Constructing Restricted Patterson Measures for Geometrically Infinite Kleinian Groups

Kurt Falk^{†}^{∗} and Bernd O. Stratmann^{‡}

‡ Mathematical Institute, University of St Andrews, North Haugh, St Andrews KY16 9SS, Scotland

† Department of Mathematics, University of Helsinki, P.O. Box 4, 00014 Helsinki, Finland

Abstract

In this paper we study exhaustions, referred to as ρ-restrictions, of arbitrary non-elementary Kleinian groups with at most finitely many bounded parabolic elements. Special emphasis is put on the geometrically infinite case, where we obtain that the limit set of each of these Kleinian groups contains an infinite family of closed subsets, referred to as ρ-restricted limit sets, such that there is a Poincaré series and hence an exponent of convergence δ_{ρ}, canonically associated to every element in this family. Generalizing concepts which are well-known in the geometrically finite case, we then introduce the notion of ρrestricted Patterson measure, and show that these measures are nonatomic, δ_{ρ}-harmonic, δ_{ρ}-subconformal on special sets and δ_{ρ}-conformal on very special sets. Furthermore, we obtain the results that each ρrestriction of our Kleinian group is of δ_{ρ}-divergence type and that the Hausdorff dimension of the ρ-restricted limit set is equal to δ_{ρ}.

∗Research supported by the Schweizer Nationalfonds No. 20-61379.00 and European Project TMR “Geometric Analysis” ACR-OFES No. UE 00.0349.

1 Introduction

In this paper we consider arbitrary non-elementary Kleinian groups with at most finitely many bounded parabolic elements. Special emphasis will be put on geometrically infinite groups (that is, groups which have fundamental domains with infinitely many faces), although strictly speaking our analysis also applies to the geometrically finite situation where it resembles well-known results from the Patterson-Sullivan theory for geometrically finite Kleinian groups. Recall that every arbitrary Kleinian group admits the construction of the classical Patterson measure, a measure which is canonically obtained from the orbit structure of the group action, and which is always supported on the limit set of the group. For geometrically finite groups this measure is well understood and has proven to be an extremely fruitful tool in the studies of Kleinian groups under various aspects, such as for instance under aspects of group cohomology, or of spectral analysis on hyperbolic manifolds, or also of fractal geometry on limit sets. However, for the class of geometrically infinite Kleinian groups, the situation appears to be completely different and less satisfying. For instance, for so called δ-convergence type Kleinian groups (that is, groups whose associated Poincaré series converge at their abzissa of convergence, and which are necessarily geometrically infinite), Sullivan has shown that the geodesic flow on the associated hyperbolic manifolds is no longer ergodic with respect to the Liouville-Patterson measure, a property which is equivalent to the fact that the Patterson measure vanishes on the set of so-called radial limit points ([19], [22]). In other words, for this class of hyperbolic manifolds the Patterson measure appears to be no longer a useful concept for studying recurrent geodesic dynamics. For this reason it is usually rather difficult, not to say impossible, to transfer the successful techniques, which have been elaborated in connection with this measure in the geometrically finite case, to the studies of geometrically infinite groups by means of their Patterson measure. In this paper we show how to avoid these difficulties by setting up a new type of Patterson measure, referred to as restricted Patterson measure. These measures are specially designed to allow investigations of recurrent geodesic dynamics of geometrically finite type within geometrically infinite hyperbolic manifolds. The idea is to consider finite volume regions of the convex core of the manifold, and to associate to each of these an orbital measure which allows to quantify their internal long range geodesic dynamics. More precisely, within the manifold we consider the family of balls of radius ρ centred at some fixed base point, where ρ is assumed to be sufficiently large such that the balls engulf the regions where the intersections of the convex core with the bounded cusps meet the thick part of the manifold. The finite volume regions, mentioned above, are then given by the so-called restricted cores, that is the intersection of the convex core with the union of the bounded cusps and the ρ-ball. By embedding these restricted cores into the universal covering of the manifold and introducing a coarse geometric way of identifying asymptotic geodesic behaviour within them, we obtain certain subsets G_{ρ} of the underlying Kleinian group G, which we refer to as ρ-restrictions. Now, each of these ρ-restrictions acts on hyperbolic space and hence, by considering the G_{ρ}-orbit of some point in hyperbolic space, it gives rise to a limit set L(G_{ρ}), referred to as ρ-restricted limit set. Clearly, each L(G_{ρ}) is by construction a closed subset of the limit set of G. Also, to each ρ-restriction, we can associate a Poincaré series P_{ρ}(s) and hence an exponent of convergence δ_{ρ}, which then allows to mimic the construction of the classical Patterson measure within this ‘restricted setting’. In this way we derive for any arbitrary observation points x in hyperbolic space our so-called ρ-restricted Patterson measure μ_{ρ}^{x} with support equal to L(G_{ρ}). The paper continues by giving a geometric analysis of the class of ρ-restricted Patterson measures. The main results of this analysis are the following (Theorem 3.3).

• The measure μ_{ρ}^{x} is non-atomic.

• G_{ρ} is of δ_{ρ}-divergence type, meaning that P_{ρ}(s) diverges for s = δ_{ρ}.

• The Hausdorff dimension of L(G_{ρ}) is equal to δ_{ρ}.

Part of the proof of this theorem is to show that the Poincaré exponent δ(G) of G and the exponents δ_{ρ} of the ρ-restrictions are related as follows. We remark that this result may be also be of independent interest in the theory of dynamical systems.

• lim_{ρ→∞} δ_{ρ} = δ(G). (Proposition 3.1)

Subsequently, we also derive the following two results, which mark the similarities between the classical and the ρ-restricted Patterson measures. In here δ_{ρ}-harmonic refers to the fact that the Radon-Nikodym derivative of μ_{ρ}^{x} with respect to μ_{ρ}^{0} can be expressed in terms of the δ_{ρ}-th power of the Poisson kernel. Note that with this notation the classical Patterson measure is in fact δ(G)-harmonic. Also, note that as a consequence of the δ_{ρ}-harmonicity we have that the measure class of μ_{ρ}^{x} is invariant under changes of the observation point x. We refer to the end of Section 3 for a more detailed definition of δ_{ρ}-harmonic and for the meaning of ‘δ_{ρ}-subconformal on special sets’ and ‘δ_{ρ}-conformal on very special sets’.

• The measure μ_{ρ}^{x} is δ_{ρ}-harmonic. (Lemma 3.4)

• The measure μ_{ρ}^{x} is δ_{ρ}-subconformal on special sets, and it is δ_{ρ}-conformal on very special sets. (Lemma 3.5)

We remark that Kleinian groups of δ(G)-convergence type provide a particularly interesting class of examples to which our analysis applies. Clearly, since ρ-restrictions are of δ_{ρ}-divergence type, for these groups we have that δ_{ρ} is always strictly less than δ(G). Note that this class of examples includes all geometrically infinite, finitely generated Kleinian groups acting in hyperbolic 3-space which have the property that the area of their limit sets vanishes. (This follows since in this case δ(G) = 2 ([4], [5]), and hence 2-divergence-type would imply that the radial limit set is of full Patterson measure ([20]), contradicting the fact that the area of the limit set vanishes. Also note that Sullivan has shown that these groups have only finitely many non-equivalent parabolic elements ([21])). - On the other hand, for geometrically finite Kleinian groups, our analysis is trivial in the sense that it leads to well-known results from the Patterson-Sullivan theory. (This follows since in this case the ρ-restricted cores which we consider are always equal to the convex core of the manifold, and therefore δ_{ρ} is always equal to δ(G)). - These observations clearly illustrate in which way the constructions in this paper generalize the classical Patterson measure for geometrically finite groups. The paper is organised as follows. In Section 2 we first introduce the basic notions in connection with our ‘concept of restrictions’, and then discuss a few technical observations which will be required later. Section 3 starts with the construction of the ρ-restricted Patterson measures. This is followed by a finer analysis of these measures, which then leads to the main results of this paper. Furthermore, the Appendix contains a detailed description of ‘the method of a slowly varying function’, which was first used by Patterson in [13] to construct orbital measures on limit sets of Fuchsian groups (and which is referred to by Sullivan as ‘nifty’ in his generalizations of the Patterson measure to Kleinian groups ([23])). We have included this description, since our construction is a slightly modified version of Patterson’s original construction, and also mainly, because the literature seems not to contain a description of this method in all its details (and hence we hope that the reader may find it helpful to see such a detailed discussion). Finally, we remark that this paper was originally inspired by the generalizations of the notion of conformal measure to Julia sets of rational maps containing critical points by Denker and Urbański (see e.g. [6], [7], [24]). However, it seems that their construction, which uses ergodic theory and in particular the thermodynamical formalism, is only vaguely connected to our construction in this paper, which is more canonical in the sense that it is purely geometric. Throughout, we use the following conventions to describe the relationship between two positive real numbers a and b. We write a ≍ b to mean that the ratio of a and b is uniformly bounded away from 0 and infinity, and we write a ≪ b if a/b is uniformly bounded from above. Furthermore, if exp(a) ≍ exp(b), then we write a ≍_{+} b.

2 Restricted limit sets and their geometry

We assume that the reader is familiar with the basic theory of Kleinian groups (see e.g. [2], [11]). Throughout, let G be a non-elementary Kleinian group with a at most finite set P = {p_{1}, …, p_{k}} of non-equivalent bounded standard parabolic fixed points (where P is allowed to be the empty set). It is well-known that to each p ∈ P we can associate a horoball H_{p} intersecting a fundamental domain at the origin such that the G-orbit G(H) := {g (H_{p}) : g ∈ G, p ∈ P } represents a packing of hyperbolic space 𝔻^{N+1} by mutually pairwise disjoint balls tangent to the boundary S^{N} of 𝔻^{N+1}. Let C = C(G) denote the convex hull within 𝔻^{N+1} of the limit set L(G) of G, and choose 0 ∈ 𝔻^{N+1} as a fixed reference point. Fix ρ_{0} such that the geodesics between different elements of P are contained in B_{ρ_{0}}(0) ∪ ⋃_{p∈P} H_{p}, and such that for the horospherical boundaries H_{p} of the horoballs H_{p} we have that ⋃_{g∈G} g(B_{ρ_{0}}(0)) ⊃ C ∩ ⋃_{p∈P} ⋃_{g∈G} g(H_{p}). (Here, B_{r}(z) refers to the open hyperbolic ball centred at z of radius r). For ρ > ρ_{0}, we define the ρ-restricted convex core of G by

C_{ρ} := C ∩ (G(H) ∪ ⋃_{g∈G} g(B_{ρ}(0))) .

Also, we introduce the notion of the core shade S_{ρ}(x, y), which is given for x, y ∈ 𝔻^{N+1} such that B_{ρ}(x) ∩ B_{ρ}(y) = ∅, by

S_{ρ}(x, y) := {z ∈ 𝔻^{N+1} \ B_{ρ}(y) : s_{w,z} ∩ B_{ρ}(y) ≠ ∅ for all w ∈ B_{ρ}(x)}.

In here, s_{x,y} refers to the geodesic segment between x and y. Note that a core shade is by definition an open subset of 𝔻^{N+1}. Furthermore, if the boundary at infinity ∂E of a set E ⊂ 𝔻^{N+1} is defined as the Euclidean interior in S^{N} of \overline{E} \ (E ∪ cl(E)), where cl(·) refers to the closure in 𝔻^{N+1}, and \overline{·} to the closure in ℝ^{N+1}, then we have for the boundary at infinity of S_{ρ}(x, y) that

∂S_{ρ}(x, y) = ⋂_{w∈B_{ρ}(x)} Π_{w}(B_{ρ}(y)).

In here, Π_{w} refers to the shadow projection based at w ∈ 𝔻^{N+1} which is given for E ⊂ 𝔻^{N+1} by Π_{w}(E) := {ξ ∈ S^{N} : s_{w,ξ} ∩ E ≠ ∅}. Finally, we define the ρ-restricted core shade

Ŝ_{ρ}(x, y) := (𝔻^{N+1} \ U_{ρ} (𝔻^{N+1} \ S_{ρ}(x, y))) ∪ \overline{∂S_{ρ}(x, y)},

where U_{ρ}(·) refers to the open hyperbolic ρ-neighbourhood of a subset of 𝔻^{N+1}. Note that Ŝ_{ρ}(x, y) is by construction a closed subset of 𝔻^{N+1}. The following definition introduces the types of limit sets which will be important throughout.

Definition 2.1. Let ρ > ρ_{0} and g ∈ G be given. A geodesic l is called (ρ, g)-visible if and only if it is fully contained in C_{ρ} and intersects B_{ρ}(g(0)). The (ρ, g)-restriction G_{ρ}^{g} is then given by

G_{ρ}^{g} := {h ∈ G : l ∩ B_{ρ}(h(0)) ≠ ∅ for some (ρ, g)−visible l} ,

and the (ρ, g)-restricted limit set L(G_{ρ}^{g}) of G is the derived set of G_{ρ}^{g}(0), which is defined by

L(G_{ρ}^{g}) := \overline{G_{ρ}^{g}(0)} \ G_{ρ}^{g}(0).

Also, we introduce the following two subsets of L(G_{ρ}^{g}).

• The set of ρ-radial limit points of G_{ρ}^{g} will be denoted by L_{r}(G_{ρ}^{g}). Where ξ is called ρ-radial if there exists a geodesic l in cl(C_{ρ}) with endpoint ξ such that l intersects cl(B_{ρ}(g(0))), and such that each ray r ⊂ l towards ξ intersects cl(B_{ρ}(h(0))) for infinitely many h ∈ G_{ρ}^{g}.

• The set of bounded parabolic fixed points of G which are contained in L(G_{ρ}^{g}) will be denoted by L_{p}(G_{ρ}^{g}).

For ease of notation, we let G_{ρ} := G_{ρ}^{{id.}}, and if a geodesic l is (ρ, {id.})-visible then we shall refer to it as ρ-visible.

We now collect a few observations which will turn out to be helpful in our measure theoretical analysis of the following section. We shall see first that L(G_{ρ}^{g}) can be written as the disjoint union of L_{p}(G_{ρ}^{g}) and L_{r}(G_{ρ}^{g}). This observation represents the analogon of the well-known corresponding result of Beardon and Maskit in the geometrically finite case (cf. [3]).

Proposition 2.2. For ρ > ρ_{0} and g ∈ G the following holds.

(i) For each ξ ∈ L(G_{ρ}^{g}) there exists a geodesic ending at ξ which is contained in cl(C_{ρ})and has non-empty intersection with cl(B_{ρ}(g(0))).

(ii) L(G_{ρ}^{g}) = L_{r}(G_{ρ}^{g}) ∪ L_{p}(G_{ρ}^{g}).

Proof. We give the proof for g = {id.} and remark that the general case follows exactly in the same way. In order to prove the statement in (i), let ξ ∈ L(G_{ρ}) be given. Then there exists a sequence (g_{n}) in G_{ρ} such that g_{n}(0) tends to ξ in the Euclidean metric. Let l_{n} denote the ρ-visible geodesic associated with g_{n} by definition of G_{ρ}, and let ξ_{n} and η_{n} denote its endpoints. Without loss of generality we can assume that ξ_{n} tends to ξ. It follows that (l_{n}) accumulates at some geodesic l which terminates at ξ. By passing to a subsequence if necessary, we can assume without loss of generality that l_{n} converges to l. Since l_{n} is contained in C_{ρ} for all n, and since C_{ρ} is open, it follows that l is contained in cl(C_{ρ}). Furthermore, each l_{n} has non-empty intersection with B_{ρ}(0), which implies that l intersects cl(B_{ρ}(0)). In order to prove the statement in (ii), let again ξ ∈ L(G_{ρ}) and let l be the associated geodesic which we obtained in the proof of (i) above. In this situation a ray towards ξ which is contained in l either intersects cl(B_{ρ}(g(0))) for infinitely many g ∈ G_{ρ}, or is eventually contained in h(H_{p}), for some h ∈ G and p ∈ P. In the first case it follows that ξ ∈ L_{r}(G_{ρ}). In the second case we conclude that ξ = h(p) ∈ L_{p}(G_{ρ}).

For the remaining part of this section we restrict the discussion to the (ρ, {id.})-restriction G_{ρ} of G. To further clarify the significance of a radial limit point, we introduce the following concept of a ρ-trace at a point in L_{r}(G_{ρ}).

Definition 2.3. For ρ > ρ_{0} and ξ ∈ L_{r}(G_{ρ}), the ρ-trace at ξ consists of the optimal sequence (g_{m}) of elements g_{m} ∈ G_{ρ} such that ξ ∈ ∂S_{ρ}(0, g_{m}(0)) for all m and such that the Euclidean diameter of ∂S_{ρ}(0, g_{m}(0)) tends to 0 monotonically, for m tending to infinity.

Lemma 2.4. There exists a ρ-trace at each ξ ∈ L_{r}(G_{ρ}).

Proof. Let ξ ∈ L_{r}(G_{ρ}) be given. Then there exists a geodesic l_{ξ} in cl(C_{ρ}) ending at ξ, and an optimal sequence (g_{m}) in G_{ρ} with g_{1} = {id.} and such that l_{ξ} has non-empty intersection with cl(B_{ρ}(g_{m}(0))), for all m. With e_{m} referring to the hyperbolic distance between g_{m}(0) and l_{ξ}, we clearly have that lim inf e_{m} ≤ ρ. We consider the cases lim inf e_{m} < ρ and lim inf e_{m} = ρ separately. For lim inf e_{m} < ρ we have that there exist ε > 0 and a ray r_{ξ} towards ξ with initial point somewhere in B_{ρ}(0), such that r_{ξ} intersects B_{ρ−ε}(g_{m}(0)) for all m sufficiently large. If we assume that there is no ρ-trace at ξ, then there exists a ray r_{ξ}^{∗} towards ξ with initial point somewhere in B_{ρ}(0), such that r_{ξ}^{∗} has empty intersection with B_{ρ}(g_{m}(0)) for all m sufficiently large. Combining these two observations, it follows that there exists a nested sequence (H_{m}(ξ)) of horospheres based at ξ such that for x_{m} := H_{m}(ξ)∩r_{ξ} and x_{m}^{∗} := H_{m}(ξ)∩r_{ξ}^{∗} we have that d(x_{m}, x_{m}^{∗}) ≥ ε for all m sufficiently large. On the other hand, recall that r_{ξ} and r_{ξ}^{∗} are asymptotic which gives that lim_{m→∞} d(x_{m}, x_{m}^{∗}) = 0, and hence leads to a contradiction. If lim inf e_{m} = ρ, then we in fact have that lim e_{m} = ρ, and we now show that this case can not occur. Let r′_{ξ} denote some ray towards ξ which starts in cl(B_{ρ}(0)) and which is contained in l_{ξ}. Let σ_{m} := r′_{ξ} ∩ cl(B_{ρ}(g_{m}(0))), and note that the hyperbolic length of σ_{m} tends to 0 for m tending to infinity. Note that eventually σ_{m} has to lie outside G(H), since otherwise r′_{ξ} would leave G(H) infinitely often and hence would intersect B_{c}(h(0)) for infinitely many h ∈ G_{ρ}, for some constant c > 0 which depends only on the distance of 0 to H. This clearly contradicts the fact that the hyperbolic length of σ_{m} tends to 0. Now, consider geodesic segments of some fixed hyperbolic length which are contained in r′_{ξ} and which are sufficiently far away from the origin. Again since the hyperbolic length of σ_{m} tends to 0, we have that the number of balls cl(B_{ρ}(g_{m}(0))) required to cover such a segment increases if the distance of the segment to the origin grows. This clearly contradicts the fact that G_{ρ} acts discontinuously on 𝔻^{N+1}, and hence the lemma follows.

Finally, we now outline a few algebraic properties of G_{ρ}. First note that since the composition of two elements of G_{ρ} is not necessarily again an element of G_{ρ}, it follows that in general we do not have that G_{ρ} is a group. Nevertheless, as we shall see now, G_{ρ} has certain weaker properties of invariance under compositions.

Lemma 2.5. With the notation above, the following holds.

(i) {id.} ∈ G_{ρ}.

(ii) g ∈ G_{ρ} if and only if g^{−1} ∈ G_{ρ}.

(iii) If g ∈ G and h ∈ G_{ρ} are such that B_{ρ}(g(0)) does not intersect B_{ρ}(0) and B_{ρ}(h(0)) is contained in S_{ρ}(0, g(0)), then g, g^{−1}h ∈ G_{ρ}.

Proof. The statements in (i), (ii) and the conclusion in (iii) that g ∈ G_{ρ} are immediate consequences of the definition of G_{ρ}. In order to prove the remaining assertion, let l be a ρ-visible geodesic which intersects B_{ρ}(h(0)). Since B_{ρ}(h(0)) is a subset of S_{ρ}(0, g(0)), we have by definition of S_{ρ}(0, g(0)) that l has non-empty intersection with B_{ρ}(g(0)). Hence, g^{−1}(l) is contained in C_{ρ} and intersects each of the balls B_{ρ}(g^{−1}(0)), B_{ρ}(0) and B_{ρ}(g^{−1}h(0)). This implies that g^{−1}h ∈ G_{ρ}.

As an immediate consequence of the preceeding lemma we have the following corollary which will be required in Section 3.

Corollary 2.6. For g ∈ G_{ρ} such that B_{ρ}(g(0)) does not intersect B_{ρ}(0), the following holds.

(i) {gh : h ∈ G_{ρ}, gh(0) ∈ Ŝ_{ρ}(g(0), 0)}
⊂ {k ∈ G_{ρ} : k(0) ∈ Ŝ_{ρ}(g(0), 0)} .

(ii) \{k ∈ G_{ρ}^{g} : \begin{matrix}there exists a geodesic l ⊂ cl(C_{ρ}) ∩ cl(S_{ρ}(g(0), 0)) \\ intersecting cl(B_{ρ}(0)), cl(B_{ρ}(g(0))) and cl(B_{ρ}(k(0))) \end{matrix} \}

= \{gh : \begin{matrix}there exists a geodesic l ⊂ cl(C_{ρ}) ∩ cl(S_{ρ}(0, g^{−1}(0))) \\ intersecting cl(B_{ρ}(0)) and cl(B_{ρ}(g^{−1}(0))), and h ∈ G_{ρ} \end{matrix} \} .

3 Restricted Patterson measures

In this section we derive the main results of this paper. First we introduce the concept of the ρ-restricted Poincaré series and their associated exponent of convergence δ_{ρ}. We then give the construction of ρ-restricted Patterson measures which is followed by an analysis of these measures. We shall see that these measures have no atoms and that they have the properties of being δ_{ρ}-harmonic, δ_{ρ}-subconformal on special sets and even δ_{ρ}-conformal on very special sets. Furthermore, we obtain the results that G_{ρ} is of δ_{ρ}-divergence type and that the Hausdorff dimension of L(G_{ρ}) is equal to δ_{ρ}.

Constructing Restricted Patterson Measures. Recall that classically one associates to a Kleinian group G its Poincaré series P(x, s), which is given for x ∈ 𝔻^{N+1} and s ≥ 0 by

P(x, s) := ∑_{g∈G} e^{−s d(x,g(0))} .

The abzissa of convergence of this series, which is referred to as the exponent of convergence of G or sometimes also as the Poincaré exponent, will be denoted as usual by δ = δ(G). More precisely, we have that

δ := inf{s ≥ 0 : P(x, s) converges}.

We remark that δ clearly does not dependent on the particular choice of x. We now modify this classical concept as follows. Let ρ > ρ_{0} be given. For s ≥ 0 and x ∈ 𝔻^{N+1}, we define the ρ-restricted Poincaré series P_{ρ}(x, s) by

P_{ρ}(x, s) := ∑_{g∈G_{ρ}} e^{−s d(x,g(0))} .

Let δ_{ρ} denote the abzissa of convergence of this series, that is

δ_{ρ} := inf{s ≥ 0 : P_{ρ}(x, s) converges}.

Again, we remark that δ_{ρ} does clearly not depend on the particular choice of x. Now, note that it is a priori not clear that P_{ρ}(x, δ_{ρ}) is infinite, or with other words that G_{ρ} is of δ_{ρ}-divergence type. In order to overcome this difficulty, we employ ‘the method of a slowly varying function’, which was first used by Patterson to construct orbital measures on limit sets of Fuchsian groups ([13]). Since our construction requires some modifications of the original construction in [13], and also since the literature seems not to contain a description of this method in all its details, we have included such a description in the appendix and we refer to it for the details. However, the essence of the method is that there exists a function ϕ_{ρ} : G_{ρ} → ℝ^{+} such that for each x ∈ 𝔻^{N+1} the abzissa of convergence of the modified ρ-restricted Poincaré series

P′_{ρ}(x, s) := ∑_{g∈G_{ρ}} (ϕ_{ρ}(g) e^{−d(x,g(0))})^{s}

stays to be equal to δ_{ρ}, whereas we have that P′_{ρ}(x, s) diverges for s = δ_{ρ}. Using this modified ρ-restricted Poincaré series, we can now mimic the construction of the classical Patterson measure in order to derive our restricted Patterson measures as follows. For s > δ_{ρ}, let μ_{ρ,s}^{x} denote the orbital measure given for E ⊂ 𝔻^{N+1} by

μ_{ρ,s}^{x}(E) := \frac{1}{P′_{ρ}(0, s)} ∑_{g∈G_{ρ}} (ϕ_{ρ}(g) e^{−d(x,g(0))})^{s} 1_{g(0)}(E) ,

where 1_{g(0)} refers to the Dirac point mass of weight one at g(0). Fix a sequence (s_{n}) such that lim_{n→∞} s_{n} = δ_{ρ} and such that s_{n} > δ_{ρ} for all n. By Helly’s Theorem we have that there exists a subsequence, which for simplicity will also be denoted by (s_{n}), such that the sequence (μ_{ρ,s_{n}}^{x}) converges weakly to some measure μ_{ρ}^{x} for n tending to infinity. Note that the measure μ_{ρ}^{x} depends on x as well as on the sequence (s_{n}). Therefore, for any given ρ, throughout we shall fix one such sequence (s_{n}) for all x ∈ 𝔻^{N+1} and will assume that it is employed universally in the construction of each μ_{ρ}^{x} (this is justified by a straight-forward adaption of an argument in the construction of the classical Patterson measure (cf. [12] Theorem 3.4.1 and/or Lemma 3.4; see also Proposition 4.5)). Clearly, since P′_{ρ}(x, s) diverges for s = δ_{ρ}, it follows that μ_{ρ}^{x} is supported on L(G_{ρ}), and we shall refer to μ_{ρ}^{x} as the ρrestricted Patterson measure associated to x and G_{ρ}. For ease of notation we shall often write μ_{ρ,s_{n}} instead of μ_{ρ,s_{n}}^{0}, and also μ_{ρ} instead of μ_{ρ}^{0}.

Non-Atomicity, δ_{ρ}-Divergence Type and Hausdorff Dimension. A first important property of ρ-restricted Patterson measures is that they are non-atomic. For the proof we require the following observation which relates δ and δ_{ρ}.

Proposition 3.1. For the critical exponent δ of G we have that δ = lim_{ρ→∞} δ_{ρ}.

Proof. Recall from [16] that an element ξ ∈ L(G) is called c-uniformly radial point of G, for some c > 0, if and only if the geodesic ray s_{ξ} from 0 to ξ is contained in ⋃_{g∈G} B_{c}(g(0)). Let L_{ur}^{c}(G) denote the set of all c-uniformly radial points of G. By generalizing an argument of Bishop and Jones([4]), we obtained in [16] that for each ε > 0 there exists a constant c(ε) such that

dim_{H} L_{ur}^{c(ε)}(G) > δ − ε.

Hence, by definition of Hausdorff dimension (cf. [8]), it follows that

∑_{g∈G_{ur}^{c(ε)}} diam(Π_{0}(B_{c(ε)}(g(0))))^{δ−ε} = ∞,

where diam(·) refers to the Euclidean diameter in S^{N}, and where we have set

G_{ur}^{c(ε)} := {g ∈ G : s_{ξ} ∩ B_{c(ε)}(g(0)) ≠ ∅ for some ξ ∈ L_{ur}^{c(ε)}(G)} .

Now note that if ε > 0 is given, then by choosing ρ sufficiently large we see that L_{ur}^{c(ε)}(G) ⊂ L_{r}(G_{ρ}) as well as G_{ur}^{c(ε)} ⊂ G_{ρ}. Combining these observations, it follows that

P_{ρ}(0, δ − ε) ≥ ∑_{g∈G_{ur}^{c(ε)}} e^{−(δ−ε) d(0,g(0))} ≍ ∑_{g∈G_{ur}^{c(ε)}} diam(Π_{0}(B_{ρ}(g(0))))^{δ−ε} = ∞,

which implies that δ_{ρ} ≥ δ − ε. The statement of the lemma now follows, since ε was chosen to be arbitrary, and since obviously δ_{ρ} ≤ δ.

We are now in the position to give some preliminary geometric measure estimates for μ_{ρ}^{x}. These estimates represent the main ingredients for proving that μ_{ρ}^{x} is non-atomic and that G_{ρ} is of δ_{ρ}-divergence type.

Proposition 3.2. For each x ∈ 𝔻^{N+1} and ρ sufficiently large, the ρ-restricted Patterson measure μ_{ρ}^{x} has the following properties.

(i) If ξ ∈ L_{r}(G_{ρ}), then for each element g_{m} in the ρ-trace at ξ we have that

μ_{ρ}^{x} (∂S_{ρ}(0, g_{m}(0))) ≪ e^{−δ_{ρ} d(0,g_{m}(0))}.

(ii) If ξ ∈ L_{r}(G_{ρ}), then ξ = g(p) for some p ∈ P and g ∈ G_{ρ}, where without loss of generality we assume that g is chosen such that d(0, g(0)) = min{d(0, gh(0)) : h ∈ Γ_{p}} (here Γ_{p} refers to the stabiliser of p in G). Then, for every ε > 0 there exists r_{ε} > 0 such that for each spherical ball b(ξ, r) ⊂ S^{N} centred at ξ of radius 0 < r < r_{ε} we have that

μ_{ρ}^{x}(b(ξ, r)) ≪ e^{−(δ_{ρ}−ε) d(0,g(0))} (re^{d(0,g(0))})^{2(δ_{ρ}−ε)−k(p)} ,

where k(p) refers to the rank of the parabolic fixed point p.

Proof. It is clearly sufficient to prove the assertions for x = 0. In order to prove (i), note that for each h ∈ G_{ρ} such that h(0) ∈ Ŝ_{ρ}(0, g_{m}(0)) the hyperbolic triangle inequality gives that d(0, h(0)) ≍_{+} d(0, g_{m}(0))+d(g_{m}(0), h(0)). Combining this with Corollary 2.6 (i) and Lemma 4.3, we obtain for all n that

∑_{\begin{matrix}h∈G_{ρ} \\ h(0)∈Ŝ_{ρ}(0,g_{m}(0)) \end{matrix}} (ϕ_{ρ}(h) e^{−d(0,h(0))})^{s_{n}}

≍ e^{−s_{n} d(0,g_{m}(0))} ∑_{\begin{matrix}h∈G_{ρ} \\ h(0)∈Ŝ_{ρ}(0,g_{m}(0)) \end{matrix}} (ϕ_{ρ}(h) e^{−d(0,g_{m}^{−1}h(0))})^{s_{n}}

≤ e^{−s_{n} d(0,g_{m}(0))} ∑_{\begin{matrix}k∈G_{ρ} \\ k(0)∈Ŝ_{ρ}(g_{m}^{−1}(0),0) \end{matrix}} (\frac{ϕ_{ρ}(g_{m}k)}{ϕ_{ρ}(k)}ϕ_{ρ}(k) e^{−d(0,k(0))})^{s_{n}}

≪ e^{−s_{n} d(0,g_{m}(0))} P′_{ρ}(0, s_{n}).

Note that in this estimate the final inequality follows since Lemma 4.3 is applicable, and hence ϕ_{ρ}(g_{m}k)/ϕ_{ρ}(k) tends to 1, for d(0, k(0)) tending to infinity. Now, if we divide the so obtained inequality by P′_{ρ}(0, s_{n}) and let s_{n} tend to δ_{ρ}, the statement in (i) follows. In order to prove (ii), let λ_{r} denote the open hyperbolic lens centred at g(p) of Euclidean radius r. That is, λ_{r} is the intersection of 𝔻^{N+1} with the open Euclidean ball in ℝ^{N+1} of radius r which is symmetric about the geodesic ray s_{g(p)} and whose boundary sphere is orthogonal to S^{N}. Since λ_{r} ∪ ∂λ_{r} is open in the relative Euclidean topology of 𝔻^{N+1} ∪ S^{N}, and since μ_{ρ,s_{n}}(λ_{r} ∪ ∂λ_{r}) = μ_{ρ,s_{n}}(λ_{r}) for all n, the weak convergence of μ_{ρ,s_{n}} to μ_{ρ} gives that

μ_{ρ}(∂λ_{r}) ≤ lim inf_{n→∞} μ_{ρ,s_{n}}(λ_{r}).  (1)

Also, for ρ sufficiently large, we have for all n that

μ_{ρ,s_{n}}(λ_{r}) ≤ μ_{ρ,s_{n}} ( ⋃_{f ∈ Γ_{g(p)}(r)}Ŝ_{ρ}(0, f(0))) + μ_{ρ,s_{n}} ( ⋃_{f ∈ Γ_{g(p)}(r)}B_{2ρ}(f(0))) , (2)

where we have set Γ_{g(p)}(r) := {ge ∈ G_{ρ} : e ∈ Γ_{p}, ∂λ_{m} ∩ ∂S_{ρ}(0, ge(0)) ≠ ∅}. In order to give estimates from above for the terms on the right hand side of the latter inequality, first note that for f, h ∈ G_{ρ} such that h(0) ∈ Ŝ_{ρ}(0, f(0)), the hyperbolic triangle inequality implies that d(0, h(0)) ≍_{+} d(0, f(0)) + d(f(0), h(0)). Using this together with Corollary 2.6 (i), we obtain

μ_{ρ,s_{n}} ( ⋃_{f ∈ Γ_{g(p)}(r)}Ŝ_{ρ}(0, f(0)))

≤ \frac{1}{P′_{ρ}(0, s_{n})} ∑_{f∈Γ_{g(p)}(r)} ∑_{\begin{matrix}h∈G_{ρ}: \\ h(0)∈Ŝ_{ρ}(0,f(0)) \end{matrix}} (ϕ_{ρ}(h) e^{−d(0,h(0))})^{s_{n}}

≍ \frac{1}{P′_{ρ}(0, s_{n})} ∑_{f∈Γ_{g(p)}(r)} e^{−s_{n} d(0,f(0))} ∑_{\begin{matrix}h∈G_{ρ}: \\ f^{−1}h(0)∈Ŝ_{ρ}(f^{−1}(0),0) \end{matrix}} (ϕ_{ρ}(h) e^{−d(0,f^{−1}h(0))})^{s_{n}}

≤ \frac{1}{P′_{ρ}(0, s_{n})} ∑_{f∈Γ_{g(p)}(r)} e^{−s_{n} d(0,f(0))} ∑_{\begin{matrix}k∈G_{ρ}: \\ k(0)∈Ŝ_{ρ}(f^{−1}(0),0) \end{matrix}} (ϕ_{ρ}(fk) e^{−d(0,k(0))})^{s_{n}} .  (3)

Now note that by Lemma 4.1 we have for each f ∈ Γ_{g(p)}(r) that

ϕ_{ρ}(fk) ≤ e^{ε d_{a}(fk(0),k(0))} ϕ_{ρ}(k),

for almost all k ∈ G_{ρ} with k(0) ∈ Ŝ_{ρ}(f^{−1}(0), 0), where ‘almost all’ refers to that there may be finitely many exceptions (depending on ε). (Here, we have used d_{a}(x, y) to denote the annular distance between two points x, y ∈ 𝔻^{N+1}, that is d_{a}(x, y) := |d(0, x) − d(0, y)|). Also note that since k(0) ∈ Ŝ_{ρ}(f^{−1}(0), 0), the hyperbolic triangle inequality gives that

d_{a}(fk(0), k(0)) = d(0, fk(0)) − d(0, k(0))
= d(0, fk(0)) − d(f(0), fk(0))
≍_{+} d(0, f(0)).

Therefore, in the final sum in (3) we have that ϕ_{ρ}(fk) ≪ e^{ε d(0,f(0))} ϕ_{ρ}(k). Using this observation, we continue the estimate in (3) as follows.

μ_{ρ,s_{n}} ( ⋃_{f ∈ Γ_{g(p)}(r)}Ŝ_{ρ}(0, f(0)))

≪ \frac{1}{P′_{ρ}(0, s_{n})} ∑_{f∈Γ_{g(p)}(r)} e^{−(s_{n}−ε) d(0,f(0))} ∑_{\begin{matrix}k∈G_{ρ}: \\ k(0)∈Ŝ_{ρ}(f^{−1}(0),0) \end{matrix}} (ϕ_{ρ}(k) e^{−d(0,k(0))})^{s_{n}}

= ∑_{f∈Γ_{g(p)}(r)}e^{−(s_{n}−ε)d(0,f(0))} μ_{ρ,s_{n}} (Ŝ_{ρ}(f^{−1}(0), 0)) ≪ ∑_{f∈Γ_{g(p)}(r)}e^{−(s_{n}−ε)d(0,f(0))}.

For the second term in (2) we obtain, using Lemma 4.1,

μ_{ρ,s_{n}} ( ⋃_{f ∈ Γ_{g(p)}(r)}B_{2ρ}(f(0))) ≪ \frac{1}{P′_{ρ}(0, s_{n})} ∑_{f∈Γ_{g(p)}(r)} (ϕ_{ρ}(f) e^{−d(0,f(0))})^{s_{n}}

≪ \frac{1}{P′_{ρ}(0, s_{n})} ∑_{f∈Γ_{g(p)}(r)} e^{−(s_{n}−ε) d(0,f(0))} .

Hence combining these two latter estimates with (1) and (2), it follows that

μ_{ρ}(∂λ_{r}) ≪ ∑_{f∈Γ_{g(p)}(r)} e^{−(δ_{ρ}−ε) d(0,f(0))} ≪ e^{−(δ_{ρ}−ε) d(0,g(0))} ∑_{f∈Γ_{g(p)}(r)} e^{−(δ_{ρ}−ε) d(g(0),f(0))} ,

where the second inequality is obtained immediately from the hyperbolic triangle inequality. In order to derive the statement in (ii), we can now proceed exactly as in the proof of the ‘global measure formula’ in the geometrically finite situation, and we refer to [18] for the details (see also [17] for the completely analogous argument in the context of rational maps).

Theorem 3.3. For each x ∈ 𝔻^{N+1} and ρ sufficiently large, the following holds.

(i) The measure μ_{ρ}^{x} is non-atomic.

(ii) G_{ρ} is of δ_{ρ}-divergence type. That is, P_{ρ}(x, s) diverges for s = δ_{ρ}. (Note, this implies that the use of the slowly varying function ϕ_{ρ} in the construction of μ_{ρ}^{x} is redundant).

(iii) For the Hausdorff dimension of L(G_{ρ}) we have

dim_{H}(L(G_{ρ})) = δ_{ρ}.

Proof. In order to prove (i) let k_{max} := max{k(p) : p ∈ P }, and recall that by a result of Beardon (cf. [1]) we have that δ > k_{max}/2, and that by Proposition 3.1 δ_{ρ} tends to δ. Combining these two facts gives that for each ε > 0 sufficiently small and for all ρ sufficiently large we have that

δ_{ρ} − ε > \frac{k_{max}}{2} .

Hence with this choice of ε and ρ, Proposition 3.2 (ii) immediately implies that μ_{ρ}^{x} does not have atoms at bounded parabolic points. On the other hand, the fact that radial limit points are no atoms is clearly a direct consequence of a combination of Lemma 2.4 and Proposition 3.2 (i). For (ii), assume that P_{ρ}(x, s) converges for s = δ_{ρ}. Since, by Lemma 2.4, L_{r}(G_{ρ}) is contained in the limsup-set of the family {∂S_{ρ}(0, g(0))}_{g∈G_{ρ}} and since e^{−d(0,g(0))} ≍ diam(∂S_{ρ}(0, g(0)) for all g ∈ G_{ρ}, the Borel-Cantelli lemma gives that μ_{ρ}(L_{r}(G_{ρ})) = 0. By Proposition 2.2 (ii) we have that L(G_{ρ}) = L_{r}(G_{ρ}) ∪ L_{p}(G_{ρ}). Hence, since L_{p}(G_{ρ}) is countable and since as we have seen above, μ_{ρ} has no atoms at parabolic points, it follows that μ_{ρ}(L(G_{ρ})) = 0. This clearly contradicts the fact that μ_{ρ} is supported on L(G_{ρ}). We are now left to prove (iii). First note that by combining the facts that δ_{ρ} is the abzissa of convergence of P_{ρ}(x, s), that e^{−d(0,g(0))} ≍ diam(∂S_{ρ}(0, g(0)) for all g ∈ G_{ρ} and that L(G_{ρ}) coincides with L_{r}(G_{ρ}) up to a countable set, we immediately obtain the upper bound dim_{H}(L(G_{ρ})) ≤ δ_{ρ}. For the lower bound we remark that the fact that μ_{ρ} is non-atomic implies that the measure estimates in Proposition 3.2 can be improved significantly. That is, knowing that μ_{ρ} has no atoms allows to establish for μ_{ρ} the same type of ‘upper global formula’ obtained in [18] for the classical Patterson measure in the geometrically finite case. The proof for μ_{ρ} is exactly the same as in the classical situation and we refer to [18] for the details. More pecisely, for each ξ ∈ L(G_{ρ}) and t > 0 we have the following ‘upper global estimate’

μ_{ρ}(b(ξ, e^{−t})) ≪ e^{−δ_{ρ}t} e^{−d(ξ_{t},G_{ρ}(0)) τ(ξ_{t})}.  (4)

Here ξ_{t} ∈ s_{ξ} is uniquely determined by d(0, ξ_{t}) = t, and if ξ_{t} ∈ g(H_{p}) for some g ∈ G_{ρ} and p ∈ P then τ(ξ_{t}) := δ_{ρ} −k(p), whereas τ(ξ_{t}) := 0 otherwise. Now, (4) immediately implies that if G has no parabolic elements, or else if G and ρ are such that δ_{ρ} ≥ k_{max}, then we have for each ξ ∈ L(G_{ρ})

lim sup_{r→0} \frac{μ_{ρ}(b(ξ, r))}{r^{δ_{ρ}}} ≪ 1.

Hence, an application of the ‘mass distribution principle for Hausdorff measures’ (cf. [9] and/or [15]) gives the lower bound δ_{ρ} for the Hausdorff dimension in this special situation. For the remaining cases we employ the following Khintchine-type argument. Let κ > 0 be given, and fix a sequence (t_{n}) such that t_{n} > (δ_{ρ}κn+2 log n)/(2δ_{ρ}− k_{max}) for all n. By Lemma 4.4 there exists n′_{ε} such that for all n ≥ n′_{ε} we have

card A_{ρ}(n) := card {g ∈ G_{ρ} : n ≤ d(0, g(0)) < n + 1} ≪ e^{(1+κ)δ_{ρ}n}.

Combining this and (4), and defining r_{p,g} := diam (g(H_{p})) for p ∈ P and g ∈ G_{ρ} such that d(0, g(0)) = min{d(0, gh(0)) : h ∈ Γ_{p}}, we obtain

∑_{p∈P} ∑_{n=n′_{κ}}^{∞} ∑_{g∈A_{ρ}(n)} μ_{ρ}(b(g(p), e^{−t_{n}}r_{p,g})) ≪ ∑_{p∈P} ∑_{n=n′_{κ}}^{∞} ∑_{g∈A_{ρ}(n)} r_{p,g}^{δ_{ρ}}e^{−t_{n}(2δ_{ρ}−k_{max})}

≪ ∑_{n=n′_{κ}}^{∞} e^{δ_{ρ}κn}e^{−t_{n}(2δ_{ρ}−k_{max})}

≪ ∑_{n=1}^{∞} n^{−2} < ∞.

Hence by applying the Borel-Cantelli lemma, we now have the following. For μ_{ρ}-almost every ξ ∈ L(G_{ρ}) there exists t^{∗} = t^{∗}(ξ) such that for each t > t^{∗} we have that if ξ_{t} ∈ g(H_{p}) for some g ∈ G_{ρ} and p ∈ P such that r_{p,g} ≍ e^{−n} for some n, then

d(ξ_{t}, G_{ρ}(0)) ≤ \frac{δ_{ρ}κn + 2 log n}{2δ_{ρ} − k_{max}}.  (5)

Note that if δ_{ρ} ≥ k(p) then there is nothing to prove. Hence, we can assume without loss of generality that δ_{ρ} < k(p). Define k_{max}^{∗} := max{k(q) : q ∈ P, δ_{ρ} < k(q)}, and fix a number σ such that

σ > \frac{(δ_{ρ}κ + 2(log n)/n)(k_{max}^{∗} − δ_{ρ})}{2δ_{ρ} − k_{max}}.

By combining (4) and (5), the following holds for μ_{ρ}-almost every ξ and for each t > t^{∗}. Here we can assume without loss of generality that ξ_{t} ∈ g(H_{p}) for some g ∈ G_{ρ} and p ∈ P such that r_{p,g} ≍ e^{−n} for some n (note that in this situation we have that t > n).

μ_{ρ}(b(ξ, e^{−t})) ≪ e^{−tδ_{ρ}} e^{−d(ξ_{t},G_{ρ}(0)) τ(ξ_{t})}
≪ e^{−tδ_{ρ}} exp (\frac{(k_{max}^{∗} − δ_{ρ})(δ_{ρ}κn + 2 log n)}{2δ_{ρ} − k_{max}})
≪ e^{−t(δ_{ρ}−σ)} e^{−tσ} r_{p,g}^{−σ} ≤ e^{−t(δ_{ρ}−σ)}.

Since κ was chosen to be arbitrary, we have that σ can be made arbitrarily small. Hence, it follows that for μ_{ρ}-almost every ξ we have

lim inf_{r→0} \frac{log μ_{ρ}(b(ξ, r))}{log r} ≥ δ_{ρ}.

Now, by employing once more ‘the mass distribution principle for Hausdorff measures’, the theorem follows.

Harmonicity and Subconformality.

We end this section by giving two results, which will mark the similarities between the classical and the ρ-restricted Patterson measures. We now first see that varying the base point x of a ρ-restricted Patterson measure μ_{ρ}^{x} does not alter its measure class. This will follow from the fact that ρ-restricted Patterson measures enjoy the property of being δ_{ρ}-harmonic, meaning that for arbitrary x, y ∈ 𝔻^{N+1} the Radon-Nikodym derivative of μ_{ρ}^{y} with respect to μ_{ρ}^{x} at some arbitrary ξ ∈ L(G_{ρ}) has the property

\frac{dμ_{ρ}^{y}}{dμ_{ρ}^{x}} (ξ) = e^{δ_{ρ} <x,y>_{ξ}}.

Here we have used the notation of Helgason for the the signed horospherical distance < x, y >_{ξ} between x and y at ξ (cf. [10]). That is, < x, y >_{ξ} refers to the hyperbolic distance of the two horospheres based at ξ containing x and y respectively, where < x, y >_{ξ} is positive if and only if the horosphere at ξ containing y is contained in the horoball bounded by the horosphere at ξ through x.

Lemma 3.4. For each x ∈ 𝔻^{N+1}, the measure μ_{ρ}^{x} is δ_{ρ}-harmonic.

Proof. Let ξ ∈ L(G_{ρ}) be fixed, and choose a sequence (λ_{m}) of hyperbolic lenses such that ξ ∈ ∂λ_{m} for all m, and such that the Euclidean diameter of ∂λ_{m} tends to 0 if m tends to infinity. For a given pair m, n ∈ ℕ, we have

\frac{μ_{ρ,s_{n}}^{y}(λ_{m})}{μ_{ρ,s_{n}}^{x}(λ_{m})} = \frac{∑_{g∈G_{ρ} : g(0)∈λ_{m}} e^{−s_{n} d(y,g(0))}}{∑_{g∈G_{ρ} : g(0)∈λ_{m}} e^{−s_{n} d(x,g(0))}}

= \frac{∑_{g∈G_{ρ} : g(0)∈λ_{m}} e^{s_{n}(d(x,g(0))−d(y,g(0)))} e^{−s_{n} d(x,g(0))}}{∑_{g∈G_{ρ} : g(0)∈λ_{m}} e^{−s_{n} d(x,g(0))}} .

Observe that if z ∈ 𝔻^{N+1} tends to ξ, then the difference d(x, z) − d(y, z) tends to < x, y >_{ξ}. Using this observation and by letting n tend to infinity, the latter estimate yields

inf_{η ∈ ∂λ_{m}} e^{δ_{ρ} <x,y>_{ξ}} ≤ \frac{μ_{ρ}^{y}(∂λ_{m})}{μ_{ρ}^{x}(∂λ_{m})} ≤ sup_{η ∈ ∂λ_{m}} e^{δ_{ρ} <x,y>_{ξ}}.

This implies that

lim_{m→∞} \frac{μ_{ρ}^{y}(∂λ_{m})}{μ_{ρ}^{x}(∂λ_{m})} = e^{δ_{ρ} <x,y>_{ξ}}.

Since the sequence (λ_{m}) was chosen to be arbitrary and since the expression e^{δ_{ρ} <x,y>_{ξ}} is continuous in ξ, the lemma follows.

Finally, we now remark that ρ-restricted Patterson measures have the following weaker properties of conformality. For keeping the exposition simple, we have restricted the discussion to the case x = 0. Also, we remark that for each g ∈ G_{ρ} we have by definition of G_{ρ}^{g} that the intersection of \overline{∂S_{ρ}(g(0), 0)} and L(G_{ρ}^{g}) is a closed subset of L(G_{ρ}).

Lemma 3.5. (i) The measure μ_{ρ} is δ_{ρ}-subconformal on special sets. That is, for each g ∈ G_{ρ} with B_{ρ}(g(0)) ∩ B_{ρ}(0) = ∅ we have for every A ⊂ \overline{∂S_{ρ}(g(0), 0)} measurable

μ_{ρ}(g^{−1}A) ≤ ∫_{A} |(g^{−1})′(ξ)|^{δ_{ρ}} dμ_{ρ}(ξ).

(ii) The measure μ_{ρ} is δ_{ρ}-conformal on very special sets. That is, for each g ∈ G_{ρ} with B_{ρ}(g(0)) ∩ B_{ρ}(0) = ∅ we have for every A ⊂ L(G_{ρ}^{g}) ∩ \overline{∂S_{ρ}(g(0), 0)} measurable

μ_{ρ}(g^{−1}A) = ∫_{A} |(g^{−1})′(ξ)|^{δ_{ρ}} dμ_{ρ}(ξ).

Proof. Fix g ∈ G_{ρ} as given in the lemma. For the proof of (i) note that Corollary 2.6 (i) implies, for each n ∈ ℕ and A′ ⊂ Ŝ_{ρ}(g(0), 0) measurable,

μ_{ρ,s_{n}}(g^{−1}A′) = \frac{1}{P_{ρ}(0, s_{n})} ∑_{h∈G_{ρ}:h(0)∈g^{−1}(A′)}e^{−s_{n} d(0,h(0))}

= \frac{1}{P_{ρ}(0, s_{n})} ∑_{h∈G_{ρ}:gh(0)∈A′} e^{−s_{n} d(g(0),gh(0))}

≤ \frac{1}{P_{ρ}(0, s_{n})} ∑_{k∈G_{ρ}:k(0)∈A′}e^{−s_{n} d(g(0),k(0))}.  (6)

Hence, by letting n tend to infinity we derive, for each A ⊂ \overline{∂S_{ρ}(g(0), 0)} measurable,

μ_{ρ}(g^{−1}A) ≤ μ_{ρ}^{g(0)}(A).

Combining this estimate and Lemma 3.4, it follows that

μ_{ρ}(g^{−1}A) ≤ ∫_{A} e^{δ_{ρ} <0,g(0)>_{ξ}}dμ_{ρ}(ξ) = ∫_{A} |(g^{−1})′(ξ)|^{δ_{ρ}}dμ_{ρ}(ξ).

For proving (ii), the idea is to restrict g^{−1} ∗ μ_{ρ,s_{n}} and μ_{ρ,s_{n}}^{g(0)} to some closed subset V_{g} of 𝔻^{N+1} ∪ S^{N+1} which is given by some suitable extension of X_{g} := L(G_{ρ}^{g}) ∩ \overline{∂S_{ρ}(g(0), 0)} to the interior of hyperbolic space. (Here g ∗ ν refers to the pull back of a measure ν, that is g ∗ ν(E) := ν(gE) for each E measurable). First note that to each ξ ∈ X_{g} we can associate a geodesic l(ξ) ending at ξ such that l(ξ) is contained in cl(C_{ρ}) and intersects cl(B_{ρ}(g(0))) as well as cl(B_{ρ}(0)). Then let r(ξ) denote some geodesic ray towards ξ which starts in cl(B_{ρ}(0)) and which is fully contained in l(ξ). Define Y_{g} to be the union of the set of balls cl(B_{ρ}(h(0))) for which h ∈ G_{ρ}^{g} such that cl(B_{ρ}(h(0))) intersects r(ξ) for some ξ ∈ X_{g}. Similarly, let Z_{g} denote the union of the set of closed horoballs which are based at elements of L_{p}(G_{ρ}^{g}) and which have non-empty intersection with r(ξ) for some ξ ∈ X_{g}. We now define the extension V_{g} which we already mentioned above by V_{g} := X_{g} ∪ Y_{g} ∪ Z_{g}. Clearly, by construction we have that V_{g} is closed in 𝔻^{N+1} ∪ S^{N+1}. Now, consider g^{−1} ∗ μ_{ρ,s_{n}}|_{V_{g}} and μ_{ρ,s_{n}}^{g(0)}|_{V_{g}}, the restrictions of g^{−1} ∗ μ_{ρ,s_{n}} and μ_{ρ,s_{n}}^{g(0)} to V_{g}. A calculation similar to (6), using (ii) of Corollary 2.6 instead of (i), yields that g^{−1} ∗ μ_{ρ,s_{n}}|_{V_{g}} = μ_{ρ,s_{n}}^{g(0)}|_{V_{g}}. Recall that in general if a sequence of measures converges weakly to some limit measure then the restrictions of the elements of this sequence to some closed set converge weakly to the limit measure restricted to the same closed set. This general observation implies that (μ_{ρ,s_{n}}^{g(0)}|_{V_{g}}) converges weakly to μ_{ρ}^{g(0)}|_{V_{g}}, and also that (g^{−1} ∗ μ_{ρ,s_{n}}|_{V_{g}}) converges weakly to g^{−1} ∗ μ_{ρ}|_{V_{g}}. Since, as we have seen above, the sequences (g^{−1} ∗ μ_{ρ,s_{n}}|_{V_{g}}) and (μ_{ρ,s_{n}}^{g(0)}|_{V_{g}}) coincide, it follows that g^{−1} ∗ μ_{ρ}|_{V_{g}} = μ_{ρ}^{g(0)}|_{V_{g}}. Hence, as in the proof of (i) above, an application of Lemma 3.4 finishes the proof of (ii).

4 Appendix

‘The Method of a Slowly Varying Function’.

With the notation of Section 3, let the elements of G_{ρ} be ordered such that

G_{ρ} = {g_{0}, g_{1}, g_{2}, … },

where d(0, g_{n}(0)) ≤ d(0, g_{n+1}(0)) for all n. Also, throughout let (ε_{n}) denote some fixed sequence of positive reals which decreases to 0 strictly monotonically. The slowly varying function ϕ_{ρ} : G_{ρ} ↦ ℝ^{+} associated with P_{ρ}(x, s) and (ε_{n}) is then defined by way of induction as follows.

Step 0: Let ϕ_{ρ}(g_{0}) := 1 and k_{0} := 0. Step 1: Let t_{0} := d(0, g_{0}(0))(= 0), and for i ∈ {k_{0} + 1, …, k_{1}} define

ϕ_{ρ}(g_{i}) := e^{ε_{0}(d(0,g_{i}(0))−t_{0})} (= e^{ε_{0}d(0,g_{i}(0))}) ,

where k_{1} is implicitly determined by

1 + (ϕ_{ρ}(g_{k_{1}}) e^{−d(0,g_{k_{1}}(0))})^{δ_{ρ}} > ∑_{i=k_{0}+1}^{k_{1}} (ϕ_{ρ}(g_{i}) e^{−d(0,g_{i}(0))})^{δ_{ρ}} ≥ 1.

(Note, the existence of k_{1} follows since δ_{ρ} is the abzissa of convergence of P_{ρ}(0, s), and hence ∑_{i∈ℕ} e^{−δ_{ρ}(1−ε_{0}) d(0,g_{i}(0))} diverges). Step 2: Let t_{1} := d(0, g_{k_{1}}(0)), and for i ∈ {k_{1} + 1, …, k_{2}} define

ϕ_{ρ}(g_{i}) := ϕ_{ρ}(g_{k_{1}})e^{ε_{1}(d(0,g_{i}(0))−t_{1})} (= e^{ε_{0}(t_{1}−t_{0})} e^{ε_{1}(d(0,g_{i}(0))−t_{1})}) ,

where k_{2} is implicitly determined by

1 + (ϕ_{ρ}(g_{k_{2}}) e^{−d(0,g_{k_{2}}(0))})^{δ_{ρ}} > ∑_{i=k_{1}+1}^{k_{2}} (ϕ_{ρ}(g_{i}) e^{−d(0,g_{i}(0))})^{δ_{ρ}} ≥ 1.

(Note, the existence of k_{2} follows since

∑_{i=k_{1}+1}^{∞} (ϕ_{ρ}(g_{k_{1}})e^{ε_{1}(d(0,g_{i}(0))−t_{1})} e^{−d(0,g_{i}(0))})^{δ_{ρ}} = ∑_{i=k_{1}+1}^{∞} e^{δ_{ρ}t_{1}(ε_{0}−ε_{1})} e^{−δ_{ρ}(1−ε_{1}) d(0,g_{i}(0))}

≥ ∑_{i=k_{1}+1}^{∞} e^{−δ_{ρ}(1−ε_{1}) d(0,g_{i}(0))},

where clearly the final series diverges and hence also the first). Step (m+1): Assume that k_{m} has been obtained in Step m. Let t_{m} := d(0, g_{k_{m}}(0)), and for i ∈ {k_{m} + 1, …, k_{m+1}} define

ϕ_{ρ}(g_{i}) := ϕ_{ρ}(g_{k_{m}})e^{ε_{m}(d(0,g_{i}(0))−t_{m})} (= e^{ε_{m}(d(0,g_{i}(0))−t_{m})} ∏_{j=0}^{m−1} e^{ε_{j}(t_{j+1}−t_{j})}) ,

where k_{m+1} is implicitly determined by

1 + (ϕ_{ρ}(g_{k_{m+1}}) e^{−d(0,g_{k_{m+1}}(0))})^{δ_{ρ}} > ∑_{i=k_{m}+1}^{k_{m+1}} (ϕ_{ρ}(g_{i}) e^{−d(0,g_{i}(0))})^{δ_{ρ}} ≥ 1.

(Note, the existence of k_{m+1} follows since

∑_{i=k_{m}+1}^{∞} (ϕ_{ρ}(g_{k_{m}})e^{ε_{m}(d(0,g_{i}(0))−t_{m})} e^{−d(0,g_{i}(0))})^{δ_{ρ}}

= ∑_{i=k_{m}+1}^{∞} e^{δ_{ρ}(ε_{0}t_{1}+ε_{1}(t_{2}−t_{1})+...+ε_{m−1}(t_{m}−t_{m−1})−ε_{m}t_{m})} e^{−δ_{ρ}(1−ε_{m}) d(0,g_{i}(0))}

= ∑_{i=k_{m}+1}^{∞} e^{δ_{ρ}(t_{1}(ε_{0}−ε_{1})+t_{2}(ε_{1}−ε_{2})+...+t_{m}(ε_{m−1}−ε_{m}))} e^{−δ_{ρ}(1−ε_{m}) d(0,g_{i}(0))}

≥ ∑_{i=k_{m}+1}^{∞} e^{−δ_{ρ}(1−ε_{m}) d(0,g_{i}(0))},

where clearly the final series diverges and hence also the first).

Before we come to the main result of this appendix, we first give four lemmata in which we collect some technical observations which will turn out to be useful. For the first result recall the notation d_{a}(z, w), which refers to the annular distance between z and w in 𝔻^{N+1}, that is d_{a}(z, w) := |d(0, z) − d(0, w)|.

Lemma 4.1. For each ε > 0 there exists n_{ε} ∈ ℕ such that for all n ≥ n_{ε} and all i ∈ ℕ we have

ϕ_{ρ}(g_{n+i}) ≤ e^{ε d_{a}(g_{n}(0),g_{n+i}(0))} ϕ_{ρ}(g_{n}).

Proof. Fix 0 < ε < 1, and let m_{0} denote the integer which is uniquely determined by ε_{m_{0}} ≤ ε < ε_{m_{0}−1}. With (k_{m}) referring to the sequence obtained in the construction of ϕ_{ρ} above, define n_{ε} := k_{m_{0}}. Then for each pair i, n ∈ ℕ such that n ≥ n_{ε}, there exists m, m′ ∈ ℕ with m ≥ m′ ≥ m_{0} such that

ϕ_{ρ}(g_{n+i}) = e^{ε_{m}(d(0,g_{n+i}(0))−t_{m})} ∏_{j=0}^{m−1} e^{ε_{j}(t_{j+1}−t_{j})} ,

ϕ_{ρ}(g_{n}) = e^{ε_{m′}(d(0,g_{n}(0))−t_{m′})} ∏_{j=0}^{m′−1} e^{ε_{j}(t_{j+1}−t_{j})} .

Now, if m = m′ then we have

\frac{ϕ_{ρ}(g_{n+i})}{ϕ_{ρ}(g_{n})} = e^{ε_{m}(d(0,g_{n+i}(0))−d(0,g_{n}(0)))} ≤ e^{ε d_{a}(g_{n+i}(0),g_{n}(0))}.

If on the other hand m > m′, then we have that

\frac{ϕ_{ρ}(g_{n+i})}{ϕ_{ρ}(g_{n})} = e^{ε_{m}(d(0,g_{n+i}(0))−t_{m})} e^{ε_{m′}(t_{m′}−d(0,g_{n}(0)))} ∏_{j=m′}^{m−1} e^{ε_{j}(t_{j+1}−t_{j})}
= e^{∑_{j=m′}^{m−1} ε_{j}(t_{j+1}−t_{j}) + ε_{m}(d(0,g_{n+i}(0))−t_{m}) + ε_{m′}(t_{m′}−d(0,g_{n}(0)))},

which proves the lemma if combined with the following estimate for the exponent in the latter expression.

∑_{j=m′}^{m−1} ε_{j}(t_{j+1} − t_{j}) + ε_{m}(d(0, g_{n+i}(0)) − t_{m}) + ε_{m′}(t_{m′} − d(0, g_{n}(0)))

≤ ∑_{j=m′}^{m−1} ε_{m′}(t_{j+1} − t_{j}) + ε_{m′}(d(0, g_{n+i}(0)) − t_{m} + t_{m′} − d(0, g_{n}(0)))
= ε_{m′} (d(0, g_{n+i}(0)) − d(0, g_{n}(0))) ≤ ε d_{a}(g_{n+i}(0), g_{n}(0)) .

The following two lemmata are immediate consequences of the previous lemma. In here A_{ρ}(n) refers to the n-annulus of G_{ρ}, which is defined for n ∈ ℕ ∪ {0} by

A_{ρ}(n) := {g ∈ G_{ρ} : n ≤ d(0, g(0)) < n + 1} .

Also, we let [σ] denote the integer part of a non-negative real number σ.

Lemma 4.2. For every ε > 0 and for all n ≥ n_{ε} and j ∈ ℕ, we have for each g ∈ A_{ρ}([d(0, g_{n}(0))] + j) that

ϕ_{ρ}(g) ≤ e^{ε(j+1)}ϕ_{ρ}(g_{n}).

Proof. By Lemma 4.1 we have that ϕ_{ρ}(g_{n+i}) ≤ e^{ε d_{a}(g_{n+i}(0),g_{n}(0))}ϕ_{ρ}(g_{n}) for all n ≥ n_{ε} and i ∈ ℕ. Hence, if g ∈ A_{ρ}([d(0, g_{n}(0))] + j) for some j, then it follows that

ϕ_{ρ}(g) ≤ e^{ε d_{a}(g(0),g_{n}(0))}ϕ_{ρ}(g_{n}).

Since in this situation d_{a}(g(0), g_{n}(0)) ≤ j + 1, the lemma follows.

Lemma 4.3. If (f_{n}) and (h_{n}) are two sequences of elements of G_{ρ} such that d(0, f_{n}(0)) ≍_{+} d(0, h_{n}(0)) uniformly for all n, and such that

lim_{n→∞} d(0, f_{n}(0)) = lim_{n→∞} d(0, h_{n}(0)) = ∞,

then it follows that

lim_{n→∞} \frac{ϕ_{ρ}(f_{n})}{ϕ_{ρ}(h_{n})} = 1.

Proof. Let ε > 0 be fixed. Using Lemma 4.1 and the assumption that d_{a}(f_{n}(0), h_{n}(0)) is uniformly bounded, we have for each n ≥ n_{ε} that

| log(ϕ_{ρ}(f_{n})) − log(ϕ_{ρ}(h_{n}))| ≤ ε d_{a}(f_{n}(0), h_{n}(0)) ≪ ε.

Since ε was chosen to be arbitrary, the lemma follows.

Finally, we state a result which is not in the context of the slowly varying function ϕ_{ρ}. Clearly, this result is an immediate consequence of the fact that δ_{ρ} is the abzissa of convergence of P_{ρ}(0, s), and we omit its proof.

Lemma 4.4. For each κ > 0 there exists n′_{κ} ∈ ℕ such that for all n ≥ n′_{κ} we have

card(A_{ρ}(n)) ≤ e^{δ_{ρ}(1+κ)(n+1)}.

Now, the main result of this appendix is stated in the following proposition.

Proposition 4.5. For each ρ > 0 and x ∈ 𝔻^{N+1} we have that δ_{ρ} is the abzissa of convergence of P′_{ρ}(x, s) such that P′_{ρ}(x, s) diverges for s = δ_{ρ}.

Proof. Since P′_{ρ}(0, s) ≍ P′_{ρ}(x, s) (where the constants involved depend on x ∈ 𝔻^{N+1}), it clearly suffices to prove the assertions for x = 0. First note that by construction of the function ϕ_{ρ} we have that

∑_{n=0}^{∞} (ϕ_{ρ}(g_{n}) e^{−d(0,g_{n}(0))})^{δ_{ρ}} ≥ 1 + ∑_{m=0}^{∞} ∑_{i=k_{m}+1}^{k_{m+1}} (ϕ_{ρ}(g_{i}) e^{−d(0,g_{i}(0))})^{δ_{ρ}} ≥ 1 + ∑_{m=0}^{∞} 1.

Hence, it follows that P′_{ρ}(0, s) diverges for s = δ_{ρ}. In order to see that P′_{ρ}(0, s) converges for each s > δ_{ρ}, let τ > 0 be given and then fix some 0 < ε < 1 and κ > 0 sufficiently small such that 2δ_{ρ}(ε + κ)/(1 − ε) < τ . Define

N(ε, κ) := max{[d(0, g_{n_{ε}}(0))], n′_{κ}},

where n_{ε} and n′_{κ} are determined by Lemma 4.1 and Lemma 4.4 respectively. The aim is to show that the tail Σ_{τ}(ε, κ) of the sum in P′_{ρ}(0, δ_{ρ} + τ) is finite, where

Σ_{τ}(ε, κ) := ∑_{n=0}^{∞} ∑_{g∈A_{ρ}(n+N(ε,κ))} (ϕ_{ρ}(g)e^{−d(0,g(0))})^{δ_{ρ}+τ} .

In order to see this we use Lemma 4.2, Lemma 4.4 and the special choice of ε and κ, which gives

Σ_{τ}(ε, κ) ≪ ∑_{n ≥ N(ε,κ)} e^{ε(n+1)(δ_{ρ}+τ)} ∑_{g∈A_{ρ}(n)} (e^{−d(0,g(0))})^{δ_{ρ}+τ}
≤ ∑_{n ≥ N(ε,κ)} e^{ε(δ_{ρ}+τ)(n+1)} e^{−(δ_{ρ}+τ)n} card(A_{ρ}(n))
≪ ∑_{n ≥ N(ε,κ)} e^{ε(δ_{ρ}+τ)(n+1) − (δ_{ρ}+τ)(n+1) + δ_{ρ}(1+κ)(n+1)}
= ∑_{n ≥ N_{ε,κ}} e^{−((1−ε)τ−δ_{ρ}(ε+κ)) (n+1)} < ∑_{n ≥ N(ε,κ)} e^{−(n+1) δ_{ρ}(ε+κ)} < ∞ .

References

[1] A. F. Beardon, ‘The exponent of convergence of Poincaré series’, Proc. London Math. Soc. 18 (1968) 461–483.

[2] A.F. Beardon, The Geometry of Discrete Groups, Springer Verlag, New York, 1983.

[3] A. F. Beardon & B. Maskit, ‘Limit points of Kleinian groups and finite sided fundamental polyhedra’, Acta Math. 132 (1974) 1–12.

[4] C. J. Bishop & P. W. Jones, ‘Hausdorff dimension and Kleinian groups’, Acta Math. 179 (1997) 1–39.

[5] R.D. Canary, ‘Ends of hyperbolic 3-manifolds’, J. Amer. Math. Soc. 6 (1993) 1–35.

[6] M. Denker & M. Urbański, ‘On Sullivan’s conformal measures for rational maps of the Riemann sphere’, Nonlinearity 4 (1991) 365–384.

[7] M. Denker & M. Urbański, ‘On the existence of conformal measures’, Transact. Amer. Math. Soc. 328 , No. 2 (1991) 563–587.

[8] K.J. Falconer, Fractal Geometry, Mathematical Foundations and Applications, J. Wiley, 1990.

[9] K.J. Falconer, Techniques in Fractal Geometry,J. Wiley, 1997.

[10] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Acad. Pres, San Diego, 1978.

[11] M. Kapovich, Hyperbolic Manifolds and Discrete Group: Lectures on Thurstons Hyperbolization, University of Utah Lecture Notes, 1999.

[12] P.J. Nicholls, The Ergodic Theory of Discrete Groups, Cambridge University Press, 1989.

[13] S. J. Patterson, ‘The limit set of a Fuchsian group’, Acta Math. 136 (1976) 241–273.

[14] B. O. Stratmann, ‘A note on counting cuspidal excursions’, Ann. Acad. Sci. Fenn. 20 (1995) 359–372.

[15] B.O. Stratmann, ‘Fractal dimensions for Jarník limit sets; the semi-classical approach’, Ark. för Mat. 33 (1995) 385–403.

[16] B. O. Stratmann, ‘The exponent of convergence of Kleinian groups; on a theorem of Bishop and Jones’, Mathematica Gottingensis 1 (2001) 1–16.

[17] B. O. Stratmann & M. Urbański, ‘The geometry of conformal measures of parabolic rational maps’, Math. Proc. Cambridge Philos. Soc. 128(1) (2000) 141–156.

[18] B. O. Stratmann & S. Velani, ‘The Patterson measure for geometrically finite groups with parabolic elements, new and old’, Proc. Lond. Math. Soc. 71 (1995) 197–220.

[19] D. Sullivan, ‘The density at infinity of a discrete group of hyperbolic motions’, IHES Publ. Math. 50 (1979) 171–202.

[20] D. Sullivan, ‘Growth of positive harmonic functions and Kleinian group limit sets of planar measure 0 and Hausdorff dimension 2’, in Geometry Symposium (Utrecht 1980), Springer Lect. Notes in Math. 894 (1981) 127–144.

[21] D. Sullivan, ‘A finiteness theorem for cusps’, Acta Math. 147 (1981) 289-299.

[22] D. Sullivan, ‘Discrete conformal groups and measurable dynamics’, Bull. Amer. Math. Soc. 6 (1982) 57–73.

[23] D. Sullivan, ‘Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups’, Acta Math. 153 (1984) 259–277.

[24] M. Urbański, ‘Measures and dimensions in conformal dynamics’, preprint, Stony Brook survey article, 1999.