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ERGODICITY OF BOWEN-MARGULIS MEASURE FOR

SOME NONSTRICTLY CONVEX HILBERT GEOMETRIES:

EXTENDED VERSION

HARRISON BRAY

Abstract. We study the geodesic flow of a class of 3-manifolds intro-duced by Benoist which have some hyperbolicity but are non-Riemannian,not CAT(0), and with non-C1 geodesic flow. The geometries are non-strictly convex Hilbert geometries in dimension three which admit com-pact quotient manifolds by discrete groups of projective transformations.We prove the Patterson-Sullivan density is canonical, with applicationsto counting, and construct explicitly the Bowen-Margulis measure ofmaximal entropy. The main result of this work is ergodicity of theBowen-Margulis measure.

Remark 0.1. This version of the paper has been expanded upon for clarityand readability, and is not for publication.

1. Introduction

In 2004, Yves Benoist released the first results on geodesic flows of com-pact quotients of properly convex domains in real projective space, provingthat strict convexity of the domain is equivalent to an Anosov geodesic flowof the quotient [Ben04]. Not long after, Benoist produced nontrivial ex-amples of nonstrictly convex domains with compact quotients in dimensionthree, and proved rigid geometric properties for these domains ([Ben06], seeTheorem 1.4).

This family of 3-manifolds, whose quotients we call the Benoist 3-manifolds,lack the Anosov property but have have similar topological properties to non-positively curved manifolds which are rank one. Hence they are promisingcandidates for studying the geodesic flow. However, the geometry is onlyFinsler and not Riemannian, meaning angles are not defined, the naturalmetric is not CAT(0), and the geodesic flow is not C1. In this work, weextend the approach of Knieper for rank one manifolds [Kni97, Kni98] andstudy the geodesic flow of properly convex domains in real projective space,known also as Hilbert geometries, without the strictly convex hypothesis forthe first time. We prove the following central result:

Theorem 1.1. The Bowen-Margulis measure is an ergodic measure of max-imal entropy for geodesic flows of the Benoist 3-manifolds.

1

2 HARRISON BRAY

In seeking the main result, we develop the asymptotic geometry andPatterson-Sullivan measures at infinity. Let δΓ denote the critical exponentof the fundamental group Γ acting on the universal cover.

Theorem 1.2. The universal cover of a Benoist 3-manifold admits a Buse-mann density of dimension δΓ called the Patterson-Sullivan density, andBusemann densities of the same dimension δ > 0 are unique up to constant.

Let SΩ(x, t) be the sphere of Hilbert radius t about x. Let vol be a Hilbertvolume form on spheres (for more on Hilbert volume, see [Mar14, Section1]). As in [Sul79], Theorem 1.2 can be applied to prove:

Theorem 1.3. Let Ω be a properly convex, indecomposable Hilbert geometryof dimension three which admits a cocompact action by a discrete, torsion-free group Γ of projective transformations. Then for all x ∈ Ω, there is aconstant a(x) > 0 such that

1

a≤ volSΩ(x, t)

eδΓt≤ a.

A corollary of Theorem 1.3 is that the group Γ is divergent.

Historical remarks. Properly convex domains in real projective space arenamed Hilbert geometries after Hilbert’s solution to his fourth problem;they are examples of affine metric spaces for which lines are always geo-desic. Though much work has been done for geodesic flows of strictly con-vex Hilbert geometries, little is known on the dynamics in the nonstrictlyconvex case [Ben04, Cra11, Cra09, Cra14b, CM14]. As alluded to in theintroduction, Benoist first proved that for any divisible properly convex do-main Ω ⊂ RPn, meaning Ω admits a discrete, cocompact action by a groupΓ of projective transformations, the following are equivalent: (i) Ω is strictlyconvex, (ii) the topological boundary ∂ Ω is C1, (iii) Γ is δ-hyperbolic, and(iv) the geodesic flow of M = Ω/Γ is Anosov [Ben04, Theorem 1.1]. Sincethe geodesic flow is also topologically transitive (in fact, mixing, [Ben04,Theorem 1.2]), it follows that there is a unique measure of maximal entropyin the strictly convex setting [Bow75, Fra77].

Benoist then constructed examples of nonstrictly convex, divisible Hilbertgeometries in dimension three which have some hyperbolicity but have iso-metrically embedded flats. These flats appear as properly embedded triangles4 in Ω, meaning 4 ⊂ Ω and ∂4 ⊂ ∂ Ω, which are isometric to R2 with thehexagonal norm in the Hilbert metric [dlH93]. Moreover, he shows that anynonstrictly convex, indecomposable, divisible Hilbert geometry must havethe same basic structure:

BOWEN-MARGULIS MEASURE FOR BENOIST 3-MANIFOLDS: EXT. VERSION 3

Theorem 1.4 ([Ben06, Theorem 1.1]). Let Γ < SL(4,R) be a discretetorsion-free subgroup which divides an open, properly convex, indecompos-able Ω ⊂ RP3, and let M = Ω/Γ. Let T denote the collection of properlyembedded triangles in Ω. Then

(a) Every subgroup in Γ isomorphic to Z2 stabilizes a unique triangle4 ∈ T .(b) If 41,42 ∈ T are distinct, then 41 ∩42 = ∅.(c) For every 4 ∈ T , the stabilizer StabΓ(4) contains an index-two Z2

subgroup.(d) The group Γ has only finitely many orbits in T .(e) The image in M of triangles in T is a finite collection Σ of disjoint tori

and Klein bottles, denoted by T . If one cuts M along each T ∈ Σ, eachof the resulting connected components is atoroidal.

(f) Every open line segment in ∂ Ω is included in the boundary of some4 ∈ T .

(g) If Ω is not strictly convex, then the set of vertices of triangles in T isdense in ∂ Ω.

This structure is essential to make the arguments needed, and we will referback to parts of this theorem throughout the paper. Since a version of thistheorem does not yet exist in higher dimensions, our arguments are validonly in dimension three. We call compact quotient manifolds of nonstrictlyconvex, properly convex, indecomposable domains the Benoist 3-manifolds.

The existing theory does not apply to studying the geodesic flows of theBenoist 3-manifolds. The geodesic flow of Ω/Γ has the same regularity of∂ Ω, hence by Benoist’s dichotomy, in the nonstrictly convex setting thegeodesic flow is not C1. Crampon’s Lyapunov exponents cannot be com-puted, and Pesin theory, which requires the flow to be C1+α, does not apply[Cra14b]. Knieper’s work uses the existence of an inner product and a notionof angle, so his work on Riemannian rank one manifolds cannot be directlyapplied [Kni97, Kni98]. The geometry is not CAT(0) because the isometri-cally embedded flats, which are properly embedded projective triangles, arenot CAT(0) so we cannot use results from the thesis of Ricks [dlH93, Ric15].

Nonetheless, we can adapt the methods of Knieper in rank one followingthe Patterson-Sullivan approach [Pat76, Sul79]. The irregularity of the ge-ometry and our techniques to manage this comprise a significant portion ofthe paper. The Bowen-Margulis measure comes from the Patterson-Sullivandensity in a natural way, and ergodicity follows a variation of the Hopfargument [Hop39]. Originally, Hopf applied smooth stable and unstablefoliations to prove ergodicity of the geodesic flow of constant curvature hy-perbolic surfaces for Liouville volume [Hop39]. This proof has been extendedto many settings, the most relevant being geometrically finite manifolds ofpinched negative curvature [Rob03], compact rank one manifolds [Kni98,Theorem 4.3], and geometrically finite rank one manifolds [LP15, Section7]. In the setting we study, the stable and unstable sets are not even locally

4 HARRISON BRAY

smooth and are not defined for a dense set of directions, but we are still ableto adapt this classical proof.

Structure of the paper. We first introduce Hilbert geometries and the centraltools in Section 2. In Section 3 we study asymptotic geometry of Benoist3-manifolds and the compatible Busemann function. We then constructthe Patterson-Sullivan density in Section 4 and prove the Shadow Lemma(Lemma 4.9), and in Section 5 we prove this construction is canonical (The-orem 1.2), with application to growth rates of volumes of spheres and di-vergence of Γ (Theorem 1.3). Lastly, in Section 6, we construct the Bowen-Margulis measure and complete the proof of the main result, Theorem 1.1.

Acknowledgements. The author is very grateful to advisor Boris Hasselblatt,to postdoc mentors Dick Canary and Ralf Spatzier, and to Aaron Brown andMickael Crampon for helpful conversations, emails, and feedback on thepaper. The author thanks the CIRM for support in the early stages of thiswork. Many thanks to the referee for valuable feedback which immenselyimproved the paper. The author was supported in part by NSF RTG grant1045119.

2. Preliminaries

We say a domain Ω ⊂ RPn is properly convex if Ω can be representedas a bounded set in some affine chart, and denote by ∂Ω the topologicalboundary of Ω in RPn. Define H to be a supporting hyperplane to aproperly convex Ω ⊂ RPn if H is a codimension 1 projective subspace ofRPn which intersects ∂ Ω but not Ω. Then a properly convex Ω is strictlyconvex if every supporting hyperplane intersects ∂ Ω at a single point.

For any properly convex domain Ω, fix an affine chart in which Ω isbounded and define the Hilbert Ω-distance between x, y ∈ Ω as follows:define dΩ(x, x) = 0 and for x 6= y, let xy denote the projective line inthis affine chart uniquely determined by x and y and take a, b ∈ ∂ Ω to bethe distinct intersection points of xy with ∂ Ω. Then dΩ(x, y) = 1

2 | log[a :x : y : b]| where [a : x : y : b] is the Euclidean cross-ratio, a projectiveinvariant. It will be useful to denote by [xy] the segment of xy between xand y in Ω, and [xy) = [xy] r y. The cross-ratio of four projective linesL1, L2, L3, L4 intersecting at one point is well-defined as [L1 : L2 : L3 : L4] =[a1 : a2 : a3 : a4] where ai ∈ Li and a1, a2, a3, a4 are collinear; this can beused to prove that dΩ is a well-defined metric. The group Aut(Ω) := g ∈PSL(n + 1,R) | gΩ = Ω is a subgroup of isometries of (Ω, dΩ). TheHilbert metric comes from a Finsler norm FΩ defined on the tangent bundleTΩ (see [Cra14a]). The norm FΩ is only Riemannian when Ω is an ellipsoidand has the same regularity as the boundary of the domain [SM00, Cra14a].

Projective lines are geodesic and are the only geodesics in the strictlyconvex case, but in general geodesics are not always lines. The metric space


(Ω, dΩ) is complete and the topology induced by the metric dΩ coincideswith the ambient Euclidean topology on Ω in this affine chart.

We say that Ω in RPn is divisible if there exists a discrete subgroup Γ ofAut(Ω) acting properly discontinuously and cocompactly on Ω. Also, Ω isdecomposable if the cone over Ω in Rn+1 is decomposable, and indecompos-able if Ω is not decomposable (see [Mar14, Section 3] for more details). LetM = Ω/Γ denote the quotient manifold.

Since geodesics are not unique for the Benoist 3-manifolds, we define thegeodesic flow to be flowing along projective lines, as is the case when Ω isstrictly convex. More formally, let Ω be a divisible properly convex domainwith dividing group Γ and quotient manifold M . Let `v : R → M be theprojective line parameterized at unit Hilbert speed, uniquely determined byv ∈ T 1M , the unit tangent bundle to M for the Finsler norm FΩ. TheFinsler unit tangent bundle to Ω is denoted T 1Ω. Then the Hilbert geodesicflow of M is ϕt : T 1M → T 1M defined by ϕt(v) = ˙

v(t). This geodesicflow has the same regularity as the boundary of the universal cover Ω (formore details, see [Cra14a, Section 2.4]).

Formally, a geodesic γ for the Hilbert metric is a path in Ω or M such thatthe length of any segment of γ is equal to distance between the endpoints.On occasion we will parameterize γ at unit Hilbert speed, and treat γ as amapping from R to Ω or M to take advantage of the parameterization.

2.1. Busemann functions. For any three points x, y, z ∈ Ω, we define theBusemann function to be

βz(x, y) = dΩ(x, z)− dΩ(y, z).

Evidently, for all z ∈ Ω, the function βz is anti-symmetric, meaningβz(x, y) = −βz(y, z), and satisfies the property of a cocycle, that is βz(x, y)+βz(y, w) = βz(x,w), for all x, y, w ∈ Ω. Also |βz(x, y)| ≤ dΩ(x, y) by the tri-angle inequality. Lastly, since Γ is acting on Ω by isometries, βγz(γx, γy) =βz(x, y) for all γ ∈ Γ. Geometrically, βz(x, y) describes the signed distancebetween the Hilbert spheres centered at z passing through x and y.

Definition 2.1. To extend the Busemann cocycles βz to ∂Ω, let ` : [0,∞)→Ω denote a projective ray, with forward endpoint `+ ∈ ∂Ω. Then define

β−ξ (x, y) = lim inf`+=ξ,t→∞

β`(t)(x, y). β+ξ (x, y) = lim sup

`+=ξ,t→∞β`(t)(x, y).

These functions exist and are bounded in absolute value by dΩ(x, y) forall x, y ∈ Ω and ξ ∈ ∂Ω. It is straightforward to verify for all x, y, w ∈ Ωand ξ ∈ ∂Ω that β−ξ (x, y) = −β+

ξ (y, x), and

β−ξ (x, y) + β−ξ (y, w) ≤ β−ξ (x,w) ≤ β+ξ (x,w) ≤ β+

ξ (x, y) + β+ξ (y, w).

Since Γ acts by isometries and takes projective lines to projective lines,β±γξ(γx, γy) = β±ξ (x, y). Then if indeed β+

ξ = β−ξ , we may define βξ = β+ξ ,

and see that the anti-symmetric, cocycle, and Γ-invariance properties of βzfor z ∈ Ω extend to βξ.

6 HARRISON BRAY

For such ξ ∈ ∂Ω, the horosphere through x ∈ Ω based at ξ is the zero setof βξ(x, ·), denoted by Hξ(x).

2.2. Boundary points. Recall that a supporting hyperplane to Ω at apoint ξ in ∂Ω is a projective hyperplane H such that H contains ξ andH ∩ Ω = ∅. Borrowing language from convex geometry, we introduce thefollowing terms:

Definition 2.2. A point ξ in ∂Ω smooth if there is a unique supportinghyperplane to Ω at ξ. The point ξ in ∂Ω is extremal if ξ is not contained inany open line segment inside ∂Ω.

Note that smooth points in ∂Ω may not be C1 points when ∂Ω is treatedas a curve in an affine chart. For the examples of interest to this work, therewill be a dense set of points in the boundary for which the derivative is notdefined. By Benoist, the complement of boundaries of properly embeddedtriangles in ∂ Ω is exactly the set of smooth extremal points:

Proposition 2.3 ([Ben06, Proposition 3.8]). Let Γ be a discrete, torsion-freesubgroup of PSL(4,R) which divides an indecomposable, divisible, properlyconvex domain in RP3. Then

a) For every nontrivial line segment σ ⊂ ∂Ω, there exists a properly embed-ded triangle 4 such that σ ⊂ ∂4.

b) A point x ∈ ∂Ω is smooth if and only if x is not the vertex of any properlyembedded triangle in Ω.

It follows from Theorem 1.4 that smooth extremal points are dense in ∂Ωin the setting of interest. We will see that these smooth extremal pointscarry the hyperbolic behavior of the dynamics, and the Busemann cocycleswill be well-defined for these points.

2.3. Busemann densities. We modify the original definition in our settingto address issues with nonsmooth points in the boundary.

Definition 2.4. An α-dimensional Busemann density or Busemann densityof dimension α > 0 for Ω is a family of finite Borel measures µxx∈Ω on ∂Ωwhich satisfy:

• (quasi-Γ-invariance) for all γ ∈ Γ, γ∗µx = µγx, and• (transformation rule) for all x, y ∈ Ω and ξ ∈ suppµy, the measuresµx and µy are absolutely continuous, and their Radon–Nikodym de-rivative satisfies

e−αβ+ξ (x,y) ≤ dµx

dµy(ξ) ≤ e−αβ

−ξ (x,y).

Note that if the Busemann function was well-defined for every point in∂Ω, we would recover the standard definition of a Busemann density. Wewill see later that there is a construction of such a family of measures withfull support on the boundary for which almost every point is smooth andextremal and therefore has a well-defined Busemann function.


2.4. Shadow topology. At times we will take advantage of the ambientEuclidean topology on Ω, represented as a bounded convex domain in anaffine chart, and the induced topology on ∂Ω and Ω = Ω ∪ ∂Ω. We defineanother topology on ∂Ω which interacts with the Hilbert geometry inside Ωas follows.

Definition 2.5. Let BΩ(x, r) be the open metric dΩ-ball about x ∈ Ω ofradius r. Then the shadow of radius r from x to y is denoted byOr(x, y), andis equal to the endpoints of projective rays based at x which pass throughBΩ(y, r). These shadows generate a possibly basepoint-dependent topologyon ∂Ω called the shadow topology based at x.

We will see in Proposition 3.3 that the shadow topology does not dependon the basepoint, because it agrees with the Euclidean topology on ∂Ω inan affine chart in which Ω is bounded.

2.5. Regular vectors. For any vector v ∈ T 1Ω, there is a unique orientedprojective line `v determined by v, and we let v− and v+ denote the inter-sections of `v in ∂Ω in backward and forward time, respectively.

Definition 2.6. A vector v ∈ T 1Ω is regular if both v− and v+ are smoothextremal points. The set of regular vectors in T 1Ω is denoted by T 1Ωreg .

Regularity is preserved by projective transformations, so a vector in T 1Mis regular if any lift in T 1Ω is regular, and T 1Mreg is the set of all regular

vectors in T 1M .

2.6. Standing assumptions. Throughout the rest of the paper, unlessotherwise indicated, we assume Ω to be a nonstrictly convex, properly con-vex, divisible, indecomposable domain in RP3 with discrete torsion-free di-viding group Γ. Then M = Ω/Γ is a Benoist 3-manifold. We fix an affinechart in which Ω is bounded, and work with Ω in this affine chart.

3. Asymptotic geometry

In this section, we will prove some lemmas on the shadow topology andBusemann functions.

3.1. The shadow topology is well-defined. We prove the shadow topol-ogy is basepoint independent and Hausdorff via Proposition 3.3.

First, for any set U in Ω represented as a bounded convex set in an affinechart, denote the convex hull of U by CU ; this set is the smallest convex setcontaining U . If Ω is a properly convex domain in RPn, represented as abounded set in an affine chart, then a face in ∂Ω is a set F contained in ∂Ωwhich is open in the subspace topology on ∂Ω and such that F ⊆ H ∩ ∂Ωwhere H is a supporting hyperplane to Ω.

This next lemma does not require that Ω be a Benoist 3-manifold, al-though the rest of the paper will use that assumption.

8 HARRISON BRAY

Lemma 3.1. If U is open in ∂Ω in the subspace topology, then either U iscontained in a face in ∂Ω or CU ∩ Ω has nonempty interior.

Proof. The open set U is contained in a face in ∂Ω if and only if CU∩Ω = ∅.Suppose there exists a point u ∈ CU ∩ Ω. Let ξ, η ∈ U be such that u ison the projective line segment (ξη), which is completely contained in Ω byconvexity. Since U is open in ∂Ω in the subspace topology, there are openneighborhoods Uξ, Uη contained in U of ξ and η, respectively. Then

V = p ∈ Ω | p ∈ (ab) for some a ∈ Uξ, b ∈ Uη

is open in the subspace topology on Ω, hence u ∈ V ∩ Ω is an open neigh-borhood of u in Ω contained in CU ∩ Ω.

Lemma 3.2. Let yn ∈ Ω be a sequence of points converging in the Euclideantopology on Ω to ξ ∈ ∂Ω. If ξ is a smooth extremal point in ∂Ω, then for allr < 0, x ∈ Ω, there exists a k ∈ R such that

∞⋂n≥kOr(x, yn) = ξ.

Proof. Let η be a point in ∂Ω distinct from ξ. Since ξ is smooth and ex-tremal, the open line segment (ηξ) is contained in Ω. Then a straightforwardcross-ratios argument confirms that the minimum Hilbert distance betweenthe projective line segment (xη) and the points yn diverge as yn converges toξ. Then for any r > 0 and x ∈ Ω, for sufficiently large n, (xη)∩BΩ(yn, r) = ∅as desired.

Proposition 3.3. For every x ∈ Ω, the shadow topology on ∂Ω based at xagrees with the Euclidean topology on ∂Ω.

Proof. Let U be open in the Euclidean topology on Ω. By Lemma 3.1, eitherU is contained in a face in ∂Ω, or CU ∩Ω has nonempty interior. In the firstcase, it is clear by definition of the Hilbert metric that one can find a pointy and a small diameter r so that the shadow Or(x, y) is contained in U .However for the Benoist 3-manifolds, this case does not apply by Theorem1.4. Thus, we can in fact fix r and let the shadows Or(x, y) vary over y ∈ Ω.

Assume U is not contained in any face in ∂Ω. Since smooth extremalpoints are dense in ∂Ω (Theorem 1.4 and Proposition 2.3), there exists asmooth extremal ξ ∈ U . We can choose a sequence of points yn alongthe projective line segment (xξ) which converge to ξ. Since CU ∩ Ω hasnonempty interior, and Hilbert metric balls are open, there is a shadowbased at x contained in U by applying Lemma 3.2 to the sequence yn.

Conversely, we prove shadows are open in the Euclidean subspace topol-ogy on ∂Ω. Consider a sequence ξn ∈ ∂Ω r Or(x, y) which converges toξ ∈ ∂Ω in the Euclidean topology. Assume by contradiction ξ is in Or(x, y).Let ` = (xξ) be the projective ray from x to ξ and `n = (xξn) for each n.By assumption, `∩BΩ(y, r) 6= ∅. It is clear that if the points ξn converge to


ξ in the boundary, then the lines `n converge to ` in the Gromov-Hausdorffsense. Then by definition of the Hilbert metric, for a point p ∈ `∩BΩ(y, r),there is a sequence of points pn ∈ `n converging to p. For sufficiently largen, we then have pn ∈ `n ∩BΩ(y, r), a contradiction.

3.2. The Busemann function and horospheres. In this subsection, weverify some regularity properties of the Busemann function.

Lemma 3.4. The Busemann function βξ is well-defined on smooth points

ξ in ∂Ω, meaning βξ = β−ξ = β+ξ if ξ ∈ ∂Ω is smooth.

More specifically, we have the following geometric description of the Buse-mann function: for any x, y ∈ ∂Ω, and any ξ ∈ ∂Ω which is smooth, let Hξ

be the supporting hyperplane to Ω at ξ, and let x−, y− be the intersectionpoints of xξ, yξ respectively with ∂Ω which are not ξ. Let q(x, y, ξ) be the

unique intersection point of x−y− with Hξ in projective space. Then

β−ξ (x, y) = β+ξ (x, y) =

1

2log[x−y− : xq : yq : ξq].

(See Figure 3.1).

Proof. First, we confirm that when ξ is smooth, the limit βξ(x, y) =limt→∞ β`(t)(x, y) exists and does not depend on the projective ray ` oreven the parameterization of ` which converges to ξ. The argument is asfollows: when x, y, `, and ξ are all collinear, the limit is either dΩ(x, y)or −dΩ(x, y). Then, if x, y, ξ are collinear, consider any sequence of realnumbers tk diverging to ∞. Then because ξ is smooth and by definition ofthe Hilbert metric, there exists a parameterization `′ of the projective linesegment (xξ) so that dΩ(`′(tk), `(tk)) → 0 as k → ∞. The limit βξ(x, y)can be verified with the triangle inequality as again being either dΩ(x, y) or−dΩ(x, y), and since x, y, ξ are all collinear, this confirms the statement ofthe lemma.

Lastly, assume x, y and ξ are not necessarily collinear. Consider a se-quence zn of points on (xξ) converging to ξ. As illustrated in Figure 3.1,construct the projective lines zny and take intersections of these lines with∂Ω to be z+

n and z−n where z−n is closer to y than to zn. Let x− be theintersection point other then ξ of xξ with ∂Ω. Then the projective lines xξand zny span some projective plane, and there exists a unique point qn intheir intersection. The point yn in the intersection of yqn with xξ is the

same distance from zn as y. As zn → ξ, by smoothness of ξ, the line z+n qn

converges in the Gromov-Hausdorff sense to a line inside of Hξ, the unique

supporting hyperplane to Ω at ξ. Then yn → y a point on xξ, and z−n → y−,where y− is the intersection of yξ with ∂Ω other than ξ. It follows thatβ±zn(x, y)→ βξ(x, y), where y = x−ξ ∩ yq.

We now conclude that βξ satisfies the anti-symmetric, cocycle, and Γ-invariance properties discussed in Definition 2.1 whenever ξ is smooth.

10 HARRISON BRAY

ξ

y−

x−

Hξq

x

yxy

ξ

z−n

z+n

x−

qn

yn

zn

y

Figure 3.1. For the proof of Lemma 3.4. In the left panel,we take a 2-dimensional slice of Ω determined by x, ξ, and y.In the right panel, we take a sequence 2-dimensional slicesof Ω determined by xξ and zny. In Lemma 3.4 we confirmthat if ξ is smooth, then βξ(x, y) = 1

2 log[x− : x : y : ξ] =12 log[y− : x : y : ξ] as pictured in the left panel.

Lemma 3.5. The Busemann function restricted to smooth points in ∂Ω iscontinuous.

Proof. Suppose ξ is a smooth point in ∂Ω. By the cocycle property, itsuffices to consider y in Ω such that βξ(x, y) = 0. Let x−, y− be the other

intersection points of ξx, ξy with ∂ Ω, respectively. Let Hξ denote the uniquesupporting hyperplane to Ω at ξ+. Since βξ(x, y) = 0, there is a point

p ∈ RP3 such that xy ∩ x−y− ∩Hξ = p by the geometric characterizationof the Busemann function (Lemma 3.4 and Figure 3.1). A 2-dimensionalversion of the set-up is pictured in Figure 3.2 for clarity.

Let ξn ∈ ∂Ω be a sequence of smooth boundary points converging to ξ andlet Hξn be the unique supporting hyperplanes at ξn, and x−n the intersection

of xξn with ∂ Ω and y−n the intersection of yξn with ∂ Ω. Let Lnxy = x−n y−n

and qn = Lnxy ∩Hξn . Then we construct a sequence yn by the intersection

of yqn with xξn.By contrapositive, suppose βξn(x, y) does not converge to 0. Then there

is a subsequence nj such that βξnj (x, y) ≥ ε > 0. Since RP3 rΩ is compact,

up to extraction of subsequences we can assume qnj → q /∈ Ω. Moreover,

ξnjqnj ⊂ RP3 r Ω accumulates on ξq in RP3 r Ω, making ξq a supporting

hyperline to Ω at ξ. Also, q is on the line x−y−.Now, as ynj = yqnj ∩ xξnj , and ξnj → ξ, we conclude ynj → y where

y = yq ∩ xξ. Since βξnj (x, y) → dΩ(x, y) ≥ ε, we have y 6= x and q 6= p.

Then ξq is a supporting hyperline to Ω at ξ which is not contained in Hξ,contradicting that ξ is smooth.

Since horospheres are zero sets of the Busemann function, we have:

Corollary 3.6. Horospheres based at smooth boundary points are globallydefined and vary continuously over smooth points.


ξ

y−

x−

Hξ p

xy

y−nx−n

ξn

qn

yn

Figure 3.2. Choose x, y so that βξ(x, y) = 0. If there existsa sequence ξn such that yn → y 6= x on (xξ), then ξ is notsmooth.

4. Patterson-Sullivan Theory

In this section, we construct the Patterson-Sullivan density, a conformalfamily of measures on ∂Ω parameterized by x ∈ Ω which are compatiblewith the group action Γ. The density is named for the independent work ofPatterson and Sullivan in negative curvature and has since been generalizedto many settings, including rank one manifolds [Pat76, Sul79, Kni97]. The-orems 1.2 and 1.3 follow the study of these measures and their properties.To generalize the results beyond dimension three, more understanding ofthe shadow topology in the general setting would be necessary, as well assome lemmas around hyperbolicity of the geometry.

4.1. Poincare Series and the critical exponent. The critical exponent,δΓ, of a group Γ acting discretely, properly discontinuously, and by isometrieson (Ω, dΩ) is the critical value of 0 ≤ s ∈ R for the Poincare series,

P (x, y, s) =∑γ∈Γ

e−sdΩ(x,γy)

The group Γ is of divergent type if P (x, y, δΓ) diverges and convergenttype if P (x, y, δΓ) converges. It is straightforward to verify that convergenceof P (x, y, s) does not depend on x or y by the triangle inequality and that

we can realize δΓ = lim supt→∞

1

tlogNΓ where NΓ := #γ ∈ Γ | dΩ(x, γx) ≤ t.

By previous work we have that δΓ > 0 [Bra17]. Recently it was shown thatδΓ ≤ dim(Ω)− 1 with equality if and only if Ω is the ellipsoid, generalizinga result of Crampon for the strictly convex case [BMZ16, Cra09].

4.2. Patterson-Sullivan densities.

Proposition 4.1. There exists a nontrivial Busemann density of dimensionδΓ > 0 on ∂Ω, called a Patterson-Sullivan density.

12 HARRISON BRAY

Proof. The construction follows Patterson and Sullivan [Pat76, Sul79]. Fors > δΓ, choose an observation point o ∈ Ω for the measures and for thevisual boundary. For each x ∈ Ω define a measure on Ω by

µx,s =1

P (o, o, s)

∑γ∈Γ

e−sdΩ(x,γo)δγo

where δp is the Dirac mass at p. Note that for s > δΓ, µx,s is supported on Ω.Also, by definition of the critical exponent, if s > δΓ then P (x, y, s) is finitefor all x, y ∈ Ω so µx,s(Ω) = P (x, o, s)/P (o, o, s) <∞. By compactness of Ωwe may take a weak limit as s decreases to δΓ to obtain a finite nontrivialmeasure,

µx = lims→δ+

Γ

µx,s.

If the Poincare series diverges at δΓ (Γ is of divergent type), then the totalmass of µx,s is pushed to ∂Ω as s decreases to δΓ and P (o, o, s) → ∞. Atthe limit, suppµx ⊂ ∂Ω. If the Poincare series converges at δΓ (Γ is ofconvergent type), then we follow Patterson’s method for Fuchsian groupswhich generalizes to any manifold group [Pat76]. First, he showed it ispossible to constuct an increasing function f : R+ → R+ with subexponentialgrowth: that is, for all ε < 0, there exists an x0(ε) > 0 such that for allx > x0, y > 0

f(x+ y) ≤ f(x)ey·ε,

and the modified Poincare series

Pf (x, y, s) =∑γ∈Γ

f(dΩ(x, γy))e−sdΩ(x,γy)

has the same critical exponent δΓ and diverges at s = δΓ [Pat76, Lemma

3.1]. Then we denote by µfx a weak limit as s decreases to δΓ of

µfx,s =1

Pf (o, o, s)

∑γ∈Γ

f(dΩ(x, γo))e−sdΩ(x,γo)δγo.

Taking f ≡ 1 recovers µx, so we will check that these measures satisfy thedefinition of a Busemann density for the case that Γ is convergent.

We remark first that Pf (o, o, s) exhibits the same convergence and di-

vergence behavior as P (o, o, s) for s 6= δΓ so µfx,s will be a finite nontrivialmeasure supported on point masses in Ω much like µx,s. Taking a weak-limit

then produces a finite nontrivial measure µfx supported on ∂Ω by the diver-gence of Pf (o, o, s) as s decreases to δΓ. Moreover, for any Borel measurable

set A ⊂ Ω,

µfx,s(γ−1A) =

1

Pf (o, o, s)

∑g∈Γ

f(dΩ(x, go))e−sdΩ(x,go)δgo(γ−1A)

=1

Pf (o, o, s)

∑γg∈Γ

f(dΩ(γx, γgo))e−sdΩ(γx,γgo)δγgo(A) = µfγx,s(A).


Then the quasi-Γ-invariance property from Definition 2.4 holds for any weak

limit µfx. Since µfx,s is supported on countably many point masses in Ω fors > δΓ, we compute

dµfx,s

dµfy,s(γo) =

µfx,s(γo)

µfy,s(γo)=f(dΩ(x, γo))e−sdΩ(x,γo)

f(dΩ(y, γo))e−sdΩ(y,γo)=f(dΩ(x, γo))

f(dΩ(y, γo))e−sβγo(x,y).

As s→ δΓ, indeed suppµfx,s, suppµfy,s is pushed to ∂Ω. By the increasingand subexponential properties of f , for all ε > 0 we have that for all γo suchthat dΩ(γo, y) is sufficiently large,

f(dΩ(x, γo)) ≤ f(dΩ(y, γo) + dΩ(x, y)) ≤ f(dΩ(y, γo)) edΩ(x,y)·ε.

The transformation rule from Definition 2.4 then follows for all ξ in thesupport of such measures, since such ξ must be accumulation points of theΓ-orbit of any point in Ω.

Remark 4.2. The measures µx are Borel measures on ∂Ω, and have full sup-port by quasi-Γ-invariance and minimality of the action of Γ on ∂Ω [Ben06,Proposition 3.10].

4.3. The Shadow Lemma and applications. In this subsection we proveSullivan’s Shadow Lemma in the setting of interest [Sul79].

4.3.1. Geometric lemmas. Define γ ∈ Aut(Ω) to be hyperbolic if γ has anattracting fixed point and a repelling fixed point in ∂ Ω, denoted γ+ andγ−, which are both smooth and extremal, and γ has no other fixed points inΩ. This definition diverges from the classical definition that the translationlength of γ is positive and realized in Ω, which is a consequence but notequivalent. We choose this definition in this setting to separate stabilizersof triangles from group elements that act hyperbolically with north-southdynamics, since both such isometries have positive translation length real-ized in Ω. We will need a proposition from the topological study of theBenoist 3-manifolds, which is straightforward given Theorem 1.4 of [Ben06]:

Proposition 4.3 ([Bra17]). If M = Ω/Γ is a Benoist 3-manifold then Γis the disjoint union of hyperbolic isometries and stabilizers of properly em-bedded triangles. There are infinitely many conjugacy classes of hyperbolicgroup elemments.

The immediate goal is to prove the following geometric proposition, sim-ilar to that in [Bal95], as needed for the Shadow Lemma.

Proposition 4.4. Fix x ∈ Ω. For any two noncommuting hyperbolic isome-tries g, h preserving Ω and O a sufficiently small neighborhood of h+, thereexists an R large and M ∈ N such that for all r ≥ R and all y ∈ Ω, eitherhMO ⊂ Or(y, x) or gMhMO ⊂ Or(y, x).

We first prove two geometric lemmas. Let CA be the convex hull of asubset A in our affine chart for Ω.

14 HARRISON BRAY

Lemma 4.5. If h is a hyperbolic isometry then for any open sets O+ ⊂ ∂ Ωcontaining h+ and O− containing h−, there exists an N ∈ N such that forall n ≥ N ,

hn(Ω r CO−) ⊂ O+ and h−n(Ω r CO+) ⊂ O−.

Proof. If h is a projective transformation preserving Ω with only two fixedpoints in ∂ Ω (and none inside Ω since we assume Γ is torsion-free), then his a biproximal matrix, so h+ is an attracting eigenline in Rn+1 and h− is arepelling eigenline. The result follows since h preserves ∂ Ω.

Lemma 4.6. Suppose h, g are hyperbolic projective transformations preserv-ing Ω such that g+ 6= h+. Then there exist neighborhoods Vg, Vh of g+, h+

such that CVg ∩ CVh = ∅ and there is no properly embedded triangle which

intersects both CVg and CVh.

Proof. Since g, h are hyperbolic, g+, h+ are smooth extremal points. Thereare disjoint open neighborhoods Vg, Vh around g+, h+ respectively in ∂Ω

whose closures are also disjoint, and for which g− 6∈ V g and h− 6∈ V h.If the lemma was false, by convexity of CgnVg, ChnVh, there would exist a

sequence of properly embedded triangles 4n such that gnVg ∩ ∂4n 6= ∅and hnVh ∩ ∂4n 6= ∅ for all n. Since the collection of properly embeddedtriangles is closed in Ω [Ben06, Proposition 3.2], the4n accumulate on some4 properly embedded in Ω. Because g, h are hyperbolic, ∩∞n=1g

nVg = g+and ∩∞n=1h

nVh = h+. Then (g+h+) ⊂ 4 which contradicts the smoothextremal property for fixed points of hyperbolic isometries.

Proof of Proposition 4.4. Applying Lemma 4.6, there are pairwise disjointneighborhoods V ±h , V

±g of h±, g± respectively such that no properly embed-

ded triangle intersects any pair of convex hulls of these neighborhoods inΩ. In particular, this means for Vi, Vj ∈ V ±h , V

±g with Vi 6= Vj , for any

x ∈ CVi and y ∈ CVj , the projective line (xy) is contained in Ω and is notcontained in any single properly embedded triangle.

By Lemma 4.5, there exists an N1 such that h−n(Ω r CV +h ) ⊂ V −h for

all n ≥ N1. Moreover, there exists an N2 such that g−n(Ω r CV +g ) ⊂ V −g ,

implying

g−n(CV +h ) ⊂ g−n(Ω r CV +

g ) ⊂ V −g ⊂ Ω r CV +h .

Then for all y ∈ Ω and all n ≥ maxN1, N2, either h−ny ∈ CV −h or

h−ng−ny ∈ CV −h . Let M = maxN1, N2.Next, we claim that for all r > 0, for γ ∈ h−M , h−Mg−M and R =

dΩ(x, γx), if γy ∈ CV −h then

γ−1V +h ⊂ γ

−1Or(γy, x) ⊂ Or+R(y, x)

which completes the proof of the lemma. Note first that the rightmost inclu-sion is true for all γ ∈ Γ: if p ∈ BΩ(x, r), then dΩ(γ−1p, x) ≤ dΩ(γ−1p, γ−1x)+


dΩ(x, γ−1x) ≤ r + R. So if a projective ray ξ with ξ(0) = γy intersectsBΩ(x, r), then γ−1ξ is a projective ray with γ−1ξ(0) = y which intersectsBΩ(x, r +R).

For the leftmost inclusion, we show that V +h ⊂ Or(γy, x) for sufficiently

large r. First, for any η ∈ V +h the projective ray (γy η) is contained in

Ω but not any properly embedded triangle by choice of V −h , V+h (Lemma

4.5). Then take r ≥ maxη∈V +

h

dΩ(x, (γy η)) and the leftmost containment is

satisfied.

4.3.2. The Shadow Lemma.

Lemma 4.7. Let µ be a nontrivial Busemann density on ∂Ω. Then suppµx =∂Ω.

Proof. Suppose there exists a ξ ∈ O ⊂ ∂Ω, with O open, such that µx(O) =0. By quasi-Γ-equivariance of µx and absolute continuity of the density, forall γ ∈ Γ,

µx(O) = µγx(γO) = 0 =⇒ µx(γO) = 0.

Since O is open, Γ.O covers ∂Ω because Γ acts on ∂Ω minimally (Remark4.2). Therefore,

µx(∂Ω) ≤∑γ∈Γ

µx(γO) = 0

which proves the lemma by contrapositive.

Lemma 4.8. For all ξ ∈ Or(x, y),

dΩ(x, y)− 2r ≤ β−ξ (x, y) ≤ β+ξ (x, y) ≤ dΩ(x, y).

Proof. The rightmost inequality is immediate from the triangle inequality.For the leftmost inequality, let z ∈ Ω converge to ξ along the projective linefrom x to ξ. Divide the projective line (xz) into two segments by its first

intersection p with the closed ball BΩ(y, r). Then by the triangle inequality,dΩ(x, y) ≤ dΩ(x, p) + r and dΩ(z, y) ≤ dΩ(z, p) + r, so

dΩ(x, y)−2r ≤ dΩ(x, p)+dΩ(p, z)−dΩ(y, z) = dΩ(x, z)−dΩ(y, z) = βz(x, y).

The lower bound follows.

Lemma 4.9 (Shadow Lemma). Let µ be a Busemann density of dimensionδ on ∂Ω. Then for every x ∈ Ω and all suffiently large r, there exists aC > 0 such that for all γ ∈ Γ,

1

Ce−δdΩ(x,γx) ≤ µx

(Or(x, γx)

)≤ Ce−δdΩ(x,γx).

16 HARRISON BRAY

Proof. We follow the elegant proof of Roblin [Rob03]. Since γ is an isometryand by quasi-Γ-invariance,

µx(Or(x, γx)) = µx(γOr(γ−1x, x)) = µγ−1x(Or(γ−1x, x)).(4.1)

By the transformation rule (Definition 2.4),

(4.2)

∫Or(γ−1x,x)

e−δβ+ξ (γ−1x,x)dµx(ξ) ≤ µγ−1x(Or(γ−1x, x))

≤∫Or(γ−1x,x)

e−δβ−ξ (γ−1x,x)dµx(ξ).

Combining Equations (4.1) and (4.2) with Lemma 4.8,

(4.3)

∫Or(γ−1x,x)

e−δdΩ(γ−1x,x)dµx(ξ) ≤ µx(Or(x, γx))

≤∫Or(γ−1x,x)

e−δ(dΩ(γ−1x,x)−2r)dµx(ξ),

so, letting ‖µx‖ := µx(∂Ω) <∞,(4.4)

e−δdΩ(γ−1x,x)µx(Or(γ−1x, x)) ≤ µx(Or(x, γx)) ≤ e−δdΩ(γ−1x,x)e2δr‖µx‖.

The rightmost inequality of Equation (4.4) gives us the rightmost inequalityof the lemma immediately. By Proposition 4.3 there exist two noncommut-ing hyperbolic isometries g, h. Then apply Proposition 4.4 to obtain opensets O1 = hMO,O2 = gMhMO ⊂ ∂Ω such that for all r sufficiently large andall γ ∈ Γ, either O1 ⊂ Or(γ−1x, x) or O2 ⊂ Or(γ−1x, x). The µx have fullsupport (Remark 4.2) so we may take 0 < 1

C < minµx(Oi) to completethe proof.

Corollary 4.10 (of Lemma 4.9, [Rob03]). For Γ < PSL(4,R) acting dis-cretely and cocompactly on a Benoist 3-manifold Ω, with critical exponentδΓ,

(a) If there exists a Busemann density of dimension δ > 0, then δ ≥ δΓ.(b) For each x ∈ Ω, there exists a C such that NΓ(x, r) ≤ CeδΓr.

4.3.3. Boundaries of flats are null sets. Let S4(x, r) := y ∈ 4 : dΩ(x, y) =r denote the sphere of Hilbert radius r about x restricted to a properlyembedded triangle, 4. Similarly, B4(x, r) is the ball of Hilbert radius rabout x restricted to the triangle 4. For a properly embedded triangle 4in Ω, let StabΓ(4) = γ ∈ Γ | γ4 = 4.

Lemma 4.11. Pick a tiling of a properly embedded triangle 4 by StabΓ(4)such that p is in the interior of a fundamental domain in the tiling. ChooseR so that the open B4(p,R) covers the compact fundamental domain con-taining p. If Nr denotes the minimal number of γ.B4(p,R) which coverS4(p, r), where γ ∈ StabΓ(4), then Nr is quasi-linear in r.


Proof. We will use De La Harpe’s planar model for (4, dΩ) where p is sentto the origin [dlH93]. The Hilbert metric becomes the hexagonal metric onR2, denoted (R2,7). Choose the unit hexagon in R2 to have vertices at(±1, 0), (±1,±1), (0,±1).

Up to index two, the action of StabΓ(4) is a Z2-action with generatingtranslations g, h and fundamental domain a Euclidean rectangle which wedenote by D, chosen such that there exists an R such that D ⊂ B4(p,R)but γD 6⊂ B4(p,R) for any γ 6= Id ([Ben06], Theorem 1.4(c)). Now we cancompute how many images of B4(p,R) ⊃ D are needed to cover the sixsides of the r-hexagon given by vertex coordinates (±r, 0), (±r,±r), (0,±r),which is the sphere of radius r about the origin in the hexagonal metric.Note that by the nature of the hexagonal geometry, it suffices to verify howmany images of B4(p,R) are needed to cover the lines from the origin toeach of (±r, 0), (±r,±r), (0,±r). Symmetry allows us to restrict attentionto a preferred line: let `r denote the line segment from the origin to (r, 0),and let N(r,0) denote the minimal number of γi ∈ 〈g, h〉 needed so that theunion of the γi.B4(p,R) cover `r.

We make a geometric observation for our counting argument. There aretwo families of parallel lines generated by 〈g, h〉. ∂ D each invariant underexactly one of g or h. The minimal number of fundamental domains neededto cover `r is exactly the number of intersections of each of these familiesof parallel lines with `r. The g-invariant lines have a constant horizontaldisplacement tg, as do the h-invariant lines (denoted th). The number ofcrossings that `r has with g-invariant lines is approximately r/tg, and simi-larly for h. Thus, we obtain a linear upper bound for the minimal numberof γi.B4(p,R) needed to cover `r:

N(r,0) ≤r

tg+r

th.

It is clear that N(r,0) is at least linear in r because R is chosen so thatB4(p,R) covers no more than one fundamental domain in the tiling. Theargument is identical for the other sides of S4(p, r) by symmetry of B4(p,R)so we conclude that Nr is quasi-linear in r.

Proposition 4.12. The boundary of any properly embedded triangle is anull set for any Busemann density of dimension δ > 0.

Proof. Choose a fundamental domain T for the action of StabΓ(4) on aproperly embedded triangle 4, a point p ∈ T and R0 as in Lemma 4.11. Letx ∈ Ω be in a fundamental domain D for the Γ-action on Ω such that T ⊂ D.Choose R large enough that D ⊂ BΩ(x,R) and B4(p,R0) ⊂ BΩ(x,R).Then the Γ.BΩ(x,R) covers Ω, and the minimal number of γ such that∪Ni=1γi.BΩ(x,R) covers S4(p, r), is bounded above by the Nr in Lemma4.11. For each r, choose a covering of S4(x, r) by Nr-many γiBΩ(x,R) andassume that γi ∈ StabΓ(4) for i = 1, . . . , Nr.

18 HARRISON BRAY

Next, we show for all large enough r, ∂4 ⊂⋃Nri=1O2R(x, γix). Let r > 2R.

Consider any projective ray η based at p such that η+ ∈ ∂4. Let ξ denoteprojective ray based at x such that ξ+ = η+. Then parameterizing ξ, η atunit speed, we have that dΩ(ξt, ηt) ≤ dΩ(x, p) ≤ R for all t ≥ 0. Sincep ∈ 4 and η+ ∈ ∂4, then η ∩ S4(p, r) 6= ∅ and there exists a γi suchthat η ∩ BΩ(γix,R) 6= ∅. Let t ≥ 0 be such that dΩ(ηt, γix) < R. ThendΩ(ξt, γix) ≤ dΩ(ξt, ηt) + dΩ(ηt, γix) ≤ 2R, and ξ+ ∈ O2R(x, γix). Lastly,for each i = 1, . . . , Nr let qi ∈ S4(p, r) ∩BΩ(γix,R) 6= ∅. Then

r = dΩ(p, qi) ≤ dΩ(p, x) + dΩ(x, γix) + dΩ(γix, qi) ≤ dΩ(x, γix) + 2R

implying −dΩ(x, γix) ≤ 2R− r for all i = 1, . . . , Nr. By Lemma 4.9,

(4.5) µx(∂4) ≤Nr∑i=1

µx(O2R(x, γix)) ≤Nr∑i=1

Ce−δdΩ(x,γix) ≤ Ceδ2Re−δrNr.

Given that Nr is quasi-linear in r by Lemma 4.11, that δ > 0, and thatEquation 4.5 holds for all r sufficiently large, we conclude µx(∂4) = 0.

Then the following corollary is immediate after Proposition 2.3:

Corollary 4.13. The set of smooth extremal points in ∂Ω is full measurefor any Busemann density of dimension δ > 0.

And the lemma follows:

Lemma 4.14. Busemann densities of dimension δ > 0 on ∂Ω have noatoms.

Proof. It suffices to check for smooth extremal points ξ ∈ ∂Ω by Proposition4.12. By cocompactness of Γ, for fixed sufficiently large R, we can cover theprojective ray (xξ) by a subexponential number of BΩ(γix,R) where γi ∈ Γ.Apply the Shadow Lemma (Lemma 4.9) and Lemma 3.2 to the shadowsOR(x, γix) to complete the proof.

5. Busemann densities are unique

In this section we complete the proof of Theorem 1.2. The arguments inthis section follow those of Sullivan and Knieper [Sul79, Kni97]. We givebrief proofs, mainly to point out when we need Corollary 4.13.

Lemma 5.1 (Local estimates). If µx is a Busemann density of dimensionδ > 0 on ∂Ω, then for all x and all sufficiently large r there exists a constantb(r) such that for y ∈ Ω with dΩ(x, y) large,

1

b(r)e−δdΩ(x,y) ≤ µx(Or(x, y)) ≤ b(r)e−δdΩ(x,y)

The precise arrangement of the bounds is to encourage the interpretationof Or(x, y) as a ball of radius approximately e−dΩ(x,y) centered at the pro-jection of y to ∂Ω from x. This interpretation is originally due to Sullivan[Sul79].


Proof. Note that if y = γx, then we apply Lemma 4.9 to obtain the result.Else, for some Γ-tiling of Ω with compact fundamental domain D, choose rlarge enough that for all x ∈ D, we have D ⊂ BΩ(x, r2). Choosing D suchthat y ∈ D, there exists a γ ∈ Γ such that γx ∈ D ⊂ BΩ(y, r2). By thetriangle inequality,

O r2(x, γx) ⊂ Or(x, y) ⊂ O 3r

2(x, γx).

We verify the claim in this version: if ξ ∈ O r2(x, γx) then there exists

p ∈ (x, ξ) ∩ BΩ(γx, r/2) and dΩ(p, y) ≤ dΩ(p, γx) + dΩ(γx, y) ≤ r2 + r

2 = r.Thus O r

2(x, γx) ⊂ Or(x, y). Similarly, if η ∈ Or(x, y) then there exists

q ∈ (xη) ∩ BΩ(y, r) and dΩ(q, γx) ≤ dΩ(q, y) + dΩ(y, γx) ≤ 3r2 , so we have

Or(x, y) ⊂ O 3r2

(x, γx).

Applying Lemma 4.9, if r is sufficiently large then there is a uniformconstant C such that

1

Ce−δdΩ(x,γx) ≤ µx(O r

2(x, γx)) ≤ µx(Or(x, y))

≤ µx(O 3r2

(x, γx)) ≤ Ce−δdΩ(x,γx).

Our final observation is that since γx ∈ BΩ(y, r/2),

−dΩ(x, y)− r

2≤ −dΩ(x, γx) ≤ −dΩ(x, y) +

r

2

and we conclude1

Ceδr/2e−δdΩ(x,y) ≤ µx(Or(x, y)) ≤ Ceδr/2e−δdΩ(x,y).

For fixed x and sufficiently large r, the Or(x, y) generate the topology of∂Ω over all y ∈ Ω (Proposition 3.3). Thus, we have the following corollaryof Lemma 5.1:

Corollary 5.2. Busemann densities of dimension δ > 0 are equivalent.

Proof. Let µ, ν be Busemann densities of dimension δ. Let ξ ∈ ∂Ω be asmooth extremal point and take a sequence yn of points in Ω converging toξ. Then for all sufficiently large n, dΩ(x, yn) is large enough to apply Lemma5.1 to both µ and ν and conclude:

1

bµ(r)bν(r)≤ µx(Or(x, yn))

νx(Or(x, yn))≤ bµ(r)bν(r).

In particular, ∩n≥kOr(x, yn) = ξ since yn → ξ and ξ is smooth and ex-tremal (Lemma 3.2). Since smooth extremal points form a set of full measurefor any δ-dimensional Busemann density by Corollary 4.13, we conclude thatµx and νx are equivalent.

We now have that for any two Busemann densities µxΩ, νxΩ of thesame dimension, the measures µx and νx are equivalent.

20 HARRISON BRAY

Proposition 5.3. If µ is a Busemann density of dimension δ > 0 on ∂Ω,then µ is ergodic for the Γ-action on ∂Ω.

Proof. Let A ⊂ ∂Ω be a Borel, Γ-invariant set with positive µx-measure.Define a new density µx(B) := µx(A ∩ B) for all x ∈ Ω. Since A is Γ-invariant and has positive measure, it suffices to show that µx is a Busemanndensity also of dimension δ. Then µx is equivalent to µx, and we concludethat µx(∂Ω rA) = µx(∂Ω rA) = 0, proving ergodicity of µx for Γ.

Now we verify that µx is a Busemann density. First, µx is clearlynontrivial and finite, since µx(∂Ω) = µx(A) > 0. Moreover, the two definingproperties of a δ-dimensional Busemann density are satisfied:

(1) (quasi-Γ-equivariance) For any Borel set B ⊂ ∂Ω,

µγx(B) = µγx(A ∩B)

= µx(γ−1(A ∩B))

= µx(A ∩ γ−1B) = µx(γ−1B).

Note that for all g ∈ Γ, g(A ∩B) ⊇ gA ∩ gB because g is invertible.The reverse containment is obvious, allowing us to conclude by Γ-invariance of A that γ−1(A ∩B) = γ−1A ∩ γ−1B = A ∩ γ−1B.

(2) (transformation rule) This is immediate by the transformation rulefor µxx∈Ω:

µx(B) = µx(A ∩B)

=

∫A∩B

e−δβξ(x,y)dµy(ξ)

=

∫Be−δβξ(x,y)IA(ξ)dµx(ξ) =

∫Be−δβξ(x,y)dµy(ξ).

where IA : ∂Ω → 0, 1 is the indicator function on A. It is evidentthat dµx/dµx(ξ) = IA(ξ).

Theorem 5.4. Busemann densities of dimension δ > 0 on ∂Ω are uniqueup to a constant.

Proof. Let µx, νx be two δ-dimensional Busemann densities. It suffices that,the Radon-Nikodym derivative dνx/dµx is Γ-invariant on the set of smoothextremal points, which are a set of full measure by Corollary 4.13. Ergodicityof µx then implies that the Radon-Nikodym derivative is constant µx-almosteverywhere. Since µx and νx are equivalent, the measures then have to agreeup to a constant.

We show Γ-invariance of the Radon-Nikodym derivative for two δ-dimensionalBusemenn densities. Let A be a Borel measurable set. Then, for ξ smooth


extremal, since these points are full measure,∫A

dµxdµx

(γ−1ξ)dµx(ξ) =

∫γA

dµxdµx

(ξ)dµx(γξ) Γ acts by homeos on ∂Ω

=

∫γA

dµxdµx

(ξ)dµγ−1x(ξ) Γ-quasi-invariance of µ

=

∫γAe−δβξ(γ

−1x,x)dµx(ξ) transformation rule of µ

=

∫γAe−δβγξ(x,γx)dµx(ξ) uniqueness of β over limit paths

=

∫Ae−δβξ(x,γx)dµγx(ξ) Γ-quasi-invariance of µ

= µx(A) transformation rule of µ.

By uniqueness of the Radon-Nikodym derivative for L1 functions up to anull set, we conclude that dµx/dµx(γ−1ξ) = dµx/dµx(ξ) almost everywhere.

Combining Proposition 4.1 and Theorem 5.4 gives us Theorem 1.2.

5.1. Volume growth and divergence of Γ. In this section, we see that Γis divergent. With all the tools is place, the proof does not differ from thatof Knieper for rank one manifolds, but we include it here for completeness[Kni97].

Proof of Theorem 1.3. Let µx denote the Patterson-Sullivan density of di-mension δΓ, existence of which we showed in Proposition 4.1. By previouswork we have that δΓ > 0 [Bra17]. Let δ = δΓ throughout the proof. LetR be sufficiently large to apply Sullivan’s Shadow Lemma (Lemma 4.9)and consequently the local estimate in Lemma 5.1. Consider r ≥ 6R. Bycompactness of SΩ(x, t), we can take xiNti=1 to be a maximal r-separatingset in SΩ(x, t). In particular, for all i 6= j, dΩ(xi, xj) ≥ r, SΩ(x, t) ⊂⋃Nti=1BΩ(xi, r), and BΩ(xi, r/3) ∩BΩ(xj , r/3) = ∅ for i 6= j.By the local estimate, there exists a b(r) such that for all xi, each of which

is distance t from x, we have

1

be−δt ≤ µx(Or(x, xi)) ≤ be−δt.

We can moreover choose the b so that this estimate applies to all r ∈[2R, 6R]. Then

µx(∂Ω) ≤ µx(

Nt⋃i=1

Or(x, xi)) ≤Nt∑i=1

µx(Or(x, xi)) ≤ Ntbe−δt,

µx(∂Ω) ≥ µx(

Nt⋃i=1

Or/3(x, xi)) =

Nt∑i=1

µx(Or/3(x, xi)) ≥Nt

be−δt.

22 HARRISON BRAY

So the local estimate for our conformal density yields a fine approximationfor the growth of Nt with t:

Nt

be−δt ≤ µx(∂Ω) ≤ Ntbe

−δt,

beδt

µx(∂Ω)≥ Nt ≥

eδt

bµx(∂Ω).

We can therefore find a b′ such that

1

b′eδt ≤ Nt ≤ b′eδt.

Now, by compactness of M and Γ-equivariance of vol on Ω, we claim thereexists an `(r) such that for all x ∈ Ω,

1

`≤ vol(BΩ(x, r) ∩ SΩ(x, t)) ≤ `.

Then we may just as easily arrange for an `′ such that

vol(SΩ(x, t)) ≤Nt∑i=1

vol(BΩ(xi, r) ∩ SΩ(x, t)) ≤ Nt · `′ ≤ `′b′eδt

and

vol(SΩ(x, t)) ≥Nt∑i=1

vol(BΩ(xi, r/3) ∩ SΩ(x, t)) ≥ Nt

`′≥ 1

`′b′eδt,

which concludes the proof of the theorem.

Corollary 5.5. Let Ω be a properly convex, divisible, indecomposable Hilbertgeometry of dimension three with dividing group Γ. Then Γ is of divergenttype.

Proof. Let D be a compact fundamental domain for the Γ-action on Ω. Thenfor s > δΓ, P (x, y, s) converges so we can apply Fubini’s Theorem to thefollowing integral:∫

D

∑γ∈Γ

e−sdΩ(x,γy)d vol(y) =∑γ∈Γ

∫γ(D)

e−sdΩ(x,γy)d vol(y)

=

∫ ∞0

e−st vol(SΩ(x, t))dt

As s decreases to δΓ, the right hand side diverges because by Theorem 1.3,∫ ∞0

e−δΓt vol(SΩ(x, t))dt ≥∫ ∞

0

1

adt

which is a divergent integral.


6. The Bowen-Margulis measure

In this section, we introduce the Γ-invariant Bowen-Margulis measureon T 1Ω, denoted µBM , following the standard construction [Sul79, Rob03,Kni97] and prove Theorem 1.1.

6.1. Definition and properties. Let µx be the Patterson-Sullivan den-sity constructed in Proposition 4.1, which is a Busemann density of dimen-sion δΓ > 0. For each x ∈ Ω and Borel set A ⊂ T 1Ω, define(6.1)

µxBM (A) =

∫v∈A

lengthΩ(`v ∩ πA)eδΓ(βv+ (x,πv)+βv− (x,πv)) dµx(v−)dµx(v+)

where π : T 1Ω → Ω is the footpoint projection and lengthΩ is the Hilbertlength of the projective line segment `v∩πA. Since the Busemann function iswell-defined almost everywhere (Lemma 3.4 and Corollary 4.13), this defini-tion is valid. Then µxBM is Γ-invariant by the definition of a δΓ-dimensionalBusemann density and the cocycle property of the Busemann function. OnT 1M the measure is finite, and we may normalize it so µxBM (T 1M) = 1.

Recall that T 1Ωreg is the set of regular vectors, which are vectors v whose

endpoints v−, v+ in ∂Ω are both smooth and extremal, and T 1Mreg is the

projection of these vectors to T 1M (Definition 2.6).The following Lemma is clear given the discussion above.

Lemma 6.1. The regular set T 1Mreg is a set of full µBM -measure.

A posteriori, since the µxBM , µyBM are equivalent by construction, they

will be equal up to a constant by ergodicity. Thus, for the remainder of thepaper, we will let µBM := µxBM for some x ∈ Ω.

6.2. Ergodicity. Define the ϕt-invariant strong unstable foliations for v ∈T 1Mreg to be

W su(v) = w ∈ T 1M | d(ϕ−tv, ϕ−tw)→ 0 as t→ +∞

and similarly for W ss(v), the strong stable foliation, which is contractedin forward time. The weak unstable set W ou(v) is the disjoint union ofW ss(ϕtv) for all t ∈ R, and similarly for the weak stable set W os(v). Thisgives us a flow-invariant foliation of the weak unstable sets by strong unsta-ble leaves, and similarly for the stable foliation.

Lemma 6.2. For all regular v, w with v 6= −w, we have W ss(v)∩W ou(w) 6=∅.

Proof. If v is regular then W ss(v) is defined by a geometric characterizationon the universal cover [Ben04, Bra17]:

W ss(v) = w ∈ T 1Ω | v+ = w+, πw ∈ Hv+(πv),

W os(v) = w ∈ T 1Ω | v+ = w+.

24 HARRISON BRAY

where Hv+(πv) is the globally defined horosphere through πv at v+ (Corol-

lary 3.6). Since Γ, hence dΓ, acts by isometries on T 1Ω, W ss(v) projects toW ss(v) in T 1M for any lift v of v. Choose lifts v, w in T 1Ω with endpoints

v+, w− smooth extremal points. Then there exists a u ∈ W ss(v) such thatu− = w− as long as w− 6= v+, which is guaranteed by the assumption that

v 6= −w. Since u− = w−, u ∈ W ou(w). Project u to T 1M and we have thedesired u ∈W ss(v) ∩W ou(w).

6.2.1. The Hopf argument. We first establish or recall basic facts which setup the ergodicity proof. Let f : T 1M → R be integrable. Then the forwardBirkhoff averages of f for ϕ are

(6.2) f+(v) = limt→+∞

1

t

∫ t

0f ϕs(v) ds.

By the Birkhoff ergodic theorem, f+ exists for µBM -almost every v ∈ T 1M(cf. [KH95, Theorem 4.1.2]). Similarly, let f− be the Birkhoff averages forϕ−1. The following lemma is straightforward to verify by compactness ofT 1M :

Lemma 6.3. Forward Birkhoff averages of continuous functions are con-stant on strong stable leaves of regular vectors and backward Birkhoff aver-ages as constant on strong unstable leaves.

Proof. Suppose f is continuous, hence uniformly continuous on the compactT 1M and thus L1-integrable. If v is forward regular, then for all w ∈W ss(v),lim

t→+∞|f(ϕt(v)− f(ϕtw)| = 0, so f+(v) = f+(w):∣∣∣∣ lim

t→+∞

1

t

∫ t

0f ϕs(v) ds− lim

t→+∞

1

t

∫ t

0f ϕs(w) ds

∣∣∣∣≤ lim

t→+∞

1

t

∫ t

0|f(ϕsv)− f(ϕsw)| ds = 0.

Since ϕ is invertible, f+ = f− µBM -almost everywhere (cf. [KH95, Propo-sition 4.1.3]).

Lemma 6.4. If f+ is constant µBM -almost everywhere for all continuousf , then every ϕt-invariant L1-integrable function is constant µBM -almosteverywhere.

Proof. Let L1(µBM ) be the L1 µBM -integrable functions on T 1M and L1(µBM , ϕ)the ϕt-invariant ones. By the Birkhoff Ergodic Theorem, Birkhoff averag-ing is a projection from L1(µBM ) to L1(µBM , ϕ). Suppose f+ is constantµBM -almost everywhere for all continuous f . Since continuous functionsare dense in L1, any g ∈ L1(µBM , ϕ) can be approximated by continuousfn. Let An be the sets of full measure such that f+

n is constant on An. Bycontinuity of the projection, f+

n → g+ = g implies g must be constant on⋂∞n=1An, also a set of full measure.


We make the arguments locally in the universal cover and conclude ergod-icity by transitivity of the flow on the quotient. We define strong unstableconditional measures as induced Patterson-Sullivan measure on strong un-stable leaves:

(6.3) µsuv (A) =

∫w∈A∩W su(v)

eδΓβw+ (x,πw) dµx(w+).

We can define the strong stable conditionals µssv on W ss(v) similarly. Notethat for w ∈W su(v), we have πw ∈ Hv−(πv) and w− = v−, hence βw+(x, πw)is constant over w ∈ W su(v) and the conditional measures will not dependthe point in a leaf of the foliation. We will say the strong unstable folia-tion is absolutely continuous if the associated strong stable conditionals areabsolutely continuous as measures.

Lemma 6.5. The strong unstable foliations are absolutely continuous forall regular points in T 1M .

Proof. For each t we have uniform contraction along flow lines:

(6.4)dϕt∗µ

suv

dµsuϕtv= e−δΓt

by the cocycle property of the Busemann function. Since δΓ > 0 this givesus absolute continuity of the strong unstable conditionals along flow lines.To do this computation, let A ⊂W su(ϕtv). Then∫Ae−δΓt dϕt∗µ

suv (w) =

∫Ae−δΓt dµsuv (ϕ−tw)

=

∫w∈A

e−δΓteδΓβw+ (x,π(ϕtw)) dµx(w+) (since w+ = (ϕtw)+)

=

∫w∈A

e−δΓteδΓ(βw+ (x,πw)+βw+ (πw,π(ϕtw))

)dµx(w+)

=

∫w∈A

e−δΓteδΓβw+ (x,πw)eδΓt dµx(w+) = µsuϕtv(A).

It remains to consider v, w regular vectors on the same strong stable leaf.This is because, given two unstable leaves W su(v1),W su(v2), there exists aw ∈ W ss(v1) ∩W ou(v2). Hence we can associate a vector in W su(v1) to avector in W su(w) and then pull back under the flow to a vector in W su(v2).To determine absolute continuity of the strong unstable conditionals, wedefine a measurable bijection h : W su(v)→W su(w) where, for u ∈W su(v),we let h(u) be the unique regular vector such that h(u)− = w−, h(u)+ = u+,and βw−(πw, πh(u)) = 0; in other words, h(u) ∈W su(w)∩W os(u) (Lemma6.2). Then we compute the density

ρv,w(u) :=dµsuwdh∗µsuv

(u) = e−δΓβu+ (πu,πh(u))

26 HARRISON BRAY

and see that 0 < ρv,w(u) < ∞ since δΓ > 0. To complete the computation,let A ⊂W su(w). Then∫

Aρv,w(u) dh∗µ

suv (u) =

∫u∈A

e−δΓβu+ (πu,πh(u))eδΓβu+ (x,πh(u)) dµx(u+)

=

∫u∈A

eδΓβu+ (x,πu) dµx(u+) = µw(A)

by the cocycle property of the Busemann function.

Remark 6.6. The final remark we make before proving ergodicity is that,locally, the µBM measure of a Borel set A agrees with the µBM -measure ofa lift of A, and we can exploit the Patterson-Sullivan product structure ofµBM on such sufficiently small neighborhoods (see Definition 6.1). We willrefer to this feature as the local product structure of µBM . Then it is clearthat, for such a small Borel measurable set N ⊂ T 1M which we identifywith a lift in T 1Ω, we have µBM (N) = 0 if and only if µsuv (N) = 0 forµBM -almost every v.

In the arguments below, we abuse notation and treat the measures asconditional measures on a small neighborhood in the universal cover.

Theorem 6.7. The Bowen-Margulis measure is ergodic for the geodesicflow.

Proof. Let f be a continuous function and Λq = v ∈ T 1M | f+(v) ≥ qfor some q ∈ Q such that µBM (Λq) > 0. Then µsuv (Λq) > 0 for µBM -almostevery v ∈ T 1Mreg by the local product structure of the Bowen-Margulis

measure and that T 1Mreg has full µBM -measure (Lemma 6.1). By Lemma6.5, the unstable conditionals are absolutely continuous for every pair ofregular vectors, so µsuv (Λq) > 0 for every regular vector v. Let G be theset of full µBM -measure on which f− = f+. Then G is also a set of fullµsuv -measure for µBM -almost every v. Then for almost every v, we haveµsuv (Λq) > 0 which implies µsuv (Λq ∩ G) > 0, and so there exists a w ∈Λq ∩ G ∩W su(v). Thus for all u ∈ G ∩W su(v), a full µsuv -measure set, wehave

f+(u) = f−(u) = f−(w) = f+(w) ≥ qsince f− is constant on strong unstable sets by ϕt-invariance of f− (Lemma6.3). Thus, u ∈ Λq and µsuv (Λq) = 1 for µBM -almost every v. This impliesµBM (Λq) = 1 by the local product structure of µBM . Since Q is dense inR we conclude f+ is constant on a set of full measure and by Lemma 6.4the proof is complete. This is because, we can let Λ+

q = Λq and Λ−q =

v ∈ T 1M | f+(v) ≤ q, and then repeat the above arguments to concludefor each q ∈ Q, either µBM (Λ+

q ) = 1 or µBM (Λ−q ) = 1, and not both if

f+ is not constant on a set of full measure. Then since f+ is a continuousprojection of a continuous function on a compact set, we have r− := infq ∈Q | µBM (Λ−q ) = 1 < supq ∈ Q | µBM (Λ+

q ) = 1 =: r+. But Q is dense


and Λ+r or Λ−r is full measure for r− < r < r+, contradicting either that r−

is an infimum or r+ is a supremum.

Remark 6.8. It suffices to prove ergodicity in a neighborhood by transitiv-ity [Bra17] and ϕt-invariance of µBM . Suppose A is a full measure subset ofN , an open neighborhood, such that f+ is constant on A. Then there existsv ∈ N with a dense orbit, so we can cover ϕ · v and hence T 1M by ϕ · N .Take a finite subcover ϕtnNkn=1 of T 1M . Then for each n = 1, · · · , k, byinvariance of µBM we have

µBM (N) = µBM (A ∩N) = µBM (ϕtnA ∩ ϕtnN) ≤ µBM (ϕtnN) = µBM (N)

(6.5)

implying ∪kn=1ϕtnA is a full measure set in ∪kn=1ϕ

tnN ⊃ T 1M on which f+

is constant.

Corollary 6.9. The µxBM are unique up to a constant.

Proof. By definition and Theorem 6.7 the µxBM and µyBM are invariant,equivalent, and ergodic, so we have the result.

Remark 6.10. An invariant measure µ for a flow ϕt : X → X where X isa probability space is mixing if for all measurable A,B ⊂ X,

µ(ϕ−t(A) ∩B)→ µ(A)µ(B) as t→∞This is equivalent to the following dual definition: for all f, g ∈ L2(X,µ),∫

f(ϕt(x)) · g(x) dµ(x)→∫f dµ ·

∫g dµ as t→∞

A generalization of the Hopf argument originally due to Babillot [Bab02,Theorem 2] for the geodesic flow of rank one manifolds yields mixing ofthe Bowen-Margulis measure as soon as noncommuting hyperbolic groupelements have noncommensurable translation lengths, which applies to theBenoist 3-manifolds [Bra17]. One is then able to apply the Hopf argumentas outlined in Lemma 6.3 and Theorem 6.7 for weak limits of f ϕt inL2(µBM ). This argument requires working with regular vectors, hence theneed for density of the hyperbolic length spectrum.

6.3. A measure of maximal entropy. In this subsection, we prove thatµBM has entropy δΓ ≥ htop(ϕ) and conclude that µBM is a measure ofmaximal entropy.

We will need some definitions from entropy theory. Let A be a finite mea-surable partition of T 1M . Then the entropy of a partitionA = A1, . . . , Amwith respect to a measure µ is

Hµ(A) =m∑i=1

−µ(Ai) log(µ(Ai)).

A refinement of a partition need only satisfy that all elements of the refine-ment are subsets of elements from the original partition. The least common

28 HARRISON BRAY

refinement of two partitions A,B is then A∨B := A∩B | A ∈ A, B ∈ B.The measure-theoretical entropy of µ with respect to the finite measureablepartition A = A1, . . . , Am, also know as the Kolmogorov-Sinai entropy, is

hµ(ϕ1,A) = limn→∞

1

nHµ(A(n)

ϕ )

where

A(n)ϕ =

n−1∨i=0

ϕ−iA = A ∨ ϕ−1A ∨ · · · ∨ ϕ−(n−1)A

is the partition consisting of all intersections⋂n−1i=0 ϕ

−iAji over all possiblej1, . . . , jn−1 ⊂ 1, . . . ,m. Then the measure-theoretic entropy of the pair(ϕ1, µ) is

hµ(ϕ1) = supAhµ(ϕ1,A).

and the entropy of µ for the geodesic flow ϕt is hµ := hµ(ϕ1). By work in[Bra17], ϕt is entropy-expansive with expansivity constant ε > 0. Then by[Bow72, Theorem 3.5], hµ = hµ(ϕ,A) for diam(A) < ε. By the variationalprinciple, we have that

htop(ϕ) = supµ ϕt-inv

hµ(ϕ),

and a measure µ which realizes the supremum is a measure of maximalentropy.

We first need:

Lemma 6.11. There exists some a > 0 such that

µBM (α) ≤ e−δΓna

for all α ∈ A(n)ϕ .

Proof. Let A be a partition with diameter less than ε, the h-expansivity

constant for hBM . Note that for all v ∈ α ∈ A(n)ϕ ,

α ⊂n−1⋂k=0

ϕ−kB(ϕkv, ε).

Let v, w ∈ α be regular vectors Then dΩ(`v(t), `w(t)) ≤ ε for all t ∈ [0, n].Choose p ∈ Ω to be the reference point for µBM . Since ε is sufficiently small,we can lift v, w to v, w ∈ Ω such that v, w, p are all in the same fundamen-tal domain with diameter diam(M). Denote by cξ,x(t) the projective linethrough any two ξ ∈ ∂ Ω and x parameterized at unit Hilbert speed suchthat cξ,x(n) = x. Choose ξ = w− and x = `v(n). Then

dΩ(cξ,x(0), p) ≤ dΩ(cξ,x(0), πv) + dΩ(πv, p)

≤ dΩ(cξ,x(0), πw) + dΩ(πw, πv) + dΩ(πv, p)

≤ ε+ ε+ diam(M) = 2ε+ diam(M).


Note that dΩ(cξ,x(0), `w(0)) < dΩ(cξ,x(n), `w(n)) ≤ ε because ξ = w− andw is a regular vector. Also, clearly w+ ∈ Oε(ξ, x). Recall the projectionp∞(v) = (v−, v+) ∈ ∂ Ω2. The above inequalities imply that

p∞(α) ⊂⋃

η∈O2ε+diamM (x,p)

η × Oε(η, x).

Now, for all η ∈ O2ε+diamM (x, p), choose q ∈ cη,x ∩ BΩ(p, 2ε + diamM).Then

dΩ(x, πv) ≤ dΩ(p, x) + dΩ(p, πv)

≤ dΩ(q, x) + dΩ(p, q) + dΩ(p, πv)

implying

dΩ(q, x) ≥ dΩ(x, πv)− dΩ(p, q)− dΩ(p, πv) ≥ n− 2ε− 2 diamM

because q ∈ BΩ(p, 2ε+ diamM) and p, πv ∈M .Now, since −dΩ(p, q) ≤ βη(p, q) for all p, q and for µBM -almost every η

(for which the Busemann function is well-defined),

µp(Oε(η, x)) = e−δΓβη(p,q)µq(Oε(η, x)) transformation rule

≤ eδΓdΩ(p,q)µq(Oε(η, x))

≤ eδΓ(2ε+diamM)be−δΓdΩ(q,x) Lemma 5.1

≤ eδΓ(2ε+diamM)be−δΓ(n−2ε−2 diamM) = be−δΓn.

Note that p∞(α) ⊂ O2ε+diamM (x, p) × Oε(η, x) anddµp

dµp⊗dµp (v−, v+) =

e2δΓ〈v−,v+〉p is bounded above by some C > 0 since 2〈v−, v+〉p = βv−(πv, p)+βv+(πv, p) ≤ 2dΩ(πv, p) ≤ diamM by choice of p in the same fundamentaldomain as πv. Then

µBM (α) ≤∫α

lengthΩ(πα ∩ `v) dµp(v)

≤∫α

diamT 1M dµp(v)

≤ C diamT 1M µp(O2ε+diamM (x, p))µp(Oε(η, x))

≤ C diamT 1M µp(∂Ω)µp(Oε(η, x))

≤ ae−δΓn.

Theorem 6.12. The Bowen-Margulis measure is a measure of maximalentropy.

30 HARRISON BRAY

Proof. By the definitions and Lemma 6.11,

HµBM (A(n)ϕ ) =

∑α∈A(n)

ϕ

−µBM (α) logµBM (α)

≥∑

α∈A(n)ϕ

−µBM (α) log(e−δΓna)

= (δΓn− log a)∑

α∈A(n)ϕ

µBM (α)

= δΓn− log a

because A(n)ϕ is a partition and we normalized µBM (T 1M) = 1. By [Bra17]

we have δΓ ≥ htop. Then by the variational principle

htop ≥ hµBM = limn→∞

1

nHµBM (A(n)

ϕ ) ≥ limn→∞

1

n(δΓn− log a) = δΓ ≥ htop.

Corollary 6.13. The critical exponent of Γ acting on Ω is equal to thetopological entropy of the geodesic flow on T 1M .

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Department of Mathematics, University of Michigan, Ann Arbor [email protected]



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