62) (cid:16) e T (1 − θ ) (cid:17) δ − k max − θ − θ (2 δ − k min − k max ) . If δ > k ( p ), then µ ( B ( z, e − T θ )) µ ( B ( z, e − T )) ≈ (cid:16) e T (1 − θ ) (cid:17) δ e ( ρ ( z,T ) − ρ ( z,T θ ))( δ − k ( p )) (cid:62) (cid:16) e T (1 − θ ) (cid:17) δ e − T θ ( δ − k ( p )) by (4 . (cid:62) (cid:16) e T (1 − θ ) (cid:17) δ − θ − θ ( δ − k min ) (cid:62) (cid:16) e T (1 − θ ) (cid:17) δ − k max − θ − θ (2 δ − k min − k max ) .
31n all cases, we have dim θ L µ (cid:62) δ − k max − θ − θ (2 δ − k min − k max )as required. δ (cid:54) ( k min + k max ) / • We show dim θ L µ = 2 δ − k max .Let p, p (cid:48) ∈ L (Γ) be two parabolic fixed points such that k ( p (cid:48) ) = k max , let f be a parabolicelement fixing p and let n be a large positive integer. Let z = f n ( p (cid:48) ) and let ε ∈ (0 , n and Lemma 4.2, we may choose T such that ρ ( z, T ) (cid:62) (1 − ε ) T and ρ ( z, T θ ) = 0 .z z T z T θ
Figure 14: Choosing the appropriate T .By Theorem 2.2, we have µ ( B ( z, e − T θ )) µ ( B ( z, e − T )) ≈ e − δT θ e − δT e − ρ ( z,T )( δ − k max ) (cid:46) (cid:16) e T (1 − θ ) (cid:17) δ e (1 − ε ) T (1 − θ )( δ − k max ) = (cid:16) e T (1 − θ ) (cid:17) δ − k max − ε ( δ − k max ) which proves dim θ L µ (cid:54) δ − k max − ε ( δ − k max ) and letting ε → L µ = min { δ − k max , k min } , so when δ (cid:54) ( k min + k max ) /
2, we have2 δ − k max = dim L µ (cid:54) dim θ L µ (cid:54) δ − k max so dim θ L µ = 2 δ − k max , as required. L (Γ) δ (cid:54) k min • We show dim θ A L (Γ) = δ + min (cid:110) , θ − θ (cid:111) ( k max − δ ).As dim θ A L (Γ) (cid:54) dim θ A µ = δ + min (cid:26) , θ − θ (cid:27) ( k max − δ ) (4.10)32hen δ (cid:54) k min , we need only prove the lower bound. To obtain this, we make use of the followingresult (see [22, Theorem 3.4.8]). Proposition 4.7.
Let F ⊆ R n and suppose that ρ = inf { θ ∈ (0 , | dim θ A F = dim A F } existsand ρ ∈ (0 , and dim L F = dim B F . Then for θ ∈ (0 , ρ ) , dim θ A F (cid:62) dim B F + (1 − ρ ) θ (1 − θ ) ρ (dim A F − dim B F ) . We note that if δ (cid:54) k min , then certainlydim L L (Γ) = min { k min , δ } = δ = dim B L (Γ) . We now show that ρ = 1 /
2. By (4.10), we have ρ (cid:62) /
2, so we need only show that ρ (cid:54) /
2. Todo this, we recall a result from [21], which states that using the orbit of a free abelian subgroupof the stabiliser of some parabolic fixed point p with k ( p ) = k max , we can find a bi-Lipschitzimage of an inverted Z k max lattice inside L (Γ) in S d , which gives usdim θ A L (Γ) (cid:62) dim θ A Z k max = min (cid:26) k max , k max − θ ) (cid:27) (4.11)with the equality coming from [25, Prop 4.5, Cor 6.4]. This proves ρ (cid:54) /
2, as required.Therefore, by Proposition 4.7, we have for θ ∈ (0 , / θ A L (Γ) (cid:62) dim B L (Γ) + (1 − ρ ) θ (1 − θ ) ρ (dim A L (Γ) − dim B L (Γ)) = δ + θ − θ ( k max − δ )as required. k min < δ < k max • Upper bound : We show dim θ A L (Γ) (cid:54) δ + min (cid:110) , θ − θ (cid:111) ( k max − δ ).The argument for the upper bound for the Assouad spectrum is nearly identical to the argumentfor the upper bound for the Assouad dimension given in [21], and so we omit any part of theargument which does not improve upon the bounds provided in the paper, in particular whenthe Assouad dimension is bounded above by δ . We also note thatdim θ A L (Γ) (cid:54) dim A L (Γ) = k max so we may assume that θ ∈ (0 , / z ∈ L (Γ), ε >
0, and T be sufficiently large suchthat T (1 − θ ) (cid:62) max { ε − , log10 } . Let { x i } i ∈ X be a centred e − T -packing of B ( z, e − T θ ) ∩ L (Γ) ofmaximal cardinality, in other words x i ∈ B ( z, e − T θ ) ∩ L (Γ) for all i ∈ X and | x i − x j | > e − T for i (cid:54) = j . Decompose X as follows X = X + X + ∞ (cid:91) n =2 X n X = { i ∈ X | ( x i ) T ∈ H p with | H p | (cid:62) e − T θ } X = { i ∈ X \ X | ρ ( x i , T ) (cid:54) εT (1 − θ ) } X n = { i ∈ X \ ( X ∪ X ) | n − < ρ ( x i , T ) (cid:54) n } . Our goal is to bound from above the cardinalities of X , X and X n separately.We start with X , 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 | (cid:62) e − T θ and H p ∩ ( ∪ i ∈ X ( x i ) T ) (cid:54) = ∅ , thenthis p must be unique, i.e. if H p ∩ ( ∪ i ∈ X ( x i ) T ) (cid:54) = ∅ and H p (cid:48) ∩ ( ∪ i ∈ X ( x i ) T ) (cid:54) = ∅ for some p, p (cid:48) ∈ P with | H p | , | H p (cid:48) | (cid:62) e − T θ , then H p and H p (cid:48) could not be disjoint. This means we can choose p ∈ P such that ( x i ) T ∈ H p for all i ∈ X , and also note that this forces z T θ ∈ H p .If δ (cid:54) k ( p ), then by Theorem 2.2 e − T θδ e − ρ ( z,T θ )( δ − k ( p )) (cid:38) µ ( B ( z, e − T θ )) (cid:62) µ ( ∪ i ∈ X B ( x i , e − T )) (cid:38) | X | min i ∈ X ( e − T ) δ e − ρ ( x i ,T )( δ − k ( p )) where the second inequality follows from the fact that { x i } i ∈ X is an e − T packing. Therefore | X | (cid:46) max i ∈ X (cid:16) e T (1 − θ ) (cid:17) δ e ( ρ ( x i ,T ) − ρ ( z,T θ ))( δ − k ( p )) (cid:54) (cid:16) e T (1 − θ ) (cid:17) δ e − T θ ( δ − k max ) = (cid:16) e T (1 − θ ) (cid:17) δ + θ − θ ( k max − δ ) . (4.12)Note that Fraser [21] estimates using ρ ( x i , T ) − ρ ( z, t ) (cid:54) T − t + 10, but we can improve this for t = T θ with θ < / ρ ( x i , T ) (cid:62) ρ ( z, T θ ) (cid:54) T θ .If δ > k ( p ), then we refer the reader to [21, 4998-5000], where it is shown that | X | (cid:46) (cid:16) e T (1 − θ ) (cid:17) δ . Therefore, we have | X | (cid:46) (cid:16) e T (1 − θ ) (cid:17) δ + θ − θ ( k max − δ ) regardless of the relationship between δ and k ( p ).For X , we have e − T θδ e − ρ ( z,T θ )( δ − k ( z,T θ )) (cid:38) µ ( B ( z, e − T θ )) (cid:62) µ ( ∪ i ∈ X B ( x i , e − T )) (cid:38) (cid:88) i ∈ X ( e − T ) δ e − ρ ( x i ,T )( δ − k ( x i ,T )) (cid:62) | X | ( e − T ) δ e − εT (1 − θ )( δ − k min ) by the definition of X and the fact that δ > k min . Therefore | X | (cid:46) (cid:16) e T (1 − θ ) (cid:17) δ e εT (1 − θ )( δ − k min ) − ρ ( z,T θ )( δ − k ( z,T θ )) (cid:54) (cid:16) e T (1 − θ ) (cid:17) δ e εT (1 − θ )( δ − k min ) − T θ ( δ − k max ) = (cid:16) e T (1 − θ ) (cid:17) δ + θ − θ ( k max − δ )+ ε ( δ − k min ) . (4.13)34inally, we consider the sets X n . If i ∈ X n for n (cid:62)
2, then ρ ( x i , T ) > n −
1, and so ( x i ) T ∈ H p forsome p ∈ P with e − T (cid:54) | H p | < e − T θ . Furthermore, we may note that B ( x i , e − T ) is containedin the shadow at infinity of the squeezed horoball 2 e − ( n − H p . Also note that as | H p | < e − T θ ,we have p ∈ B ( z, e − T θ ). For each integer k ∈ [ k min , k max ] define X kn = { i ∈ X n | k ( x i , T ) = k } . Then µ (cid:91) i ∈ X kn B ( x i , e − T ) (cid:54) µ (cid:91) p ∈ P ∩ B ( z, e − Tθ )10 e − Tθ > | H p | (cid:62) e − T k ( p )= k Π (cid:16) e − ( n − H p (cid:17) (cid:46) (cid:88) p ∈ P ∩ B ( z, e − Tθ )10 e − Tθ > | H p | (cid:62) e − T k ( p )= k µ (Π(2 e − ( n − H p )) (cid:46) e − n (2 δ − k ) (cid:88) p ∈ P ∩ B ( z, e − Tθ )10 e − Tθ > | H p | (cid:62) e − T | H p | δ by Lemma 4 . (cid:46) e − n (2 δ − k ) ( T (1 − θ ) + log10) µ ( B ( z, e − T θ )) by Lemma 4 . (cid:46) e − n (2 δ − k ) T (1 − θ ) e − T θδ e − ρ ( z,T θ )( δ − k ( z,T θ )) by Theorem 2 . (cid:46) e − n (2 δ − k ) ε − ne − T θδ e T θ ( k max − δ ) where the last inequality uses the fact that εT (1 − θ ) < ρ ( x i , T ) (cid:54) n as i / ∈ X . On the otherhand, using the fact that { x i } i ∈ X kn is an e − T packing, µ (cid:91) i ∈ X kn B ( x i , e − T ) (cid:62) (cid:88) i ∈ X kn µ ( B ( x i , e − T )) (cid:38) | X kn | e − T δ e − n ( δ − k ) using n − < ρ ( x i , T ) (cid:54) n . Therefore | X kn | (cid:46) ε − ne − nδ (cid:16) e T (1 − θ ) (cid:17) δ e T θ ( k max − δ ) = ε − ne − nδ (cid:16) e T (1 − θ ) (cid:17) δ + θ − θ ( k max − δ ) which implies | X n | = k max (cid:88) k = k min | X kn | (cid:46) ε − ne − nδ (cid:16) e T (1 − θ ) (cid:17) δ + θ − θ ( k max − δ ) . (4.14)Combining (4.12),(4.13) and (4.14), we have | X | = | X | + | X | + ∞ (cid:88) n =2 | X n | (cid:46) (cid:16) e T (1 − θ ) (cid:17) δ + θ − θ ( k max − δ )+ ε ( δ − k min ) + ∞ (cid:88) n =2 ε − ne − nδ (cid:16) e T (1 − θ ) (cid:17) δ + θ − θ ( k max − δ ) (cid:46) (cid:16) e T (1 − θ ) (cid:17) δ + θ − θ ( k max − δ )+ ε ( δ − k min ) + ε − (cid:16) e T (1 − θ ) (cid:17) δ + θ − θ ( k max − δ ) θ A L (Γ) (cid:54) δ + θ − θ ( k max − δ ) + ε ( δ − k min )and letting ε → • Lower bound : We show dim θ A L (Γ) (cid:62) δ + min (cid:110) , θ − θ (cid:111) ( k max − δ ).Note that the case when θ (cid:62) / θ ∈ (0 , / p ∈ L (Γ) be a parabolic fixed pointwith k ( p ) = k max , then we switch to the upper half-space model H d +1 and assume that p = ∞ .Recalling the argument from [21, Section 5.1.1], we know that there exists some subset of L (Γ)which is bi-Lipschitz equivalent to Z k max . Therefore, we may assume that we have a Z k max lattice, denoted by Z , as a subset of L (Γ), and then use the fact that the Assouad spectrum isstable under bi-Lipschitz maps.We now partition this lattice into { Z k } k ∈ N , where Z k = { z ∈ Z | k − (cid:54) | z | < k } . We note that | Z k | ≈ k k max − . (4.15)Let φ : H d +1 → D d +1 denote the Cayley transformation mapping the upper half space model tothe Poincar´e ball model, and consider the family of balls { B ( z, / } z ∈ Z . Taking the φ -imageof Z yields an inverted lattice and there are positive constants C and C such that if z ∈ Z k for some k , then1 C k (cid:54) | φ ( z ) − p | (cid:54) C k and 1 C k (cid:54) | φ ( B ( z, / | (cid:54) C k , where p = φ ( ∞ ).Let T >
0. We now choose a constant C small enough which satisfies the following: • The set of balls (cid:83) k ∈ N (cid:83) z ∈ Z k ( B ( φ ( z ) , C /k )) are pairwise disjoint. • If z ∈ Z k and | φ ( z ) − p | < e − T θ , then B ( φ ( z ) , C /k ) ⊂ B ( p , e − T θ ).This gives us N e − T ( B ( p , e − T θ ) ∩ L (Γ)) (cid:38) (cid:88) z ∈ Z k C /k >e − T | φ ( z ) − p |
2, this means that balls with centres in Y \ Y cannot carry a fixed proportion of µ ( B ( z, e − T θ )) for sufficiently large T . It follows that µ (cid:16) ∪ i ∈ Y B ( y i , e − T ) (cid:17) ≈ µ (cid:0) ∪ i ∈ Y B ( y i , e − T ) (cid:1) (cid:62) µ ( B ( z, e − T θ )) / e − T θδ e − ρ ( z,T θ )( δ − k ( z,T θ )) (cid:46) µ ( B ( z, e − T θ )) (cid:54) µ ( ∪ i ∈ Y B ( y i , e − T )) (cid:46) (cid:88) i ∈ Y ( e − T ) δ e − ρ ( y i ,T )( δ − k ( y i ,T )) (cid:54) | Y | ( e − T ) δ e εT (1 − θ )( k max − δ ) using ρ ( y i , T ) (cid:54) εT (1 − θ ) for i ∈ Y and δ < k max . Therefore | Y | (cid:62) | Y | (cid:38) (cid:16) e T (1 − θ ) (cid:17) δ e εT (1 − θ )( δ − k max ) e − T θ ( δ − k min ) = (cid:16) e T (1 − θ ) (cid:17) δ − θ − θ ( δ − k min )+ ε ( δ − k max ) . (4.20)At least one of (4.19) and (4.20) must hold, so we have | Y | (cid:62) | Y | + | Y | (cid:38) (cid:16) e T (1 − θ ) (cid:17) δ − θ − θ ( δ − k min )+ ε ( δ − k max ) which proves dim θ L L (Γ) (cid:62) δ − θ − θ ( δ − k min ) + ε ( δ − k max )and letting ε → Upper bound : We show dim θ L L (Γ) (cid:54) δ − min (cid:110) , θ − θ (cid:111) ( δ − k min ).In order to obtain the upper bound in this case, we require the following technical lemma dueto Bowditch [10]. Switch to the upper half space model H d +1 , and let p ∈ P be a parabolic fixedpoint with rank k min . We may assume by conjugation that p = ∞ . A consequence of geometricfiniteness is that the limit set can be decomposed into the disjoint union of a set of conicallimit points and a set of bounded parabolic fixed points (this result was proven in dimension3 partially by Beardon and Maskit [6] and later refined by Bishop [7], and then generalised tohigher dimensions by Bowditch [10]). In particular, p = ∞ is a bounded parabolic point, andapplying [10, Definition, Page 272] gives the following lemma. Lemma 4.8.
There exists λ > and a k min -dimensional linear subspace V ⊆ R d such that L (Γ) ⊆ V λ ∪ {∞} , where V λ = { x ∈ R d | inf y ∈ V || x − y || (cid:54) λ } . Note that by Lemma 4.8, we immediately get dim θ L L (Γ) (cid:54) k min for θ (cid:62) /
2, so we may assumethat θ ∈ (0 , / p ∈ L (Γ) be a parabolicfixed point with k ( p ) = k min , and then switch to the upper half-space model H d +1 , and assumethat 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 Z k = { z ∈ Z | k − (cid:54) | z | < k } noting that | Z k | ≈ k k min − . (4.21)We let φ denote the Cayley transformation mapping the upper half space model to the Poincar´eball model. Then, as before, there are positive constants C and C such that if z ∈ Z k for some k , then 1 C k (cid:54) | φ ( z ) − p | (cid:54) C k and 1 C k (cid:54) | φ ( B ( z, / | (cid:54) C k , where p = φ ( ∞ ). We let T >
0, and then by Lemma 4.8, we may choose a constant C suchthat B ( p , e − T θ ) ⊆ (cid:91) z ∈ Z k C /k >e − T | φ ( z ) − p |
2, then we have | X n | (cid:46) ε − ne − nδ (cid:18) e − t e − T (cid:19) δ .
43t follows that | X | = | X | + | X | + ∞ (cid:88) n =2 | X n | (cid:46) (cid:18) e − t e − T (cid:19) δ + ε ( δ − k min ) + ∞ (cid:88) n =2 ε − ne − nδ (cid:18) e − t e − T (cid:19) δ (cid:46) (cid:18) e − t e − T (cid:19) δ + ε ( δ − k min ) + ε − (cid:18) e − t e − T (cid:19) δ = (cid:18) C /k e − T (cid:19) δ + ε ( δ − k min ) + ε − (cid:18) C /k e − T (cid:19) δ and so by (4.22), M e − T ( B ( p , e − T θ ∩ L (Γ)) (cid:46) (cid:88) z ∈ Z k C /k >e − T | φ ( z ) − p |
Lemma 5.1.
Let ξ ∈ J ( T ) and R > r > . For R sufficiently small, we have (cid:88) c ( ω ) ∈ I ∩ B ( ξ,R ) R>r c ( ω ) (cid:62) r r hc ( ω ) (cid:46) log( R/r ) m ( B ( ξ, R )) where I := (cid:83) n (cid:62) T − n ( I ( ω )) for some ω ∈ Ω .Proof. The approach here is similar to the proof of Lemma 4.6 given in [21, Lemma 5.2]. By[46, Theorem 3.1], there exists a constant κ > T such that for each ω ∈ Ω and for sufficiently small r >
0, we have J ( T ) ⊆ (cid:91) c ( ω ) ∈ Ir c ( ω ) (cid:62) r B ( c ( ω ) , κr p ( ω ) / (1+ p ( ω )) c ( ω ) r / (1+ p ( ω )) )with multiplicity (cid:46)
1. In particular, for
R > r > R sufficiently small, the set (cid:91) c ( ω ) ∈ I ∩ B ( ξ,R ) R>r c ( ω ) (cid:62) r B ( c ( ω ) , κr p ( ω ) / (1+ p ( ω )) c ( ω ) r / (1+ p ( ω )) )45as multiplicity (cid:46) B ( ξ, ( κ + 1) R ). Therefore, we have m ( B ( ξ, ( κ + 1) R )) (cid:38) (cid:88) c ( ω ) ∈ I ∩ B ( ξ,R ) R>r c ( ω ) (cid:62) r m (cid:16) B ( c ( ω ) , κr p ( ω ) / (1+ p ( ω )) c ( ω ) r / (1+ p ( ω )) ) (cid:17) (cid:38) (cid:88) c ( ω ) ∈ I ∩ B ( ξ,R ) R>r c ( ω ) (cid:62) r (cid:16) r p ( ω ) / (1+ p ( ω )) c ( ω ) r / (1+ p ( ω )) (cid:17) h r p ( ω ) / (1+ p ( ω )) c ( ω ) r / (1+ p ( ω )) r l ( ξ ) ( h − p ( ω ) (cid:38) r h (cid:88) c ( ω ) ∈ I ∩ B ( ξ,R ) R>r c ( ω ) (cid:62) r (cid:16) r c ( ω ) r (cid:17) p ( ω ) / (1+ p ( ω )) (cid:38) r h (cid:88) c ( ω ) ∈ I ∩ B ( ξ,R ) R>r c ( ω ) (cid:62) r r c ( ω ) ≈ r l . Therefore (cid:88) c ( ω ) ∈ I ∩ B ( ξ,R ) R>r c ( ω ) (cid:62) r r hc ( ω ) (cid:54) (cid:88) m ∈ Z ∩ [0 , log( R/r )] (cid:88) c ( ω ) ∈ I ∩ B ( ξ,R ) e m +1 r>r c ( ω ) (cid:62) e m r r hc ( ω ) (cid:46) (cid:88) m ∈ Z ∩ [0 , log( R/r )] (cid:88) c ( ω ) ∈ I ∩ B ( ξ,R ) e m +1 r>r c ( ω ) (cid:62) e m r r h e mh (cid:46) (cid:88) m ∈ Z ∩ [0 , log( R/r )] r h e mh (cid:88) c ( ω ) ∈ I ∩ B ( ξ,R ) R>r c ( ω ) (cid:62) e m r (cid:46) (cid:88) m ∈ Z ∩ [0 , log( R/r )] r h e mh (cid:16) ( re m ) − h m ( B ( ξ, ( κ + 1) R )) (cid:17) by (5.1) (cid:46) log( R/r ) m ( B ( ξ, R ))where the last inequality uses the fact that m is a doubling measure.We also require the following key lemma. Lemma 5.2.
Let ξ ∈ J r ( T ) and R > be sufficiently small. Then there exists some c ( ω ) ∈ J p ( T ) and some constant C (cid:62) such that B ( ξ, R ) ⊂ B ( c ( ω ) , Cφ ( ξ, R ) h − p ( ω ) r c ( ω ) ) . Proof.
It was shown in [18, Section 5] that B ( ξ, R ) ⊂ T − nξ ( B ( ω, Cφ ( ξ, R ) h − p ( ω ) r ω ))for some ω ∈ Ω and uniform C (cid:62) T − nξ is an appropriately chosen holomorphic inversebranch of T n . By definition T − nξ ( ω ) ∈ J p ( T ), and then the result is an immediate consequence ofthe fact that ( T − nξ ) (cid:48) ( ω ) r ω ≈ r c ( ω ) , using the Koebe distortion theorem and the fact that r ω ≈ Ω is a finite set (see [18, 46]). 46 .2 The box dimension of m • We show dim B m (cid:54) max { h, h + ( h − p max } .Note that when h (cid:62)
1, we havedim B m (cid:54) dim A m (cid:54) h + ( h − p max (see Section 5.3) so we assume h <
1. Then note that for any ξ ∈ J ( T ) and r < | J ( T ) | m ( B ( ξ, r )) ≈ r h φ ( ξ, r ) (cid:62) r h which proves dim B m (cid:54) h , as required. • We show dim B m (cid:62) max { h, h + ( h − p max } .Suppose h (cid:62)
1. Let ξ ∈ J p ( T ) with associated terminating optimal sequence ( n j ( ξ )) j =1 ,...,l andhyperbolic zooms ( r j ( ξ )) j =1 ,...,l such that T n l ( ξ ) ( ξ ) = ω for some ω ∈ Ω with p ( ω ) = p max . Thenusing Theorem 2.3, we have for all sufficiently small r > m ( B ( ξ, r )) (cid:46) r h +( h − p max which proves dim B m (cid:62) h + ( h − p max , as required.If h <
1, then we have dim B m (cid:62) dim B J ( T ) = h , as required. m The lower bound will follow from our lower bound for the Assouad spectrum of m , see Section5.5. Therefore we only need to prove the upper bound. • We show dim A m (cid:54) max { , h + ( h − p max } . We only argue the case where ξ ∈ J r ( T ), and note that the case when ξ ∈ J p ( T ) follows similarly.We make extensive use of Theorem 2.3 throughout.Suppose ξ ∈ J r ( T ), with associated optimal sequence ( n j ( ξ )) j ∈ N and hyperbolic zooms ( r j ( ξ )) j ∈ N .Suppose that r j +1 ( ξ ) (cid:54) r < R < r j ( ξ ) and that T n j ( ξ )+1 ( ξ ) ∈ U ω = B ( ω, r ω ) for some ω ∈ Ω ,and let r m = r j ( ξ ) ( r j +1 ( ξ ) /r j ( ξ )) p ( ω ) .If r > r m , then m ( B ( ξ, R )) m ( B ( ξ, r )) ≈ (cid:18) Rr (cid:19) h ( R/r j ( ξ )) ( h − p ( ω ) ( r/r j ( ξ )) ( h − p ( ω ) (cid:54) (cid:18) Rr (cid:19) h +( h − p max . If R < r m , then m ( B ( ξ, R )) m ( B ( ξ, r )) ≈ (cid:18) Rr (cid:19) h ( r j +1 ( ξ ) /R ) h − ( r j +1 ( ξ ) /r ) h − = (cid:18) Rr (cid:19) . r (cid:54) r m (cid:54) R , then m ( B ( ξ, R )) m ( B ( ξ, r )) ≈ (cid:18) Rr (cid:19) h ( R/r j ( ξ )) ( h − p ( ω ) ( r j +1 ( ξ ) /r ) h − = R h +( h − p ( ω ) rr j ( ξ ) ( h − p ( ω ) r j +1 ( ξ ) h − . (5.2)If we assume that h (cid:62)
1, then note that (cid:16) rR (cid:17) h +( h − p ( ω ) m ( B ( ξ, R )) m ( B ( ξ, r )) ≈ r ( h − p ( ω )) r j ( ξ ) ( h − p ( ω ) r j +1 ( ξ ) h − is maximised when r = r m . Therefore (cid:16) rR (cid:17) h +( h − p ( ω ) m ( B ( ξ, R )) m ( B ( ξ, r )) (cid:46) r m ( h − p ( ω )) r j ( ξ ) ( h − p ( ω ) r j +1 ( ξ ) h − = r j ( ξ ) ( h − p ( ω )) (cid:16) r j +1 ( ξ ) r j ( ξ ) (cid:17) h − r j ( ξ ) ( h − p ( ω ) r j +1 ( ξ ) h − = 1which proves that m ( B ( ξ, R )) m ( B ( ξ, r )) (cid:46) (cid:18) Rr (cid:19) h +( h − p ( ω ) (cid:54) (cid:18) Rr (cid:19) h +( h − p max . Similarly, if h <
1, then by (5.2) (cid:16) rR (cid:17) m ( B ( ξ, R )) m ( B ( ξ, r )) ≈ R ( h − p ( ω )) r j ( ξ ) ( h − p ( ω ) r j +1 ( ξ ) h − is maximised when R = r m . Therefore, as above (cid:16) rR (cid:17) m ( B ( ξ, R )) m ( B ( ξ, r )) (cid:46) m ( B ( ξ, R )) m ( B ( ξ, r )) (cid:46) (cid:18) Rr (cid:19) . This covers all cases when r j +1 ( ξ ) (cid:54) r < R < r j ( ξ ).Now, we consider the cases when r j +1 ( ξ ) (cid:54) R < r j ( ξ ), r l +1 ( ξ ) (cid:54) r < r l ( ξ ), with l > j , T n j +1( ξ ) ( ξ ) ∈ U ω and T n l +1( ξ ) ( ξ ) ∈ U ω for some ω , ω ∈ Ω .Let r m = r j ( ξ ) ( r j +1 ( ξ ) /r j ( ξ )) p ( ω and r n = r l ( ξ ) ( r l +1 ( ξ ) /r l ( ξ )) p ( ω . Case 1 : R > r m , r > r n .We have m ( B ( ξ, R )) m ( B ( ξ, r )) ≈ (cid:18) Rr (cid:19) h ( R/r j ( ξ )) ( h − p ( ω ) ( r/r l ( ξ )) ( h − p ( ω ) . (5.3)If h <
1, then (cid:16) rR (cid:17) m ( B ( ξ, R )) m ( B ( ξ, r )) (cid:46) R ( h − p ( ω )) r h − r j ( ξ ) ( h − p ( ω ) (cid:54) r ( h − p ( ω )) m r j +1 ( ξ ) h − r j ( ξ ) ( h − p ( ω ) = r j ( ξ ) ( h − p ( ω )) (cid:16) r j +1 ( ξ ) r j ( ξ ) (cid:17) h − r j +1 ( ξ ) h − r j ( ξ ) ( h − p ( ω ) = 148nd if h (cid:62)
1, then by (5.3) m ( B ( ξ, R )) m ( B ( ξ, r )) (cid:46) (cid:18) Rr (cid:19) h (cid:18) r l ( ξ ) r (cid:19) ( h − p ( ω ) (cid:54) (cid:18) Rr (cid:19) h +( h − p ( ω ) using r l ( ξ ) < R . Case 2 : R (cid:54) r m , r (cid:54) r n .We have m ( B ( ξ, R )) m ( B ( ξ, r )) ≈ (cid:18) Rr (cid:19) h ( r j +1 ( ξ ) /R ) h − ( r l +1 ( ξ ) /r ) h − . (5.4)If h <
1, then m ( B ( ξ, R )) m ( B ( ξ, r )) ≈ (cid:18) Rr (cid:19) r j +1 ( ξ ) h − r l +1 ( ξ ) h − (cid:54) (cid:18) Rr (cid:19) and if h (cid:62)
1, then by (5.4) m ( B ( ξ, R )) m ( B ( ξ, r )) (cid:46) R h rr l +1 ( ξ ) h − and therefore (cid:16) rR (cid:17) h +( h − p ( ω ) m ( B ( ξ, R )) m ( B ( ξ, r )) (cid:46) R (1 − h ) p ( ω ) r (1 − h )(1+ p ( ω )) r l +1 ( ξ ) h − (cid:54) r l ( ξ ) (1 − h ) p ( ω ) r (1 − h )(1+ p ( ω )) n r l +1 ( ξ ) h − = r l ( ξ ) (1 − h ) p ( ω ) r l ( ξ ) (1 − h )(1+ p ( ω )) (cid:16) r l +1 ( ξ ) r l ( ξ ) (cid:17) − h r l +1 ( ξ ) h − = 1 . Case 3 : R > r m , r (cid:54) r n .We have m ( B ( ξ, R )) m ( B ( ξ, r )) ≈ (cid:18) Rr (cid:19) h ( R/r j ( ξ )) ( h − p ( ω ) ( r l +1 ( ξ ) /r ) ( h − = R h +( h − p ( ω ) rr j ( ξ ) ( h − p ( ω ) r l +1 ( ξ ) h − . (5.5)If h <
1, then (cid:16) rR (cid:17) m ( B ( ξ, R )) m ( B ( ξ, r )) ≈ R ( h − p ( ω )) r j ( ξ ) ( h − p ( ω ) r l +1 ( ξ ) h − (cid:54) r j ( ξ ) ( h − p ( ω )) (cid:16) r j +1 ( ξ ) r j ( ξ ) (cid:17) h − r j ( ξ ) ( h − p ( ω ) r j +1 ( ξ ) h − = 1 . If h (cid:62) p ( ω ) (cid:62) p ( ω ), then by (5.5) (cid:16) rR (cid:17) h +( h − p ( ω ) m ( B ( ξ, R )) m ( B ( ξ, r )) ≈ r ( h − p ( ω )) r j ( ξ ) ( h − p ( ω ) r l +1 ( ξ ) h − (cid:54) r l ( ξ ) ( h − p ( ω )) (cid:16) r l +1 ( ξ ) r l ( ξ ) (cid:17) ( h − p ( ω p ( ω r l ( ξ ) ( h − p ( ω ) r l +1 ( ξ ) h − = (cid:18) r l ( ξ ) r l +1 ( ξ ) (cid:19) ( h − − p ( ω p ( ω ) (cid:54) h (cid:62) p ( ω ) < p ( ω ), then by (5.5) (cid:16) rR (cid:17) h +( h − p ( ω ) m ( B ( ξ, R )) m ( B ( ξ, r )) ≈ r ( h − p ( ω )) R ( h − p ( ω ) − p ( ω )) r j ( ξ ) ( h − p ( ω ) r l +1 ( ξ ) h − (cid:54) r ( h − p ( ω )) n R ( h − p ( ω ) − p ( ω )) r l ( ξ ) ( h − p ( ω ) r l +1 ( ξ ) h − = (cid:18) Rr l ( ξ ) (cid:19) ( h − p ( ω ) − p ( ω )) (cid:54) . Case 4 : R (cid:54) r m , r > r n .This gives m ( B ( ξ, R )) m ( B ( ξ, r )) ≈ (cid:18) Rr (cid:19) h ( r j +1 ( ξ ) /R ) ( h − ( r/r l ( ξ )) ( h − p ( ω ) . (5.6)If h <
1, then (cid:16) rR (cid:17) m ( B ( ξ, R )) m ( B ( ξ, r )) (cid:46) r (1 − h )(1+ p ( ω )) r j +1 ( ξ ) − h r l ( ξ ) (1 − h ) p ( ω ) (cid:54) r l ( ξ ) (1 − h )(1+ p ( ω )) r l ( ξ ) − h r l ( ξ ) (1 − h ) p ( ω ) = 1and if h (cid:62)
1, then by (5.6) m ( B ( ξ, R )) m ( B ( ξ, r )) (cid:46) (cid:18) Rr (cid:19) h (cid:18) r l ( ξ ) r (cid:19) ( h − p ( ω ) (cid:54) (cid:18) Rr (cid:19) h +( h − p ( ω ) . In all possible cases, we have m ( B ( ξ, R )) m ( B ( ξ, r )) (cid:46) (cid:18) Rr (cid:19) max { ,h +( h − p ( ω ) ,h +( h − p ( ω ) } (cid:54) (cid:18) Rr (cid:19) max { ,h +( h − p max } which proves the desired upper bound. m The upper bound will follow from our upper bound for the lower spectrum of m , see Section 5.6.Therefore we only need to prove the lower bound. The lower bound for dim L m follows similarlyto the argument for the upper bound for dim A m , and so we omit some details. • We show dim L m (cid:62) min { , h + ( h − p max } .We make extensive use of Theorem 2.3 throughout. Let ξ ∈ J r ( T ) with associated optimalsequence ( n j ( ξ )) j ∈ N and hyperbolic zooms ( r j ( ξ )) j ∈ N . The case ξ ∈ J p ( T ) is similar and omitted.First, assume that r j +1 ( ξ ) (cid:54) r < R < r j ( ξ ) with T n j ( ξ )+1 ( ξ ) ∈ ω for some j ∈ N , ω ∈ Ω , andlet r m = r j ( ξ ) ( r j +1 ( ξ ) /r j ( ξ )) p ( ω ) .The cases when r > r m and R < r m are trivial, and so we assume r (cid:54) r m (cid:54) R . This gives m ( B ( ξ, R )) m ( B ( ξ, r )) ≈ (cid:18) Rr (cid:19) h ( R/r j ( ξ )) ( h − p ( ω ) ( r j +1 ( ξ ) /r ) h − = R h +( h − p ( ω ) r r j ( ξ ) ( h − p ( ω ) r j +1 ( ξ ) h − . h (cid:62)
1, then (cid:16) rR (cid:17) m ( B ( ξ, R )) m ( B ( ξ, r )) ≈ R ( h − p ( ω )) r j ( ξ ) ( h − p ( ω ) r j +1 ( ξ ) h − (cid:62) r ( h − p ( ω )) m r j ( ξ ) ( h − p ( ω ) r j +1 ( ξ ) h − = 1and if h <
1, then (cid:16) rR (cid:17) h +( h − p ( ω ) m ( B ( ξ, R )) m ( B ( ξ, r )) ≈ r ( h − p ( ω )) r j ( ξ ) ( h − p ( ω ) r j +1 ( ξ ) h − (cid:62) r ( h − p ( ω )) m r j ( ξ ) ( h − p ( ω ) r j +1 ( ξ ) h − = 1so in either case we have m ( B ( ξ, R )) m ( B ( ξ, r )) (cid:38) (cid:18) Rr (cid:19) min { ,h +( h − p ( ω ) } (cid:62) (cid:18) Rr (cid:19) min { ,h +( h − p max } . We now consider the case when r j +1 ( ξ ) (cid:54) R < r j ( ξ ), r l +1 ( ξ ) (cid:54) r < r l ( ξ ), with l > j , T n j +1( ξ ) ( ξ ) ∈ U ω and T n l +1( ξ ) ( ξ ) ∈ U ω for some ω , ω ∈ Ω .Let r m = r j ( ξ ) ( r j +1 ( ξ ) /r j ( ξ )) p ( ω and r n = r l ( ξ ) ( r l +1 ( ξ ) /r l ( ξ )) p ( ω . Case 1 : R > r m , r > r n .If h <
1, then by (5.3) m ( B ( ξ, R )) m ( B ( ξ, r )) (cid:38) (cid:18) Rr (cid:19) h (cid:18) r l ( ξ ) r (cid:19) ( h − p ( ω ) (cid:62) (cid:18) Rr (cid:19) h +( h − p ( ω ) and if h (cid:62)
1, then by (5.3) (cid:16) rR (cid:17) m ( B ( ξ, R )) m ( B ( ξ, r )) (cid:38) R ( h − p ( ω )) r h − r j ( ξ ) ( h − p ( ω ) (cid:62) r ( h − p ( ω )) m r j +1 ( ξ ) h − r j ( ξ ) ( h − p ( ω ) = 1 . Case 2 : R (cid:54) r m , r (cid:54) r n .If h <
1, then by (5.4) m ( B ( ξ, R )) m ( B ( ξ, r )) (cid:38) R h rr l +1 ( ξ ) h − and therefore (cid:16) rR (cid:17) h +( h − p ( ω ) m ( B ( ξ, R )) m ( B ( ξ, r )) (cid:38) r ( h − p ( ω ) R ( h − p ( ω ) r l +1 ( ξ ) h − (cid:62) r ( h − p ( ω )) n r l ( ξ ) ( h − p ( ω ) r l +1 ( ξ ) h − = 1and if h (cid:62)
1, then by (5.4) m ( B ( ξ, R )) m ( B ( ξ, r )) ≈ (cid:18) Rr (cid:19) (cid:18) r l +1 ( ξ ) r j +1 ( ξ ) (cid:19) h − (cid:62) (cid:18) Rr (cid:19) . Case 3 : R > r m , r (cid:54) r n . 51f h < p ( ω ) (cid:62) p ( ω ), then by (5.5) (cid:16) rR (cid:17) h +( h − p ( ω ) m ( B ( ξ, R )) m ( B ( ξ, r )) ≈ r ( h − p ( ω )) r j ( ξ ) ( h − p ( ω ) r l +1 ( ξ ) h − (cid:62) r ( h − p ( ω )) n r l ( ξ ) ( h − p ( ω ) r l +1 ( ξ ) h − = (cid:18) r l ( ξ ) r l +1 ( ξ ) (cid:19) ( h − − p ( ω p ( ω ) (cid:62) h < p ( ω ) < p ( ω ), then by (5.5) (cid:16) rR (cid:17) h +( h − p ( ω ) m ( B ( ξ, R )) m ( B ( ξ, r )) ≈ r ( h − p ( ω )) R ( h − p ( ω ) − p ( ω )) r j ( ξ ) ( h − p ( ω ) r l +1 ( ξ ) h − (cid:62) r ( h − p ( ω )) n R ( h − p ( ω ) − p ( ω )) r l ( ξ ) ( h − p ( ω ) r l +1 ( ξ ) h − = r l ( ξ ) ( h − p ( ω ) − p ( ω )) R ( h − p ( ω ) − p ( ω )) (cid:62) . If h (cid:62)
1, then by (5.5) (cid:16) rR (cid:17) m ( B ( ξ, R )) m ( B ( ξ, r )) ≈ R ( h − p ( ω )) r j ( ξ ) ( h − p ( ω )) r l +1 ( ξ ) h − (cid:62) r ( h − p ( ω )) m r j ( ξ ) ( h − p ( ω )) r j +1 ( ξ ) h − = 1 . Case 4 : R (cid:54) r m , r > r n .If h <
1, then by (5.6) m ( B ( ξ, R )) m ( B ( ξ, r )) (cid:38) (cid:18) Rr (cid:19) h (cid:18) r l ( ξ ) r (cid:19) ( h − p ( ω ) (cid:62) (cid:18) Rr (cid:19) h +( h − p ( ω ) and if h (cid:62)
1, then by (5.6) (cid:16) rR (cid:17) m ( B ( ξ, R )) m ( B ( ξ, r )) (cid:38) r (1 − h )(1+ p ( ω )) r j +1 ( ξ ) − h r l ( ξ ) (1 − h ) p ( ω ) (cid:62) r l ( ξ ) (1 − h )(1+ p ( ω )) r l ( ξ ) − h r l ( ξ ) (1 − h ) p ( ω ) = 1 . In all cases, we have m ( B ( ξ, R )) m ( B ( ξ, r )) (cid:38) (cid:18) Rr (cid:19) min { ,h +( h − p ( ω ) ,h +( h − p ( ω ) } (cid:62) (cid:18) Rr (cid:19) min { ,h +( h − p max } which proves the desired lower bound. m h < J ( T ), seeSection 5.9.1. Therefore it remains to prove the upper bound. • We show dim θ A m (cid:54) h + min (cid:110) , θp max − θ (cid:111) (1 − h ).The case when θ (cid:62) / (1 + p max ) follows easily, asdim θ A m (cid:54) dim A m (cid:54) ,
52o we assume θ < / (1 + p max ). Let ξ ∈ J r ( T ), and assume that r j +1 ( ξ ) (cid:54) r θ < r j ( ξ ), r l +1 ( ξ ) (cid:54) r < r l ( ξ ), with l (cid:62) j , T n j +1( ξ ) ( ξ ) ∈ U ω and T n l +1( ξ ) ( ξ ) ∈ U ω for some ω , ω ∈ Ω . The case ξ ∈ J p ( T ) is similar and omitted. Let r m = r j ( ξ ) ( r j +1 ( ξ ) /r j ( ξ )) p ( ω . If r θ > r m , then byTheorem 2.3 m ( B ( ξ, r θ )) m ( B ( ξ, r )) (cid:46) (cid:18) r θ r (cid:19) h (cid:18) r θ r j ( ξ ) (cid:19) ( h − p ( ω ) (cid:54) (cid:18) r θ r (cid:19) h r θ ( h − p ( ω ) r ( ξ ) (1 − h ) p ( ω ) (cid:46) (cid:18) r θ r (cid:19) h + θp max1 − θ (1 − h ) and if r θ (cid:54) r m , then by Theorem 2.3 m ( B ( ξ, r θ )) m ( B ( ξ, r )) (cid:46) (cid:18) r θ r (cid:19) h (cid:18) r j +1 ( ξ ) r θ (cid:19) h − (cid:54) (cid:18) r θ r (cid:19) h r θ ( h − p ( ω )) r θ ( h − r j ( ξ ) (1 − h ) p ( ω ) (cid:18) using r θ /r j ( ξ ) p ( ω p ( ω (cid:54) r j +1 ( ξ ) p ( ω (cid:19) (cid:46) (cid:18) r θ r (cid:19) h + θp max1 − θ (1 − h ) . In either case, we have m ( B ( ξ, r θ )) m ( B ( ξ, r )) (cid:46) (cid:18) r θ r (cid:19) h + θp max1 − θ (1 − h ) which proves dim θ A m (cid:54) h + θp max − θ (1 − h )as required. h (cid:62) • We show dim θ A m = h + ( h − p max .This follow easily, since h + ( h − p max = dim B m (cid:54) dim θ A m (cid:54) dim A m (cid:54) h + ( h − p max . m h < • We show dim θ L m = h + ( h − p max .Note that dim θ L m (cid:62) dim L m (cid:62) h + ( h − p max so we need only prove the upper bound. To do this, let ξ ∈ J p ( T ) with associated terminatingoptimal sequence ( n j ( ξ )) j =1 ,...,l and hyperbolic zooms ( r j ( ξ )) j =1 ,...,l such that T n l ( ξ ) ( ξ ) = ω forsome ω ∈ Ω with p ( ω ) = p max . Then using Theorem 2.3, we have for all sufficiently small r > m ( B ( ξ, r θ )) m ( B ( ξ, r )) (cid:46) (cid:18) r θ r (cid:19) h (cid:18) r θ r (cid:19) ( h − p ( ω ) = (cid:18) r θ r (cid:19) h +( h − p max θ L m (cid:54) h + ( h − p max , as required. h (cid:62) J ( T ), see Section5.10.1. Therefore it remains to prove the lower bound. • We show dim θ L m (cid:62) h + min (cid:110) , θp max − θ (cid:111) (1 − h ).The case when θ (cid:62) / (1 + p max ) follows easily, as dim θ L m (cid:62) dim L m (cid:62) , so we assume θ < / (1 + p max ). Let ξ ∈ J r ( T ), and assume that r j +1 ( ξ ) (cid:54) r θ < r j ( ξ ), r l +1 ( ξ ) (cid:54) r < r l ( ξ ), with l (cid:62) j , T n j +1( ξ ) ( ξ ) ∈ U ω and T n l +1( ξ ) ( ξ ) ∈ U ω for some ω , ω ∈ Ω . The case ξ ∈ J p ( T ) issimilar and omitted. Let r m = r j ( ξ ) ( r j +1 ( ξ ) /r j ( ξ )) p ( ω . If r θ > r m , then by Theorem 2.3 m ( B ( ξ, r θ )) m ( B ( ξ, r )) (cid:38) (cid:18) r θ r (cid:19) h (cid:18) r θ r j ( ξ ) (cid:19) ( h − p ( ω ) (cid:62) (cid:18) r θ r (cid:19) h r θ ( h − p ( ω ) r ( ξ ) (1 − h ) p ( ω ) (cid:38) (cid:18) r θ r (cid:19) h + θp max1 − θ (1 − h ) and if r θ (cid:54) r m , Theorem 2.3 gives m ( B ( ξ, r θ )) m ( B ( ξ, r )) (cid:38) (cid:18) r θ r (cid:19) h (cid:18) r j +1 ( ξ ) r θ (cid:19) h − (cid:62) (cid:18) r θ r (cid:19) h r θ ( h − p ( ω )) r θ ( h − r j ( ξ ) (1 − h ) p ( ω ) (cid:18) using r θ /r j ( ξ ) p ( ω p ( ω (cid:54) r j +1 ( ξ ) p ( ω (cid:19) (cid:38) (cid:18) r θ r (cid:19) h + θp max1 − θ (1 − h ) . In either case, we have m ( B ( ξ, r θ )) m ( B ( ξ, r )) (cid:38) (cid:18) r θ r (cid:19) h + θp max1 − θ (1 − h ) which proves dim θ L m (cid:62) h + θp max − θ (1 − h )as required. J ( T ) The lower bound will follow from our lower bound for the Assouad spectrum of J ( T ), see Section5.9. Therefore we only need to prove the upper bound. • We show dim A J ( T ) (cid:54) max { , h } .Note that when h (cid:54)
1, we have dim A J ( T ) (cid:54) dim A m (cid:54) , so throughout we assume that h > ξ ∈ J ( T ), ε >
0, and
R > r >
R/r (cid:62) max { e ε − , } . Let { B ( x i , r ) } i ∈ X be a54entred r -packing of B ( ξ, R ) ∩ J ( T ) of maximal cardinality. We assume for convenience thateach x i ∈ J r ( T ), which we may do since J r ( T ) is dense in J ( T ). This is not really necessary butallows for efficient application of Lemma 5.2. Each x i has a particular ω = ω ( i ) ∈ Ω associatedwith it, coming from the global measure formula for m ( B ( x i , r )). In particular, x i belongs toan associated canonical ball B ( c ( ω ) , r c ( ω ) ). Decompose X as X = X ∪ X ∪ ∞ (cid:91) n =2 X n where X = { i ∈ X | x i ∈ B ( c ( ω ) , r c ( ω ) ) with r c ( ω ) (cid:62) R } X = { i ∈ X \ X | φ ( x i , r ) (cid:62) ( r/R ) ε } and X n = { i ∈ X \ ( X ∪ X ) | e − n (cid:54) φ ( x i , r ) < e − ( n − } . To study those i ∈ X , we decompose X further as X = X ∪ ∞ (cid:91) n =1 X n where X = { i ∈ X | φ ( x i , r ) (cid:62) φ ( ξ, R ) } X n = { i ∈ X | e − n φ ( ξ, R ) (cid:54) φ ( x i , r ) < e − ( n − φ ( ξ, R ) } . If i ∈ X , then by Theorem 2.3 R h φ ( ξ, R ) (cid:38) m ( B ( ξ, R )) (cid:38) m (cid:16) ∪ i ∈ X B ( x i , r ) (cid:17) (cid:38) min i ∈ X | X | r h φ ( x i , r ) (cid:62) | X | r h φ ( ξ, R )which implies that | X | (cid:46) (cid:18) Rr (cid:19) h . Turning our attention to X n , for c ( ω ) ∈ J p ( T ), write X n ( c ( ω )) to denote the set of all i ∈ X n which are associated with c ( ω ) (that is, φ ( x i , r ) is defined via c ( ω ) in the context of Theorem2.3). In particular, Lemma 5.2 ensures that B ( x i , r ) ⊆ B ( c ( ω ) , C ( φ ( ξ, R ) e − ( n − ) h − p ( ω ) r c ( ω ) )for all i ∈ X n ( c ( ω )), where C (cid:62) c ( ω ) ∈ J p ( T ) such that X n ( c ( ω )) (cid:54) = ∅ . We have | c ( ω ) − z | ≈ φ ( z, ρ ) h − p ( ω ) r c ( ω ) (5.7)for all z ∈ J ( T ) and ρ > φ ( z, ρ ) is defined via c ( ω ) in the context of Theorem 2.3.This is proved in [18]. Specifically, [18, Equations (5.3) and (5.5)] give φ ( z, ρ ) h − p ( ω ) ≈ | T n ( z ) − ω | n ∈ N is such that T n ( c ( ω )) = ω and | ( T n ) (cid:48) ( c ( ω )) | ≈ r − c ( ω ) (we note that the φ used in [18]is different than the notation we are using, so we have translated the equation into our notationwhich is consistent with [45],[46]). Then, applying the Koebe Distortion Theorem, | T n ( z ) − ω | ≈ r − c ( ω ) | z − c ( ω ) | establishing (5.7).Suppose i ∈ X N ( c ( ω )) for some large N , which implies φ ( x i , r ) (cid:54) e − ( N − φ ( ξ, R ). Recall that r c ( ω ) (cid:62) R . Let j ∈ N be such that n j +1 = n j +1 ( x i ) satisfies | ( T n j +1 ) (cid:48) ( x i ) | − = r j +1 ( x i ) (cid:54) r < r j ( x i ) and T n j +1 − ( x i ) ∈ U ω . Moreover, for n such that T n ( c ( ω )) = ω and | ( T n ) (cid:48) ( c ( ω )) | ≈ r − c ( ω ) , we have T n ( ξ ) ∈ U ω . Then, since | ( T n j +1 ) (cid:48) ( x i ) | − = r j +1 ( x i ) (cid:54) r < R (cid:54) r c ( ω ) / ≈| ( T n ) (cid:48) ( c ( ω )) | − , we have T k ( ξ ) ∈ U ω for all n < k < n j +1 and hence φ ( ξ, aR ) is also defined via c ( ω ) in the context of Theorem 2.3 for some a ≈
1. Then by (5.7) | ξ − c ( ω ) | (cid:38) φ ( ξ, aR ) h − p ( ω ) r c ( ω ) (cid:38) φ ( ξ, R ) h − p ( ω ) r c ( ω ) since m is doubling, and | x i − c ( ω ) | (cid:46) φ ( x i , r ) h − p ( ω ) r c ( ω ) (cid:46) e − N ( h − p ( ω ) φ ( ξ, R ) h − p ( ω ) r c ( ω ) and therefore (see Figure 17) R (cid:62) | ξ − c ( ω ) | − | x i − c ( ω ) | (cid:38) φ ( ξ, R ) h − p ( ω ) r c ( ω ) (5.8)for N chosen large enough depending only on various implicit constants. We fix such N in thefollowing discussion. x i ξc ( ω ) R (cid:38) φ ( ξ, R ) h − p ( ω ) r c ( ω ) (cid:46) e − N ( h − p ( ω ) φ ( ξ, R ) h − p ( ω ) r c ( ω ) Figure 17: Bounding R from below.We may assume X n ( c ( ω )) (cid:54) = ∅ for some n (cid:62) N , since otherwise φ ( x i , r ) (cid:38) φ ( ξ, R ) for all i ∈ X and the argument bounding | X | also applies to bound | X | .56y Theorem 2.3 m (cid:91) i ∈ X n ( c ( ω )) B ( x i , r ) (cid:46) m (cid:16) B ( c ( ω ) , C ( φ ( ξ, R ) e − ( n − ) h − p ( ω ) r c ( ω ) ) (cid:17) (cid:46) ( e − n φ ( ξ, R )) h ( h − p ( ω ) r hc ( ω ) e − n φ ( ξ, R ) . In the other direction, as { x i } i ∈ X n ( c ( ω )) is an r -packing, m (cid:91) i ∈ X n ( c ( ω )) B ( x i , r ) (cid:62) (cid:88) i ∈ X n ( c ( ω )) m ( B ( x i , r )) (cid:38) | X n ( c ( ω )) | r h e − n φ ( ξ, R )and so | X n ( c ( ω )) | (cid:46) ( e − n φ ( ξ, R )) h ( h − p ( ω ) r hc ( ω ) r − h (cid:46) e − nh ( h − p max (cid:18) Rr (cid:19) h by (5.8). The number of distinct squeezed canonical balls giving rise to non-empty X n ( c ( ω ))with n (cid:62) N is (cid:46) φ ( ξ, aR ) is defined via c ( ω ) for some a ≈ c ( ω ) (see theargument leading up to (5.7)). Therefore | X n | (cid:46) e − nh ( h − p max (cid:18) Rr (cid:19) h . Pulling these estimates together, we get | X | = | X | + ∞ (cid:88) n =1 | X n | (cid:46) (cid:18) Rr (cid:19) h + ∞ (cid:88) n =1 e − nh ( h − p max (cid:18) Rr (cid:19) h (cid:46) (cid:18) Rr (cid:19) h . (5.9)If i ∈ X , then R h φ ( ξ, R ) (cid:38) m ( B ( ξ, R )) (cid:38) min i ∈ X | X | r h φ ( x i , r ) (cid:38) | X | r h (cid:16) rR (cid:17) ε which proves | X | (cid:46) (cid:18) Rr (cid:19) h + ε . (5.10)Finally, we turn our attention to X n . If i ∈ X n for n (cid:62)
2, then φ ( x i , r ) < e − ( n − , and thereforeby Lemma 5.2 the ball B ( x i , r ) is contained in the squeezed canonical ball B ( c ( ω ) , Ce − ( n − h − p ( ω ) r c ( ω ) )for some c ( ω ) ∈ J p ( T ). Therefore, r/C (cid:54) r c ( ω ) < R < CR and, noting that h > | c ( ω ) − ξ | (cid:54) | c ( ω ) − x i | + | x i − ξ | (cid:54) Cr c ( ω ) + R (cid:54) CR + R (cid:54) CR and so c ( ω ) ∈ B ( ξ, CR ). For p ∈ { p min , . . . , p max } , let X pn = { i ∈ X n | p ( ω ) = p } I p = (cid:91) ω ∈ Ω p ( ω )= p I where I is defined in the same way as in Lemma 5.1. Then we have m (cid:91) i ∈ X pn B ( x i , r ) (cid:54) m (cid:91) c ( ω ) ∈ I p ∩ B ( ξ, CR )6 CR>r c ( ω ) (cid:62) r/C B ( c ( ω ) , Ce − ( n − h − p r c ( ω ) ) (cid:54) (cid:88) c ( ω ) ∈ I p ∩ B ( ξ, CR )6 CR>r c ( ω ) (cid:62) r/C m (cid:18) B ( c ( ω ) , Ce − ( n − h − p r c ( ω ) ) (cid:19) (cid:46) (cid:88) c ( ω ) ∈ I p ∩ B ( ξ, CR )6 CR>r c ( ω ) (cid:62) r/C e − nh ( h − p r hc ( ω ) φ ( c ( ω ) , Ce − ( n − h − p r c ( ω ) ) by Theorem 2.3 (cid:46) (cid:88) c ( ω ) ∈ I p ∩ B ( ξ, CR )6 CR>r c ( ω ) (cid:62) r/C e − nh ( h − p r hc ( ω ) e − ( n − by Theorem 2.3 (ii) (cid:46) e − n e − nh ( h − p (cid:0) log( R/r ) + log(6 C ) (cid:1) m ( B ( ξ, R )) by Lemma 5.1 (cid:46) e − n e − nh ( h − p log( R/r ) R h φ ( ξ, R ) (cid:46) e − n e − nh ( h − p ε − nR h . The final line uses the estimate ( r/R ) ε > e − n which holds whenever X n is non-empty. On theother hand, by Theorem 2.3 m (cid:91) i ∈ X pn B ( x i , r ) (cid:62) (cid:88) i ∈ X pn m ( B ( x i , r )) (cid:38) | X pn | r h e − n . Therefore | X pn | (cid:46) ε − ne − nh ( h − p (cid:18) Rr (cid:19) h (cid:46) ε − ne − nh ( h − p max (cid:18) Rr (cid:19) h which gives | X n | (cid:54) p max (cid:88) p = p min | X pn | (cid:46) ε − ne − nh ( h − p max (cid:18) Rr (cid:19) h . (5.11)Combining (5.9),(5.10) and (5.11), we have | X | = | X | + | X | + ∞ (cid:88) n =2 | X n | (cid:46) (cid:18) Rr (cid:19) h + (cid:18) Rr (cid:19) h + ε + ∞ (cid:88) n =2 ε − ne − nh ( h − p max (cid:18) Rr (cid:19) h (cid:46) (cid:18) Rr (cid:19) h + ε + ε − (cid:18) Rr (cid:19) h which proves that dim A J ( T ) (cid:54) h + ε , and letting ε → A J ( T ) (cid:54) h , as required.58 .8 The lower dimension of J ( T ) The upper bound will follow from our upper bound for the lower spectrum of J ( T ), see Section5.10. Therefore we only need to prove the lower bound. • We show dim L J ( T ) (cid:62) min { , h } .Note that when h (cid:62)
1, we have dim L J ( T ) (cid:62) dim L m (cid:62) h < ξ ∈ J ( T ), and R > r >
R/r (cid:62)
10. Let { B ( y i , r ) } i ∈ Y be a centred r -covering of B ( ξ, R ) ∩ J ( T ) of minimal cardinality. We assume for convenience that each y i ∈ J r ( T ), whichwe may do since J r ( T ) is dense in J ( T ). Each y i has a particular ω = ω ( i ) ∈ Ω associatedwith it, coming from the global measure formula for m ( B ( x i , r )). In particular, y i belongs to anassociated canonical ball B ( c ( ω ) , r c ( ω ) ).Decompose Y as Y = Y ∪ Y where Y = { i ∈ Y | y i ∈ B ( c ( ω ) , r c ( ω ) ) with r c ( ω ) (cid:62) R } Y = Y \ Y . As { B ( y i , r ) } i ∈ Y is a covering of B ( ξ, R ) ∩ J ( T ), we have m ( B ( ξ, R )) (cid:54) m ( ∪ i ∈ Y B ( y i , r ))= m ( ∪ i ∈ Y B ( y i , r )) + m ( ∪ i ∈ Y B ( y i , r )) (5.12)and therefore one of the terms in (5.12) must be at least ( m ( ξ, R )) / Y is at least ( m ( ξ, R )) /
2. Then we write Y = { i ∈ Y | φ ( y i , r ) (cid:54) Kφ ( ξ, R ) } where K >
Lemma 5.3.
We may choose
K > independently of R and r sufficiently large such that m (cid:16) ∪ i ∈ Y \ Y B ( y i , r ) (cid:17) m ( B ( ξ, R )) (cid:54) . (5.13) Proof.
Write Y ( c ( ω )) to denote the set of all i ∈ Y \ Y such that B ( y i , r ) ⊆ B ( c ( ω ) , Cφ ( y i , r ) h − p ( ω ) r c ( ω ) ) ⊆ B ( c ( ω ) , C ( Kφ ( ξ, R )) h − p ( ω ) r c ( ω ) ) (5.14)using the definition of Y \ Y and where C (cid:62) i ∈ Y \ Y belong to some Y ( c ( ω )). Consider non-empty Y ( c ( ω )). Since r c ( ω ) (cid:62) R wemay follow the proof of (5.8) to show that, provided Y ( c ( ω )) (cid:54) = ∅ , φ ( ξ, aR ) is defined via c ( ω )in the context of Theorem 2.3 for some a ≈ K can be chosen large enough such that | c ( ω ) − ξ | (cid:46) R . Then, since m is doubling, by Theorem 2.3 R h φ ( ξ, R ) ≈ m ( B ( ξ, R )) ≈ m ( B ( c ( ω ) , R )) ≈ R h φ ( c ( ω ) , R ) ≈ R h (cid:18) Rr c ( ω ) (cid:19) ( h − p ( ω ) r c ( ω ) φ ( ξ, R ) h − p ( ω ) ≈ R. (5.15)Applying Theorem 2.3 and (5.14), m (cid:0) ∪ i ∈ Y ( c ( ω )) B ( y i , r ) (cid:1) m ( B ( ξ, R )) (cid:46) (cid:16) ( Kφ ( ξ, R )) h − p ( ω ) (cid:17) h +( h − p ( ω ) r hc ( ω ) φ ( ξ, R ) R h = K h ( h − p ( ω ) φ ( ξ, R ) h − p ( ω ) r c ( ω ) R h (cid:46) K h ( h − p ( ω ) by (5.15). Note that the number of distinct squeezed canonical balls giving rise to non-empty Y ( c ( ω )) is (cid:46)
1. This is because φ ( ξ, aR ) is defined via c ( ω ) for some a ≈ c ( ω ).Using the general bound h > p max / (1 + p max ), we see 1 + h/ (( h − p ( ω )) <
0, and therefore wemay choose K large enough to ensure (5.13).Applying (5.13), we have m (cid:16) ∪ i ∈ Y B ( y i , r ) (cid:17) ≈ m ( ∪ i ∈ Y B ( y i , r )) (cid:62) m ( B ( ξ, R )) / R h φ ( ξ, R ) (cid:46) m ( B ( ξ, R )) (cid:46) m (cid:16) ∪ i ∈ Y B ( y i , r ) (cid:17) (cid:46) | Y | r h φ ( ξ, R )where the last inequality uses the definition of Y . Therefore | Y | (cid:62) | Y | (cid:38) (cid:18) Rr (cid:19) h . (5.16)Now, suppose that the second term of (5.12) is at least m ( B ( ξ, R )) /
2. Let ε > Y = (cid:26) i ∈ Y | φ ( y i , r ) (cid:54) (cid:18) Rr (cid:19) ε (cid:27) . If i ∈ Y \ Y , then this implies that φ ( y i , r ) > ( R/r ) ε , and therefore by Lemma 5.2 the ball B ( y i , r ) is contained in the squeezed canonical ball B (cid:32) c ( ω ) , C (cid:18) Rr (cid:19) ε ( h − p ( ω ) r c ( ω ) (cid:33) for some c ( ω ) ∈ J p ( T ). Therefore, recalling the definition of Y , r/C (cid:54) r c ( ω ) < R < CR and,using h < | c ( ω ) − ξ | (cid:54) | c ( ω ) − y i | + | y i − ξ | (cid:54) Cr c ( ω ) + R (cid:54) CR + R (cid:54) CR c ( ω ) ∈ B ( ξ, CR ). Therefore m (cid:16) ∪ i ∈ Y \ Y B ( y i , r ) (cid:17) (cid:46) (cid:88) c ( ω ) ∈ J p ( T ) ∩ B ( ξ, CR )6 CR>r c ( ω ) (cid:62) r/C m (cid:32) B ( c ( ω ) , C (cid:18) Rr (cid:19) ε ( h − p ( ω ) r c ( ω ) ) (cid:33) (cid:46) (cid:88) c ( ω ) ∈ J p ( T ) ∩ B ( ξ, CR )6 CR>r c ( ω ) (cid:62) r/C (cid:18) Rr (cid:19) εh ( h − p ( ω ) r hc ( ω ) φ (cid:32) c ( ω ) , (cid:18) Rr (cid:19) ε ( h − p ( ω ) r c ( ω ) (cid:33) (cid:46) (cid:88) c ( ω ) ∈ J p ( T ) ∩ B ( ξ, CR )6 CR>r c ( ω ) (cid:62) r/C (cid:18) Rr (cid:19) εh ( h − p ( ω ) r hc ( ω ) (cid:18) Rr (cid:19) ε (since c ( ω ) ∈ J p ( T )) (cid:46) (cid:0) log( R/r ) + log(6 C ) (cid:1) (cid:18) Rr (cid:19) ε ( h +( h − p max)( h − p max m ( ξ, R )where the last inequality uses Lemma 5.1 and the fact that m is doubling. Note that the exponentof the term involving R/r is negative, recalling that h > p max / (1 + p max ), so for sufficiently large R/r , balls with centres in Y \ Y cannot carry a fixed proportion of m ( B ( ξ, R )), and so m (cid:16) ∪ i ∈ Y B ( y i , r ) (cid:17) ≈ m ( ∪ i ∈ Y B ( y i , r )) (cid:62) m ( B ( ξ, R )) / . Therefore, we have R h φ ( ξ, R ) (cid:46) m ( B ( ξ, R )) (cid:46) m (cid:16) ∪ i ∈ Y B ( y i , r ) (cid:17) (cid:46) | Y | r h (cid:18) Rr (cid:19) ε where the last inequality uses the definition of Y . Therefore | Y | (cid:62) | Y | (cid:38) (cid:18) Rr (cid:19) h − ε φ ( ξ, R ) (cid:62) (cid:18) Rr (cid:19) h − ε . (5.17)We have proven that at least one of (5.16) and (5.17) must hold, and therefore | Y | = | Y | + | Y | (cid:38) (cid:18) Rr (cid:19) h − ε which proves that dim L J ( T ) (cid:62) h − ε , and letting ε → J ( T ) h < • We show dim θ A J ( T ) = h + min (cid:110) , θp max − θ (cid:111) (1 − h ).The upper bound follows fromdim θ A J ( T ) (cid:54) dim θ A m (cid:54) h + min (cid:26) , θp max − θ (cid:27) (1 − h )61nd for the lower bound, we can apply Proposition 4.7. As h <
1, we have dim L J ( T ) =dim B J ( T ) = h and the upper bound for dim θ A J ( T ) shows that ρ (cid:62) / (1 + p max ), so we needonly show that ρ (cid:54) p max ). To do this, let ω ∈ Ω such that p ( ω ) = p max , and recall thatthere exists some sufficiently small neighbourhood U ω = B ( ω, r ω ) such that there exists a uniqueholomorphic inverse branch T − ω of T on U ω such that T − ω ( ω ) = ω . Using [17, Lemma 1], wehave that for ξ ∈ U ω ∩ (cid:0) J ( T ) \ { ω } (cid:1) and n ∈ N , | T − nω ( ξ ) − ω | ≈ n − p max which, using T -invariance of J ( T ) and bi-Lipschitz stability of the Assouad spectrum, impliesthat dim θ A J ( T ) (cid:62) dim θ A { n − /p max : n ∈ N } = min (cid:26) , p max (1 + p max )(1 − θ ) (cid:27) by [25, Corollary 6.4]. This is enough to ensure ρ (cid:54) / (1 + p max ). Therefore, by Proposition4.7, for θ ∈ (0 , / (1 + p max )) we havedim θ A J ( T ) (cid:62) h + (1 − / (1 + p max )) θ (1 − θ ) / (1 + p max ) (1 − h ) = h + θp max − θ (1 − h )as required. h (cid:62) • We show dim θ A J ( T ) = h .This follows easily, since h = dim B J ( T ) (cid:54) dim θ A J ( T ) (cid:54) dim A J ( T ) = h. J ( T ) h < • We show dim θ L J ( T ) = h .This follows easily, since h = dim L J ( T ) (cid:54) dim θ L J ( T ) (cid:54) dim B J ( T ) = h. h (cid:62) • We show dim θ L J ( T ) = h + min (cid:110) , θp max − θ (cid:111) (1 − h ).Note that we have dim θ L J ( T ) (cid:62) dim θ L m (cid:62) h + min (cid:26) , θp max − θ (cid:27) (1 − h )and so it suffices to prove the upper bound. To do this, we require the following technicallemma, which is a quantitative version of the Leau-Fatou flower theorem (see [30, 325–363] and[40]). This was not known to us initially, but seems to be standard in the complex dynamicscommunity. We thank Davoud Cheraghi for explaining this version to us. We note that thenon-quantitative version, e.g. that stated in [1, 18], is enough to bound the lower dimensionfrom above, but not the lower spectrum. 62 emma 5.4. Let ω ∈ Ω be a parabolic fixed point with petal number p ( ω ) . Then there existsa constant C > such that for all sufficiently small r > , B ( ω, r ) ∩ J ( T ) is contained in a Cr p ( ω ) -neighbourhood of the set of p ( ω ) lines emanating from ω in the repelling directions.Proof. We only sketch the proof. We may assume via standard reductions that ω = 0 and thatthe repelling directions are e n πi/p ( ω ) for n = 0 , , . . . , p ( ω ) −
1. By the (non-quantitative) Leau-Fatou flower theorem, B (0 , r ) ∩ J ( T ) is contained in a cuspidal neighbourhood of the repellingdirections. Apply the coordinate transformation z (cid:55)→ /z p ( ω ) which sends the fixed point toinfinity and the repelling directions to 1. The linearisation of the conjugated map at infinity isa (real) translation and this linearisation can be used to show that the Julia set is contained ina half-infinite horizontal strip of bounded height. It is instructive to compare this to Bowditch’stheorem (Lemma 4.8). The pre-image of this strip under the coordinate transformation consistsof cuspidal neighbourhoods of the p ( ω ) th roots of unity, and an easy calculation gives the desiredresult.We can use our work on the lower spectrum of m to show that the exponent used in Lemma5.4 is sharp. Again, this seems to be well-known in the complex dynamics community (even astronger form of sharpness than we give) but we mention it since we provide a new approach. Corollary 5.5.
In the case where ω ∈ Ω is of maximal rank, the expression Cr p ( ω ) in Lemma5.4 cannot be replaced by Cr p ( ω )+ ε for any ε > .Proof. Suppose that such an ε -improvement was possible for some ω ∈ Ω of maximal rank. Then,taking efficient r -covers of the improved cuspidal neighbourhood of B ( ω, r p max + ε ) ∩ J ( T ), wewould obtain dim θ L J ( T ) (cid:54) θ > / (1 + p max + ε ). This contradicts the lower bound forthe lower spectrum of m , proved in Section 5.6.2.We can now prove the upper bound. Note that when θ (cid:62) / (1 + p max ), we immediately havedim θ L J ( T ) (cid:54) θ ∈ (0 , / (1 + p max )). Let ω ∈ Ω be suchthat p ( ω ) = p max , and let r > N r ( B ( ω, r θ ) ∩ J ( T ))by first covering B ( ω, r θ ) ∩ J ( T ) with balls of radius r θ (1+ p max ) , and then covering each of thoseballs with balls of radius r . Using the fact that dim A J ( T ) = h , we have N r ( B ( ω, r θ ) ∩ J ( T )) (cid:46) N r θ (1+ p max) ( B ( ω, r θ ) ∩ J ( T )) (cid:32) r θ (1+ p max ) r (cid:33) h (cid:46) r θ r θ (1+ p max ) (cid:32) r θ (1+ p max ) r (cid:33) h by Lemma 5.4= r − θp max + h ( θ (1+ p max ) − = (cid:16) r θ − (cid:17) h + θp max θ − ( h − which proves that dim θ L J ( T ) (cid:54) h + θp max − θ (1 − h )as required. 63 cknowledgements. The authors thank Davoud Cheraghi and Mariusz Urba´nski for helpfuldiscussions.
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