Diophantine Extremality of the Patterson Measure

B. O. Stratmann, M. Urbański ^{∗}

Abstract

We derive universal Diophantine properties for the Patterson measure μ_{Γ} associated with a convex cocompact Kleinian group Γ acting on (n + 1) - dimensional hyperbolic space. We show that μ_{Γ} is always a S -friendly measure, for every (Γ, μ_{Γ}) -neglectable set S , and deduce that if Γ is of non-Fuchsian type then μ_{Γ} is an absolutely friendly measure in the sense of Pollington and Velani. Consequently, by a result of Kleinbock, Lindenstrauss and Weiss, μ_{Γ} is strongly extremal which means that μ_{Γ} -almost every point is not very well multiplicatively approximable. This is remarkable, since by a well-known result in classical metric Diophantine analysis the set of very well multiplicatively approximable points is of n -dimensional Lebesgue measure zero but has Hausdorff dimension equal to n.

1 Introduction and statements of results

Recall that in classical Diophantine analysis a point ξ = (ξ_{1}, …, ξ_{n}) ∈ ℝ^{n} is called very well multiplicatively approximable if for some τ > 0 there exist infinitely many (p_{1}, …, p_{n}) ∈ ℤ^{n} and q ∈ ℕ such that

(∏_{i=1}^{n} |ξ_{i} − \frac{p_{i}}{q} |)^{\frac{1}{n}} < \frac{1}{q^{1+ \frac{1}{n} +τ}} .

Likewise, the point ξ is called badly approximable if there exists c > 0 such that for all (p_{1}, …, p_{n}) ∈ ℤ^{n} and q ∈ ℕ we have

∑_{i=1}^{n} |ξ_{i} − \frac{p_{i}}{q} | > \frac{c}{q^{1+ \frac{1}{n}}} .

It is well-known that the set of very well multiplicatively approximable numbers W as well as the set of badly approximable numbers B are both of n -dimensional Lebesgue measure λ_{n} equal to zero but have Hausdorff dimension dim_{H} equal to n ([8]). A particular outcome of this paper will be to relate these classical Diophantine sets to the limit set L(Γ) of an arbitrary convex cocompact Kleinian group Γ of non-Fuchsian type acting on (n + 1) -dimensional hyperbolic space ℍ^{n+1} such that L(Γ) ⊂ ℝ^{n} (in here ‘non-Fuchsian’ refers to that L(Γ) is not contained in a finite set of spheres or affine hyperplanes of codimension at least 1). Namely, a consequence of the results in this paper will be that the Patterson measure μ_{Γ} of such a Kleinian group is an absolutely friendly measure in the sense of [7]. Combining this with recent results of [4] and the well-known fact that the Patterson measure μ_{Γ} of a convex cocompact Kleinian group is Ahlfors-regular, which is an immediate consequence of Sullivan’s Shadow Lemma, it follows that

∗Research partially supported by the NSF Grant DMS 0400481.

dim_{H} (B ∩ L(Γ)) = dim_{H}(L(Γ)).

Furthermore, using a result of [2], absolute friendliness of μ_{Γ} implies that μ_{Γ} -almost every element of L(Γ) is not very well multiplicatively approximable, that is

μ_{Γ}(W) = 0.

We remark that the latter result is closely related to work of Kleinbock and Margulis in [3] which lead to a confirmation of a conjecture of Sprindzuk [9]. This conjecture states that λ_{n} is strongly extremal for each non-degenerate submanifold of ℝ^{n}, where a measure ν is strongly extremal if ν(W) = 0 (cf. [2]). In this terminology our aforementioned result therefore says that for every convex cocompact Kleinian group of non-Fuchsian type the Patterson measure is strongly extremal. In order to state the results in greater detail let us first fix some notation. For convenience, throughout we shall work with the Poincaré model (𝔻^{n+1}, d) of (n+1) - dimensional hyperbolic space. In this model the boundary at infinity of hyperbolic space is given by the n -sphere 𝕊^{n} := {ξ ∈ ℝ^{n+1} : ||ξ|| = 1}, which is conformally equivalent to the one-point compactification of ℝ^{n}. A Kleinian group G is a discrete subgroup of Con(n) , where Con(n) refers to the group of isometries of 𝔻^{n+1} . Let L(G) denote the limit set of G, that is the smallest closed G -invariant subset of 𝕊^{n}. We always assume that G is non-elementary convex cocompact, which means that G has a fundamental domain with finitely many sides, does not have parabolic elements, and L(G) is an uncountable proper subset of 𝕊^{n}. Also, by Ahlfors’s Lemma we can assume without loss of generality that G has no elliptic elements. It is well-known that in this situation dim_{H}(L(G)) = δ > 0 , where δ = δ(G) := inf {s ≥ 0 : ∑_{g∈G} exp(−sd(0, g(0))) < ∞} refers to the exponent of convergence of G (see [1], [11]). Finally, recall that to each G we can associate a canonical measure, namely its Patterson measure μ_{G} . This measure is always a probability measure with support equal to L(G) . Furthermore, μ_{G} is δ -conformal, which means that for each Borel set S ⊂ 𝕊^{n} and for all g ∈ G,

μ_{G}(g(S)) = ∫_{S} |g′(ξ)|^{δ} dμ_{G}(ξ).

Note that we shall mainly work in the Poincaré model of hyperbolic space. Nevertheless, in order to relate our results to the results in classical metric Diophantine analysis, we occasionally require our Kleinian group to act on the closure of the Klein model ℍ^{n+1} of (n + 1) -dimensional hyperbolic space. We then let Γ refer to the conjugate of our Kleinian group G under the Cayley transformation (for which it is well-known that it switches between these two models of hyperbolic space such that hyperbolic distances are preserved). Clearly, Γ acts on the closure of ℍ^{n+1} , and without loss of generality we will always assume that {∞} is not an element of the limit set L(Γ) of Γ , that is L(Γ) ⊂ ℝ^{n}. For the purposes of this paper the following two definitions will be crucial. For this recall that a measure ν on ℝ^{n} is called doubling (or Federer) if there exists a constant c > 1 such that for each ξ ∈ ℝ^{n} and r > 0 we have ν(B(ξ, 2r)) ≤ cν(B(ξ, r)) , where B(ξ, r) refers to the open n -ball centred at ξ of radius r. Also, throughout we shall use the notation a ≍ b for two positive reals a, b to denote that a/b is uniformly bounded away from zero and infinity. We write a ≪ b if a/b is uniformly bounded away from infinity. If not stated otherwise, the bounds will always depend exclusively on the given Kleinian group G.

Definition 1 Let G < Con(n) be a Kleinian group, and let ν be a finite Borel measure on 𝕊^{n}. A set S of Borel subsets of 𝕊^{n} is called (G, ν) -neglectable if and only if the following three conditions are satisfied.

• S is compact with respect to the Hausdorff topology in 𝕊^{n};

• ν(S) = 0 for all S ∈ S ;

• g(S) ∈ S for all g ∈ G, S ∈ S .

Definition 2 Let S be a given (G, ν) -neglectable set. A finite Borel measure ν is called S -friendly if ν is a doubling measure and there exist σ, r_{0} > 0 such that for all ξ ∈ L(G), S ∈ S, r ∈ (0, r_{0}) and s > 0 ,

\frac{ν (N_{s r}(S) ∩ B(ξ, r))}{ν (B(ξ, r))} ≪ s^{σ}.

In here, N_{t}(S) := ⋃_{ξ∈S} B(ξ, t) refers to the t -neighbourhood of S in 𝕊^{n}.

The following theorem gives the first main result of this paper.

Theorem 1 For every convex cocompact Kleinian group G < Con(n) and for each (G, μ_{G}) -neglectable set S , we have that the Patterson measure μ_{G} is S -friendly.

Subsequently, we show that Theorem 1 is applicable in the special situation of [2], [4]. For this we consider the set H(G) of intersections of L(G) with spheres in 𝕊^{n} of codimension at least 1. That is, we let

H(G) := {H ∩ L(G) : H a sphere in 𝕊^{n} of codimension at least 1}.

Our second main result is stated in the following theorem. In here, ‘ G of non-Fuchsian type’ refers to that L(G) is not a subset of a finite union of elements of H(G) .

Theorem 2 For every convex cocompact Kleinian group G < Con(n) which is of non-Fuchsian type we have that the Patterson measure μ_{G} is H(G) -friendly.

In [2] and [4] it was shown that if a Ahlfors-regular measure ν in ℝ^{n} is absolutely friendly then it is strongly extremal and dim_{H}(B ∩ supp(ν)) = dim_{H}(supp(ν)) . In here, supp(ν) refers to the support of ν, and absolutely friendly means that ν is friendly with respect to the set of intersections of supp(ν) with affine hyperplanes in ℝ^{n} of codimension 1 . Since the Cayley transformation maps the set H(G) to the set of intersections of L(Γ) with the set of spheres and affine hyperplanes in ℝ^{n} of codimension at least 1 , the following result is an immediate corollary of Theorem 2. (Note that Γ , as the conjugate of G under the Cayley transformation, is acting on the Klein model, and hence for Γ the condition ‘non-Fuchsian type’ means that L(Γ) is not contained in a finite set of spheres or affine hyperplanes of codimension at least 1).

• For each convex cocompact Kleinian group Γ of non-Fuchsian type, the Patterson measure is strongly extremal and dim_{H}(B ∩ L(Γ)) = dim_{H}(L(Γ)) .

Remark: We remark that recent work in [12] of the second author shows that for an irreducible finite conformal iterated function system satisfying the open set condition, the Gibbs measure associated with some Hölder family of functions is always absolutely friendly and hence in particular strongly extremal.

Acknowledgement: We should like to thank I.H.E.S. at Bures-sur-Yvette and the Stochastic Institute at the University of Göttingen for their warm hospitality and excellent working conditions when writing this paper.

2 Friendliness of the Patterson measure

2.1 Convex cocompact geometry

We briefly recall some of the geometry which will be important in this paper. For the proofs we refer to [5], [6], [10] and [11]. The shadow map Π : 𝔻^{n+1} → 𝕊^{n} is given for A ⊂ 𝔻^{n+1} by Π(A) := {ξ ∈ 𝕊^{n} : s_{ξ} ∩ A ≠ ∅}. In here s_{ξ} refers to the hyperbolic ray in 𝔻^{n+1} starting at the origin 0 ∈ 𝔻^{n+1} and terminating at ξ. We then have the following three well-known elementary facts from hyperbolic geometry (see [10]). Elementary Shadow Lemma Let ρ > 0 be given. For each z ∈ 𝔻^{n+1} such that d(0, z) > ρ, we have for the spherical diameter diam (Π(b(z, ρ))) of the shadow of the open hyperbolic ball b(z, ρ) centred at z and of radius ρ,

diam (Π(b(z, ρ))) ≍_{ρ} e^{−d(0,z)}.

In here, ≍_{ρ} indicates that the involved constants depend on ρ. Light Cone Lemma Let ρ > log 2 be given. For each g ∈ Con(n) non-elliptic such that d(0, g(0)) > ρ, we have

𝕊^{n} \ B (Π(g^{−1}(0)), 2πe^{−ρ}) ⊂ g^{−1} (Π(b(g(0), ρ))) ⊂ 𝕊^{n} \ B (Π(g^{−1}(0)), e^{−ρ}) .

Geometric Distortion Lemma Let ρ > 0 be given. For each g ∈ Con(n) non-elliptic such that d(0, g(0)) > ρ, we have

|(γ^{−1})′ (ξ)|^{−1} ≍_{ρ} diam (Π(b(g(0), ρ))) for each ξ ∈ Π(b(g(0), ρ)).

It is well-known that the convex core C(G) of a convex cocompact Kleinian group G is a compact subset of 𝔻^{n+1} , where C(G) is defined as the convex hull of the intersection of a fundamental domain of G with the set of geodesics connecting different elements of L(G) . Throughout, let κ > 0 refer to the hyperbolic diameter of C(G) , and let τ := exp(−3κ) . Also, we partition G into ‘annuli’ as follows

A_{k} := {g ∈ G : 3κk ≤ d(0, g(0)) < 3κ(k + 1)} for k ∈ ℕ.

Combining the compactness of C(G) with the fact that for a convex cocompact Kleinian group the injective radius is uniformly bounded away from zero, we obtain the following. Hedlund’s Lemma Let G < Con(n) be a convex cocompact Kleinian group. For each ξ ∈ L(G) and k ∈ ℕ, there exists g ∈ A_{k} such that ξ ∈ Π(b(g(0), κ)) . Recall that an element ξ ∈ L(G) is called uniformly radial if s_{ξ} ⊂ ⋃_{g∈G} b(g(0), ρ) , for some ρ > 0 . Therefore, an immediate consequence of Hedlund’s Lemma is that for convex cocompact groups every element of L(G) is uniformly radial. Using the fact that the injective radius is uniformly bounded away from zero, a further immediate implication is that for each k ∈ ℕ the set {Π(b(g(0), κ)) : g ∈ A_{k}} is a covering of L(G) of multiplicity uniformly bounded from above. A combination of the Elementary Shadow Lemma, the Geometric Distortion Lemma and the Light Cone Lemma with the δ -conformality of the Patterson measure then gives rise to the following well-known result ([11]). Sullivan’s Shadow Lemma Let G < Con(n) be a Kleinian group, and let ρ > 0 be fixed. For each g ∈ G we have

μ_{G} (Π(b(g(0), ρ))) ≍_{ρ} (diam (Π(b(g(0), ρ))))^{δ} ≍_{ρ} e^{−δd(0,g(0))}.

For a convex cocompact Kleinian group G Sullivan’s Shadow Lemma then immediately implies that the Patterson measure μ_{G} is a doubling measure which is δ -Ahlfors regular, in the sense that μ_{G} (B(ξ, r)) ≍ r^{δ} , for all ξ ∈ L(G) and 0 < r < 1 .

2.2 Friendliness with respect to neglectable sets

Throughout this section let G < Con(n) be a convex cocompact Kleinian group and let μ_{G} be the associated Patterson measure. For a set S of subsets of 𝕊^{n}, we define the function Δ_{S} for t > 0 by

Δ_{S}(t) := sup {μ_{G} (N_{t}(S)) : S ∈ S} .

Proposition 1 There exists τ_{0} > 0 such that for each G -invariant set S of subsets of 𝕊^{n} we have

Δ_{S}(s · t) ≪ Δ_{S}(s) Δ_{S}(t) for all s, t ∈ (0, τ_{0}).

Proof: Let s, t > 0 be given such that τ^{m+1} ≤ s < τ^{m} and τ^{l+1} ≤ t < τ^{l}, for m, l ∈ ℕ. Let S ∈ S be arbitrary, and define for k ∈ ℕ,

A_{k}(S) := {g ∈ A_{k} : S ∩ Π(b(g(0), κ)) ≠ ∅}.

By the Elementary Shadow Lemma we have that if g ∈ A_{k}(S) then diam (Π(b(g(0), κ))) ≍ τ^{k}. Also, if g ∈ A_{m}(S) and h ∈ A_{m+l}(S) such that Π(b(g(0), κ)) ∩ Π(b(h(0), κ)) ≠ ∅ then the Geometric Distortion Lemma implies diam (g^{−1}Π(b(h(0), κ))) ≍ τ^{l}. Using these observations together with Sullivan’s Shadow Lemma, the G -invariance of S , and the fact that {Π(b(g(0), κ)) : g ∈ A_{k}(S)} is a covering of S of multiplicity uniformly bounded from above, we obtain

μ_{G} (N_{st}(S)) ≍ ∑_{h∈A_{m+l}(S)} μ_{G} (Π(b(h(0), κ)))
≪ ∑_{g∈A_{m}(S)} ∑_{f∈A_{l}(g^{−1}(S))} τ^{(m+l)δ}
≪ ∑_{g∈A_{m}(S)} τ^{mδ} ∑_{f∈A_{l}(g^{−1}(S))} τ^{lδ}
≪ ∑_{g∈A_{m}(S)} τ^{mδ}Δ_{S}(t) ≪ Δ_{S}(s)Δ_{S}(t).

Since S ∈ S was chosen to be arbitrary, the assertion of the proposition follows. □

Corollary 1 There exists σ, τ_{1} > 0 such that for each (G, μ_{G}) -neglectable set S we have

Δ_{S}(t) ≤ t^{σ} for all t ∈ (0, τ_{1}).

Proof: Since S is compact and μ_{G}(S) = 0 for all S ∈ S , we have that lim_{t→0} Δ_{S}(t) = 0. Hence, there exists 0 < τ′_{1} < 1 such that Δ_{S}(t) < 1 and such that the inequality in Proposition 1 holds, for all t ∈ (0, τ′_{1}) . Let us consider the function Φ : (− log τ′_{1}, ∞) → ℝ^{+} given by

Φ(x) := − log Δ_{S} (e^{−x}) .

Clearly, Φ is a strictly positive function which is unbounded, and by Proposition 1 we have that there exists a constant c > 0 such that

Φ(x + y) > Φ(x) + Φ(y) − c for all x, y ∈ (− log τ′_{1}, ∞).

This implies that there exist σ, τ_{1} > 0 such that

Φ(x) ≥ σx for all x ∈ (− log τ_{1}, ∞).

By rewriting this in terms of Δ_{S} , the corollary follows.

□

Proof of Theorem 1 Let S be a given (G, μ_{G}) -neglectable set. Consider S ∈ S , 0 < r < r_{0}, s > 0 and ξ ∈ L(G) . Since ξ is uniformly radial, the Geometric Distortion Lemma gives that there exists g ∈ G such that ξ ∈ Π(b(g(0), κ)) and such that |(g^{−1})′| ≍ 1/r on Π(b(g(0), κ)) . Clearly, this also holds for B(ξ, r) , that

is we have |(g^{−1})′| ≍ 1/r on B(ξ, r) . Using the δ -conformality of μ_{G} , Sullivan’s Shadow Lemma and Corollary 1, it then follows

μ_{G} (B(ξ, r) ∩ N_{sr}(S)) ≍ r^{δ} μ_{G} (g^{−1} (B(ξ, r) ∩ N_{sr}(S)))
≪ r^{δ} μ_{G} (N_{s}(g^{−1}(S))) ≪ r^{δ} Δ_{S}(s)
≪ μ_{G}(B(ξ, r)) s^{σ}.
□

2.3 Friendliness with respect to H(G)

Proposition 2 For every convex cocompact Kleinian group G < Con(n) of non-Fuchsian type we have that the set H(G) is (G, μ_{G}) -neglectable.

Proof: As already mentioned in section 2.1, if G is convex cocompact then μ_{G} is a doubling measure. Also, we clearly have that H(G) is G -invariant and compact with respect to the Hausdorff topology in 𝕊^{n}. Therefore, it remains to show that μ_{G}(H) = 0 for all H ∈ H(G) . For this, assume by way of contradiction that there exists H ∈ H(G) such that μ_{G}(H) > 0 , and then let ξ be a μ_{G} -density point of H ∩ L(G) . Since G is convex cocompact, ξ is a uniformly radial limit point, and therefore s_{ξ} ⊂ ⋃_{g∈G} b(g(0), κ) . Hence, there exists a sequence (g_{m})_{m∈ℕ} of elements g_{m} ∈ G such that ξ ∈ Π (b(g_{m}(0), κ)) and d(0, g_{m}(0)) < d(0, g_{m+1}(0)) , for all m. Since ξ ∈ H ∩ L(G) is a μ_{G} -density point, it follows

lim_{m→∞} \frac{μ_{G} (Π (b(g_{m}(0), 2κ)) ∩ H)}{μ_{G} (Π (b(g_{m}(0), 2κ)))} = 1.

Now let ϵ > 0 be fixed. We then have, for each m sufficiently large,

μ_{G} (Π (b(g_{m}(0), 2κ)) \ H) < ϵμ_{G} (Π (b(g_{m}(0), 2κ))) .

Using the δ -conformality of μ_{G} , Sullivan’s Shadow Lemma, and the fact that |(g_{m}^{−1})′| is comparable to exp(d(0, g_{m}(0)) on Π(b(g_{m}(0), 2κ)) , we have

μ_{G} (g_{m}^{−1} (Π (b(g_{m}(0), 2κ)) \ H)) ≍ e^{δd(0,g_{m}(0))}μ_{G} (Π (b(g_{m}(0), 2κ)) \ H)
< ϵe^{δd(0,g_{m}(0))}μ_{G} (Π (b(g_{m}(0), 2κ)))
≪ ϵ.

We then make the following three observations. First, by compactness of L(G) there exists η ∈ L(G) such that on a subsequence (m_{k}) we have

lim_{k→∞} g_{m_{k}}^{−1}(ξ) = η.

Secondly, since H(G) is G -invariant and compact, there exists H_{0} ∈ H(G) such that on a subsequence (m′_{k}) of (m_{k}) we have, where convergence is with respect to the Hausdorff topology,

lim_{k→∞} g_{m′_{k}}^{−1}(H) = H_{0}.

Third, we clearly have there exists a constant R > 0 such that the boundaries of g_{m′_{k}}^{−1}Π(b(g_{m′_{k}}(0), κ)) and g_{m′_{k}}^{−1}Π(b(g_{m′_{k}}(0), 2κ)) are uniformly separated by 2R.

Therefore, since lim g_{m′_{k}}^{−1}(ξ) = η and g_{m′_{k}}^{−1}(ξ) ∈ g_{m′_{k}}^{−1}Π(b(g_{m′_{k}}(0), κ)) , it follows for all k ∈ ℕ,

B(η, R) ⊂ g_{m′_{k}}^{−1}Π(b(g_{m′_{k}}(0), 2κ).

Combining this observation with the measure estimate above, we obtain μ_{G} (B(η, R) \ H_{0}) ≪ ϵ. Since this estimate holds for all ϵ, it follows

μ_{G} (B(η, R) \ H_{0}) = 0.

Now, if L(G) ∩ (B(η, R) \ H_{0}) ≠ ∅ then L(G) ∩ (B(η, R) \ H_{0}) is a nonempty open subset of L(G) , and hence μ_{G} (B(η, R) \ H_{0}) > 0 . Clearly, this gives a contradiction, and therefore

B(η, R) ∩ L(G) ⊂ H_{0}.

Then, since L(G) = ⋃_{g∈G} g (B(η, R) ∩ L(G)) , compactness of L(G) gives that there exist h_{1}, …, h_{l} ∈ G such that

L(G) ⊂ ⋃_{i=1}^{l} h_{i} (H_{0}) .

This contradicts the assumtion that G is of non-Fuchsian type, and hence the assertion follows. □

Proof of Theorem 2: Combine Theorem 1 and Proposition 2. □

Remark: We remark that in the proof of Proposition 2, in order to deduce that μ_{G}(H) = 0 for all H ∈ H(G) , we did not use the fact that the elements of H(G) are intersections of spheres with L(G) . Hence, a straight forward adaptation of the proof shows that if S is any G -invariant set of subsets of 𝕊^{n} which is closed with respect to the Hausdorff topology, then μ_{G}(S) = 0 for all H ∈ S , unless finitely many elements of S contain L(G) . Consequently, Theorem 2 admits the following generalization.

• Let G < Con(n) be a convex cocompact Kleinian group, and let S be a G - invariant set of subsets of 𝕊^{n}, which is compact with respect to the Hausdorff topology, and which has the property that L(G) is not contained in a finite union of elements of S . We then have that S is (G, μ_{G}) -neglectable, and hence Theorem 1 implies that μ_{G} is a S -friendly measure.

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B.O. Stratmann, Mathematical Institute, University of St Andrews, St Andrews KY16 9SS, Scotland. e-mail: bos@maths.st-and.ac.uk

M. Urbański, Dept. of Mathematics, University of North Texas, Denton, TX 76203-5118, USA. e-mail: urbanski@unt.edu