Entropy Rigidity and Hilbert Volume

For a closed, strictly convex projective manifold of dimension $n\geq 3$ that admits a hyperbolic structure, we show that the ratio of Hilbert volume to hyperbolic volume is bounded below by a constant that depends only on dimension. We also show that for such spaces, if topological entropy of the geodesic flow goes to zero, the volume must go to infinity. These results follow from adapting Besson--Courtois--Gallot's entropy rigidity result to Hilbert geometries.


Introduction
A(strictly) convex real projective orbifold is a quotient Ω/Γ, where Ω is an open, properly (strictly) convex subset of RP n and Γ < PGL(n + 1, R) is a discrete subgroup of projective transformations that preserves Ω. A subset Ω ⊂ RP n is proper if it is bounded in some affine patch; convex if its intersection with any projective line is connected; and strictly convex if, moreover, its topological boundary in an affine patch does not contain an open line segment. An orbifold is a manifold if Γ contains no elements of finite order.
Any properly convex set Ω admits a complete Finsler metric called the Hilbert metric. This is the Klein model of the hyperbolic metric when Ω is the interior of a round ball. Hence, the first examples of projective orbifolds are hyperbolic orbifolds. By Mostow rigidity, a hyperbolic structure on a closed manifold of dimension greater than or equal to 3 is unique, up to isometry. However, the dimension of the deformation space of strictly convex projective structures on some closed manifolds can be large. For instance, in dimension two, there is a 16g − 16 dimensional deformation space of strictly convex real projective structures on a closed surface of genus g [Gol90]. This space need not contain a hyperbolic point -there exist closed strictly convex manifolds that do not admit a hyperbolic structure.
It is of interest to characterize a hyperbolic structure, when it exists, among all strictly convex projective structures a manifold admits. This article addresses that question in terms of volume and entropy and derives a pair of results on the geometry and dynamics of these spaces. The Finsler structure on Ω provides a natural volume form on Y referred to as Hilbert volume. Let Vol(Y, g 0 ) and Vol(Y, F Ω ) denote the hyperbolic volume and the Hilbert volume, respectively, of a closed manifold Y . Our main result on Hilbert volume is given below. depending only on dimension, such that Vol(Y, F Ω ) Vol(Y, g 0 ) ≥ D.
A consequence of the Margulis lemma [BGS85] is that there exists a positive lower bound for the volume of a hyperbolic n-manifold, for each n. This gives the following corollary.
Corollary 1.2. Let Y be a closed strictly convex projective manifold of dimension n ≥ 3 which admits a hyperbolic structure. Then, there exists a constant e > 0, depending only on dimension, such that Results bounding Hilbert area in dimension 2 can be found in [CVV04] and [AC16].
It is important to underscore that there are strictly convex real projective manifolds which do not admit a hyperbolic metric. Coxeter group examples exist in dimension four [Ben06] and Gromov-Thurston examples exist for each dimension greater than three [Kap07]. However, our theorem holds in several contexts.
Every closed strictly convex projective 3-manifold admits a hyperbolic structure; by Benoist's dichtomoy [Ben04, Theorem 1.1], strict convexity of Ω is equivalent to Gromov hyperbolicity of Γ which implies the quotient admits a hyperbolic structure by geometrization. There are also examples of flexible closed hyperbolic 3-manifolds that have nontrivial projective deformations [CLT07]. In the same work, Cooper-Long-Thistlethwaite conjecture that all hyperbolic 3-manifolds are virtually flexible. Furthermore, any hyperbolic n-manifold with a totally geodesic submanifold admits the projective bending deformation of Thurston [JM87].
Our second main result concerns the dynamics of the geodesic flow for Y = Ω/Γ. Definition 1.3. Let g be a (Finsler or Riemannian) metric on a compact manifold Y . Let y ∈Ỹ be any point in the universal cover of Y and B g (R, y) the radius R ball around y with respect to g. The volume growth entropy of g is, For a nonpositively curved Riemannian metric, the volume growth entropy is equal to the topological entropy for the geodesic flow [Man79]. This result can be generalized to non-Riemannian settings under some mild conditions mimicking nonpositive curvature (see [Leu06]). Verification of these conditions in the present setting can be found in [Cra09,§8].
We prove the following relationship between this dynamical quantity and the Hilbert volume: Theorem 1.4. Let Y t = Ω t /Γ t be a family of strictly convex real projective structures on a manifold of dimension at least 2 which supports a hyperbolic metric.
Remark 1.5. Examples of manifolds in dimensions 2, 3, and 4, for which the entropy of strictly convex projective structures goes to zero are given in [Nie15]. In these cases, volume is known to grow without bound as entropy decreases. Moreover, in dimension 2, Zhang proves the entropy of strictly convex real projective structures on any closed surface can be made arbitrarily small [Zha15]. Though it is plausible the work of Zhang shows the volume will simultaneously diverge to infinity, it is not immediate. Theorem 1.4 states that this phenomenon will hold generally in any dimension with a short proof.
Theorem 1.1 and the n ≥ 3 case of Theorem 1.4 are consequences of an entropy rigidity theorem. This theorem follows a line of results beginning with the celebrated work of Besson, Courtois and Gallot in [BCG95,BCG96]. They prove the following theorem using the 'barycenter method' -the technique we will also use.
Definition 1.6. The normalized entropy functional of (Y n , g) is the quantity ent(Y, g) = h(g) n Vol(Y, g).
Theorem 1.7 (see Théorème Principal [BCG95]). If (Y, g) is a compact, oriented, Riemannian manifold of dimension n ≥ 3 homotopy equivalent to a negatively curved locally symmetric space (X, g 0 ), then with equality if and only if (Y, g) and (X, g 0 ) are isometric, up to a homothety.
Remark 1.8. Theorem 1.7 has a number of important consequences, including a proof of Mostow's rigidity theorem (see [BCG95,BCG96]). The barycenter method has been employed many times, including the work of Connell and Farb on higherrank symmetric spaces [CF03a,CF03b]. See [CF03c] for a survey.
Our paper closely follows the work of Boland and Newberger in [BN01], which adapts the Besson-Courtois-Gallot result to compact Finsler manifolds of negative flag curvature. For a C 2 -Finsler metric F on a manifold, Boland and Newberger define the eccentricity factor, denoted by N (F ). See Section 3.1.4 for the definiton, but note here that N (F ) is equal to 1 when F is Riemannian and is strictly greater than 1 otherwise. (The terminology 'eccentricity factor' is coined in [CF03c].) Their Finsler extension of Theorem 1.7 is as follows.
Theorem 1.9 (see Main Theorem [BN01]). Let (M, F ) be a compact, reversible, C 2 -Finsler manifold of negative flag curvature and dimension ≥ 3 with the same homotopy type as the compact, negatively curved, locally symmetric space (X, g 0 ).
(ii) Equality holds above if and only if (M, F ) is Riemannian and homothetic to (X, g 0 ).
We extend this result to the Hilbert geometry setting: Theorem 1.10. Let (Y, F Ω ) be a compact strictly convex real projective manifold of dimension ≥ 3. Let (X, g 0 ) be a hyperbolic structure on the same underlying manifold. Then there is a number N (F Ω ) ≥ 1, such that with equality if and only if (Y, F Ω ) is isometric to (X, g 0 ).
Our modifications to the work of Boland and Newberger revolve around the following point: If F Ω is a Finsler metric defined on a strictly convex domain Ω ⊂ RP n , one can verify (see Section 2.1) that F Ω is C 2 if and only if ∂Ω, the boundary of Ω, is C 2 . If ∂Ω is C 2 , then ∂Ω is in fact an ellipsoid (originally due to [Ben60,Theorem C], see also [Cra13, Section 3.2] for an expository note in English). Hence, any corresponding Y = Ω/Γ with the induced metric is hyperbolic. If F Ω is not C 2 , then ∂Ω is only C 1+α for some 0 < α < 1 [Ben04, Theorem 1.3]. The failure of C 2 regularity in general leads us to substitute the Blaschke metric, a particular Riemannian metric associated to Ω, for the family of reference metrics used in [BN01] at a key point in the proof. The C 2 rigidity for Hilbert geometries also allows us to reach the rigidity conclusion of Theorem 1.10 with a shorter argument.
Remark 1.11. It is conjectured in [BCG96] that Theorem 1.7 remains true in the class of Finsler metrics. This is equivalent to a restatement of Theorem 1.9 without the presence of the N (F ) factor. Following suite, we make the conjecture that Theorem 1.10 is valid without the N (F Ω ) term.
Outline of the paper. Section 2 provides the necessary background information on the Hilbert metric and Hilbert volume.
Section 3 contains the proof of Theorem 1.10. The 'natural map' between the Hilbert geometry and its hyperbolic counterpart is constructed in Section 3.1. The Jacobian of the natural map is bounded, and the inequality statement of Theorem 1.10 is deduced in Section 3.2. Finally, the rigidity statement of Theorem 1.10 is proved in Section 3.3.
Section 4 recalls basic properties of the Blaschke metric for Hilbert geometries, in particular a relationship between the Hilbert and Blaschke metrics due to Benoist and Hulin [BH13]. We then prove Theorem 1.1 and the n ≥ 3 case of Theorem 1.4. We conclude by proving Theorem 1.4 for 2-dimensional Hilbert geometries. Since our previous results require dimension greater than two, this argument uses the well-known result of Katok on entropy rigidity for surfaces [Kat82], as well as the particularly nice behavior of the Blaschke metric in dimension 2 due again to Benoist-Hulin [BH14]. Ralf Spatzier for many useful discussions. The second author was supported in part by NSF RTG grant 1045119.

Hilbert Geometry
This section provides the basic definitions for Hilbert geometry. For more details the reader may consult [BK53] and [CLT15].
2.1. Definition of a Hilbert geometry. Let Ω be a properly convex domain of RP n , as defined in the Introduction. Let ∂Ω denote the boundary of Ω. The Hilbert metric d Ω on Ω is given by q are the intersection points, in the chosen affine patch, of ∂Ω with the projective line containing x and y, and · denotes Euclidean distance.
Elements of PGL(n + 1, R) that preserve Ω are isometries of the Hilbert metric, since projective transformations preserve cross-ratio. If Ω is strictly convex, these elements constitute the full group of isometries, denoted by PGL(Ω) [dlH93]. A Hilbert geometry is a triple (Ω, d Ω , PGL(Ω)).
The Hilbert geometry where Ω is the interior of an ellipsoid is the Klein model of hyperbolic n-space, H n . The factor of 1/2 in the definition of d Ω ensures constant curvature −1.
The Hilbert metric on a properly convex domain Ω induces the Finsler structure where v − , v + are the points of intersection of the projective line through y in the direction of v with ∂Ω. Note that it is immediate from the definition that C rregularity of F Ω is equivalent to C r -regularity of ∂Ω.
2.2. Hilbert Volume. A group Γ < PGL(Ω) is discrete in PGL(n + 1, R) if and only if Γ acts properly discontinuously on Ω [CLT15]. Therefore, for such a Γ the projective and Finsler structures of Ω descend to the quotient orbifold Y = Ω/Γ.
The Finsler structure on Y provides for a natural definition of volume. Fix any Riemannian metric g on Y and let B g (1, y) and B FΩ (1, y) denote balls of radius 1 in T y Y with respect to g and F Ω , respectively. Then for any y ∈ Y , the Finsler volume element is It is easy to check that the definition is independent of the choice of g. The volume of Y with respect to Finsler volume will be referred to as the Hilbert volume of Y .

Proof of Theorem 1.10
In this section we prove Theorem 1.10. As mentioned in the Introduction, the argument closely follows Boland-Newberger's adaptation of the Besson-Courtois-Gallot result to the Finsler setting. For completeness, we present the argument here, highlighting the modifications we make and referring the reader to the original papers for the details which are unchanged.

Busemann functions.
Let Ω be a properly convex domain equipped with Hilbert metric d Ω . For p, z ∈ Ω the Busemann function B Ω p,z : Ω → R is defined by where z → ξ along any path. Geometrically, B Ω p,ξ (q) is the signed distance between horospheres based at ξ passing through p and q. If p is fixed as a basepoint, then B Ω p,ξ (q) can be viewed as a family of functions mapping Ω to R that is parametrized by elements in ∂Ω. With this notation, Busemann functions onX will be denoted by B E p,ξ .

Patterson-Sullivan measures.
The visual boundary of a convex domain Ω is the space of all geodesic rays based at a point modulo bounded equivalence. If Ω is strictly convex with C 1 -boundary, the visual boundary of Ω coincides with ∂Ω. Suppose, furthermore, that Ω admits a cocompact action by a discrete group of projective transformations. In this case we can define the Patterson-Sullivan density, a family of measures {µ p } p∈Ω on the boundary of Ω. The defining properties of the Patterson-Sullivan density are the following: where h is the topological entropy of the Hilbert geodesic flow or, equivalently in our setting, the volume entropy of (Ω, F Ω ).
The construction of the Patterson-Sullivan measures originates with the work of Patterson for Fuchsian groups [Pat76] and Sullivan for convex cocompact actions on spaces of constant negative curvature [Sul79]. The concept has been extended to many settings, the most relevant being compact negatively curved manifolds and CAT(−1) metric spaces [Kai90,Rob03]. For the familiar reader, we remark that although non-Riemannian Hilbert geometries are not negatively curved in the classical sense and are not even CAT (0) It is a straightforward exercise to check that the barycenter of λ is Γ-equivariant, that is, that bar(γ * λ) = γ · bar(λ) for all γ ∈ Γ.
3.1.4. The natural map. Let f : ∂Ỹ → ∂X be the Γ-equivariant homeomorphism induced by the identification of fundamental groups. A natural Γ-equivariant map fromỸ toX is constructed by associating to each y ∈Ỹ the barycenter inX of the push-forward of the Patterson-Sullivan measure µ y under the map f . That is, Φ :Ỹ →X is given byΦ (y) = bar(f * µ y ).
The Γ-equivariance ofΦ follows from the Γ-equivariance of f and bar, and so it descends to a map Φ : Y → X. This 'natural map' is at the heart of the Besson-Courtois-Gallot approach to entropy rigidity. Theorem 1.10 will be proved by bounding the Jacobian ofΦ.
Remark 3.1. Boland and Newberger assume their Finsler manifold has negative flag curvature. This ensures thatỸ is diffeomorphic to R n . In our setting,Ỹ is equal to Ω, a bounded domain in projective space, by assumption. Therefore,Φ(y) satisfies Changing variables by setting β = f −1 (α), and by the transformation rule, we have for a fixed p ∈Ỹ and for all y ∈Ỹ , This expression allows us to verify thatΦ is differentiable. Let Clearly v, and therefore F , is smooth in its first variable. Since ∂Ω is C 1+α , B Ω p,β (y), and therefore F , is differentiable in y (see [Ben04,§3.24]). F implicitly definesΦ by F (Φ(y), y) = 0 and the Implicit Function Theorem implies thatΦ is differentiable.
Taking the differential of this expression with respect to y we have for all w ∈ TΦ (y)X and all u ∈ T yỸ .
Let K and H be the endomorphisms on TΦ (y)X defined by and The reader can verify (or see the nice exposition in [Fer96]) that H and K are symmetric, tr(H) = 1, and (sinceX is hyperbolic space) that K = I − H. Then from equation 3.1 and an application of Cauchy-Schwarz, we have Since the goal is to bound |Jac(Φ)| from above, we assume without loss of generality that D yΦ has full rank. Fix a basis {e i } which is orthonormal with respect to the hyperbolic metric and in which the matrix for H is diagonal. Fix a Riemannian metric g r on Y ; we will specify a choice for this metric in Section 4.
Let v ′ i = (K • D yΦ ) −1 (e i ) and apply Gram-Schmidt to get a basis {v i } for T yỸ which is orthogonal with respect to g r . Then is upper-triangular with respect to this basis.
We have where the determinant is computed for the matrix with respect to the bases {e i } and {v i }. Since {v i } is orthonormal for g r , span{v i } has g r -volume 1. Hence, by the definition of dF Ω , Vol FΩ (span{v i }) = Vol gr (B gr (1, y)) Vol gr (B FΩ (1, y)) := ρ(y, g r ).
Since |det(K • D yΦ )| = | Jac(Φ)(y)| · ρ(y, g r ) · | det K|, we can use equation (3.2), the fact that K • D yΦ is upper-triangular, and the fact that H is diagonal to compute as follows: (3.3) At the last step we use the fact that the arithmetic mean bounds the geometric mean above. Since the {v i } are orthonormal for g r , Combining this with equation (3.3), and noting that for an F Ω -unit vector likev, D y B Ω p,β (v) 2 ≤ 1, we have proven If F Ω were Riemannian, we could set g r = F Ω and the third term above would be equal to 1. As F Ω may not be Riemannian, Boland-Newberger make the following definition: Definition 3.2 (compare with [BN01, p. 3]). Let (Y, F ) be any Finsler manifold, and let g r be a Riemannian metric on Y . Then let Vol gr (B gr (1, y)) .
Note that Vol gr (B gr (1, y)) is a constant depending only on the dimension of Y . It is also easy to check that N (F ) is unchanged by scaling the Riemannian metric g r . The following lemma is a straight-forward exercise: Returning to our bounds in equation (3.5) and using Definition 3.2, for all y ∈ Y , Remark 3.4. A careful reading of [BN01] shows that, rather than a single Riemannian metric g r , one can run the argument above using a family of Riemannian metrics {g u } parametrized by F Ω -unit tangent vectors u. (The definition of N (F Ω ) is adjusted accordingly.) We do not exploit this additional flexibility here. Boland-Newberger use {g u } defined by This 'direction-dependent' inner product on T yỸ is a standard tool in Finsler geometry (see [BCS00, §1.2 B]), but it requires at least C 2 regularity of F 2 , which (as noted in the Introduction) we do not have unless (Ỹ , d Ω ) immediately reduces to the hyperbolic case [Ben60]. A key step in our argument (see Section 4) is finding a good replacement for {g u }.
The following lemma is where n ≥ 3 is required for the proof of Theorem 1.10. Its proof is an optimization exercise. We remark that volume entropy for the hyperbolic metric, h(g 0 ), is equal to n − 1. Applying Lemma 3.5 to equation (3.6) and then integrating the result over y ∈ Y gives which is equivalent to the inequality statement in Theorem 1.10.
3.3. Rigidity. We now turn to the rigidity part of Theorem 1.10. Suppose that equality holds in (3.7). This forces the equality case of Lemma 3.5, i.e. K = h(g 0 ) n I and H = 1 n I.

Then equation (3.2) gives
for all u ∈ T yỸ and w ∈ TΦ (y)X . Solving for | DΦ(u), w | and taking the supremum over all w ∈ T 1 Φ(y)X gives for all u ∈ TỸ . Let L = (D yΦ ) * • (D yΦ ), where A * denotes the transpose (with respect to g r and g 0 ) of a linear map A : T yỸ → TΦ (y)X . Fix a g r -orthonormal basis {u i }. We then calculate: where we apply the same reasoning following equation (3.4) to the last line. Equality in equation (3.5) together with K = h(g0) n I and H = 1 (3.9) using (3.8). Equality must hold throughout, in particular when we invoke (det L) 1/n ≤ tr(L)/n. Equality implies that Recalling the definition of L, this implies that for all y, D yΦ : (T yỸ , F Ω ) → (TΦ (y)X , g 0 ) is an isometry composed with a homothety.
We can now conclude the proof of Theorem 1.10 with a short argument, using special rigidity properties of Hilbert geometries. That D yΦ is a homothety implies that S FΩ (1, y) is an ellipsoid; in particular it is smooth. Using the definition of F Ω , this implies that ∂Ω is smooth. Then by [Ben60] The eccentricity factor of Definition 3.2 for the Hilbert metric with respect to the Blaschke metric is .

It follows that
since, by Theorem 4.1, B F H Ω (1, y) ⊂ B F B Ω (K n , y) and S F B Ω (1, y) ⊂ B F H Ω (K n , y) for all y ∈ Ω.
Theorem 1.4 immediately follows from equation (4.1) and Theorem 1.10 for dimensions n ≥ 3 since h(g 0 ) Vol(X, g 0 ) is constant. We treat the n = 2 case separately in the next section.

Entropy and volume in dimension 2.
Theorem 4.2. Let Y t = Ω t /Γ t be a family of convex projective manifolds homeomorphic to a closed surface Σ of negative Euler characteristic. Then In this section, we write Vol d * Ω for the volumes on Ω induced by the two metrics F * Ω , and we write Vol F * Ω for the volumes these metrics induce on the tangent space at a point. Proof. Since the Blaschke metric is Riemannian we may use it for the definition of the Finsler volume. Then By Theorem 4.1 and basic properties of any volume form on T x Ω, Ω (1, x)). Equation (4.2) follows with V n = K n n . The last piece needed to prove Theorem 4.2 is another result of Benoist and Hulin: Lemma 4.4 ([BH14, Proposition 3.3]). The curvature of the Blaschke metric on a properly convex Ω ⊂ RP 2 is bounded between −1 and 0.
By [Man79], this implies that topological entropy of the Blaschke geodesic flow of the quotient manifold is equal to the volume growth entropy of the Blascke metric.