Random k-surfaces

François LABOURIE ^{∗}

September 20, 2000

1 Introduction

We associated in [1] a compact space laminated by 2-dimensional leaves, to every compact 3-manifold N with curvature less than -1. Considered as a “dynamical system”, its properties generalise those of the geodesic flow.

In this introduction, I will just sketch the construction of this space, being more precise in section 2. Let k ∈]0, 1[. A k-surface is an immersed surface in N, such that the product of the principal curvatures is k. If N has constant curvature K, A k-surface has curvature K + k. Analytically, k-surfaces are described by elliptic equations.

When dealing with ordinary differential solutions, one is lead to introduce the phase space consisting of pairs (γ, x) where γ is a trajectory solution of the O.D.E, and x is a point on γ. We recover the dynamical picture by moving x along γ.

We can mimick this construction in our situation in which a P.D.E replaces the O.D.E. More precisely, we can consider the space of pairs (Σ, x) where Σ is a k-surface, and x a point of Σ.

We proved in [4] that this construction actually makes sense. More precisely, we proved the space we just described can be compactified by a space, called the space of k-surfaces. Furthermore, the boundary is finite dimensional and related in a simple way to the geodesic flow. This space, which we denote by N, is laminated by 2-dimensional leaves, in particular by those obtained by moving x along a k-surface Σ. A lamination means that the space has a local product structure.

∗L’auteur remercie l’Institut Universitaire de France.

The purpose of this article is to study transversal measures, ergodic and of full support on this space of k-surfaces. At the present stage, let’s just to notice that since many leaves are hyperbolic (cf theorem 2.2.1), one cannot produce transversal measures by Plante’s argument. Our strategy will be to “code” by a combinatorial model on which it will be easier to build transversal measures.

This article is organised as follows.

2. The space of all k-surfaces. We describe more precisely the space of k-surfaces, we are going to work with.

3. Transversal measures. We present our main result, theorem 3.2.1, discuss other constructions and questions, and sketch the main construction.

4. A combinatorial model. In this section, we explain a combinatorial construction. Starting from a configuration data, we consider “configuration spaces”. These are spaces of mappings from ℚℙ^{1} to a space W. We produce invariant and ergodic measures under the action of P SL(2, ℤ) by left composition.

5. Configuration data and the boundary at infinity of a hyperbolic 3-manifold. We exhibit a combinatorial model associated to hyperbolic manifolds. In this context, the previous W is going to be ℂℙ^{1}.

6. Convex surfaces and configuration data. We prove here that the combinatorial model constructed in the previous section actually codes for the space of k-surfaces.

7. Conclusion. We summarize our constructions and prove our main result, theorem 3.2.1.

I would like to thank W. Goldman for references about ℂℙ^{1}-structures, and R. Kenyon for discussions.

2 The space of all k-surfaces

The aim of this section is to present in a little more details the space “of k-surfaces” that we are going to work with.

Let N be a compact 3-manifold with curvature less than −1. Let k ∈]0, 1[ be a real number. All definitions and results are expanded in [1].

2.1 k-surfaces, tubes

If S is an immersed surface in N, it carries several natural metrics. By definition, the u-metric is the metric induced from the immersion in the unit bundle given by the Gauss map. We shall say a surface is u-complete if the u-metric is complete.

A k-surface is an immersed u-complete connected surface such that the determinant of the shape operator (i.e. the product of the principal curvatures) is constant and equal to k.

We described in [1] various ways to construct k-surfaces. In section 6.3, we summarize results of [1] which allow us to obtain k-surfaces as solutions of an asymptotic Plateau problem.

Since k-surfaces are solutions of an elliptic problem, the germ of a ksurface determines the k-surface. It follows that a k-surface is determined by its image, up to coverings. More precisely, for every k-surface S immersed by f in N, there exists a unique k-surface Σ, the representative of S, immersed by ϕ, such that for every k-surface S̄ immersed by f̄ satisfying f(S) = f̄(S̄), there exists a covering π : S̄ → Σ such that f̄ = ϕ ∘ π.

By a slight abuse of language, the expression “k-surface” will generally mean “representative of a k-surface”.

The tube of a geodesic is the set of normal vectors to this geodesic. It is a 2-dimensional submanifold of the unit bundle.

2.2 The space of k-surfaces

The space of k-surfaces is the space of pairs (Σ, x) where x ∈ Σ and Σ is either the representative of a k-surface or a tube. We denote it by N. It inherits a topology coming from the topology of pointed immersed 2-manifolds in the unit bundle (cf section 2.3 of [1]). Each k-surface (or tube) S_{0} determines a leaf L_{S_{0}} defined by

L_{S_{0}} = {(S_{0}, x)/x ∈ S_{0}}.

We proved in [4] that N is compact. Furthermore, the partition of N into leaves is a lamination, i.e. admits a local product structure. Notice that N has two parts:

(1) a dense set which turns out to be infinite dimensional, and which truely consists of k-surfaces,

(2) a “boundary” consisting of the reunion of tubes, closed, finite dimensional, and which is a S^{1} fiber bundle over the geodesic flow.

Therefore, in some sense, N is an extension of the geodesic flow. To eneforce this analogy, one should also notice taht the 1-dimensional analogue, namely the space of curves of curvature k in a hyperbolic surface is precisely the geodesic flow.

The main theorem of [1] which we quote now shows that N, as a dynamical system, enjoys the chaotic properties of the geodesic flow:

Theorem 2.2.1 Let k ∈]0, 1[. Let N be a compact 3-manifold. Let h be a riemannian metric on N with curvature less than −1. Let N_{h} be the space of k-surface of N. Then

(i) a generic leaf of N_{h} is dense,

(ii) for every positive number g, the union of compact leaves of N_{h} of genus greater than g is dense,

(iii) if h̄ is close to h, then there exists a homeomorphism from N_{h} to Nh̄ sending leaves to leaves.

This last property will be called the stability property.

To conclude this presentation, we show yet another point of view on this space, which will make it belong to a family of more familiar spaces. Assume N has constant curvature, and, for just a moment, let’s vary k between 0 and ∞, the range for which the associated P.D.E is elliptic.

For k > 1, k-surfaces are geodesic spheres, therefore the space of ksurfaces is just the unit bundle, foliated by unit spheres.

For k = 1, k-surfaces are either horospheres, or equidistant surfaces to a geodesic. The space of 1-surfaces is hence described the following way: first we take the S^{1}-bundle over the unit bundle, where the fiber over uis the set of unit vectors othogonal u. This space is foliated by 2-dimensional leaves which are inverse images of geodesics. Then, we take the product of this space by [0, ∞[. The number r ∈ [0, ∞[ represents the distance to the geodesic. We complete now the space by adding horospheres, when r goes to infinity.

Our construction allows us to continue deforming k below 1. However passing through this barrier, the space of k-surfaces undergoes dramatic change; in particular, it becomes infinite dimensional and “chaotic” as we just said.

3 Transversal measures

Let N be a compact 3-manifold with curvature less than −1. Let k ∈]0, 1[ be a real number. Let N be the space of k-surfaces of N.

3.1 First examples

Let’s first show some simple examples of natural transversal measures on N. The first three ones are ergodic. They all come from the existence of natural finite dimensional subspaces in N.

- Dirac measures supported by closed leaves. By theorem 2.2.1 (ii), there are plenty of them.

- Ergodic measures for the geodesic flow. Indeed, ergodic and invariant measures for the geodesic flow give rise to transversal measures on the space of tubes, hence on the space of k-surfaces.

- Haar measures for totally geodesic planes. Assume N has constant curvature. Then, the space of oriented totally geodesic planes carries a transverse invariant measure. Indeed, the Haar measure for SL(2, ℂ)/π_{1}(N) is invariant under the SL(2, ℝ) action. But every oriented totally geodesic plane gives rise to a k-surface, namely the equidistant one to the geodesic plane. This way, we can construct an ergodic transversal measure on N, when N has constant curvature. Its’s support is finite dimensional.

- Measures on spaces of ramified coverings. We sketch briefly here a construction yielding transversal, but non ergodic, measures on N. Let ∂_{∞}M be the boundary at infinity of the universal cover M of N. Let Σ be an oriented surface of genus g. Let π be a a topological ramified covering, defined up to homeomorphism of the source, of Σ into ∂_{∞}M. Let S_{π} be the set of singular points of π and s_{π} its cardinal. Let S be a set of extra marked points of cardinal s. Assume 2g + s_{π} + s. One can show following the ideas of the proof of theorem 7.3.3 of [1] that such a ramified covering can be represented by a k-surface. More precisely, there exists a solution to the asymptotic Plateau problem (as decribed in paragraph 6.3) represented by (π, Σ \ (S_{π} ∪ S)). Let now [π] be the space of ramified coverings equivalent up to homeomorphisms of the target, to π. The group π_{1}(N) acts properly on [π], and explicit invariant measures can be obtained using equivariant family of measures (cf section 5.1.1) and configuration spaces of finite points. Since [π]/π_{1}(N) is a space of leaves of N, this yields transversal measures on this latter space.

None of these examples have full support, and they all have finite dimensional support. So far, apart from these and the construction I will present in this article, I do not know of other examples of transversal measures easy to construct.

3.2 Main theorem

We now state our main theorem

Theorem 3.2.1 Let N be a compact 3-manifold with curvature less than −1. Assume the metric on N can be deformed, through negatively curved metrics, to a constant curvature one. Then the space of k-surfaces admits infinitely many mutually singular, ergodic tranversal finite measures of full support.

3.3 First remarks

3.3.1 Restriction to the constant curvature case

The restriction upon the metric is a severe one. Actually, thanks to the stability property (iii) of theorem 2.2.1, in order to prove our main result, it suffices to show the existence of transversal ergodic finite measures of full support in the case of constant curvature manifolds.

3.3.2 Choices made in the construction

The measure we construct on N depends on several choices, and various choices lead to mutually singular measures.

We describe now one of the crucial choice needed in the construction. Let M be the universal cover of N. Let ∂_{∞}M be its boundary at infinity.

Let P(∂_{∞}M) be the space of probability radon measures on ∂_{∞}M. Let

O_{3} = {(x, y, z) ∈ ∂_{∞}M^{3}/x ≠ y ≠ z ≠ x}.

The construction requires a map ν, invariant under the natural action of π_{1}(N),

O_{3} ⟶^{ν} P(∂_{∞}M).

Here, ν(x, y, z) is assumed to be of full support, and to fall in the same measure class, independantly of (x, y, z). Such maps are easily obtained through equivariant family of measures (also described in F. Ledrappier’s article [5] as Gibbs current, crossratios etc) and a barycentric construction as shown in paragraph 5.1.

3.4 Strategy of proof

As we said in the introduction, the construction is obtained through a coding of the space of k-surfaces. We give now a heuristic, non rigourous, outline of the proof, which is completed in the last section.

From the stability property, we can assume N has constant curvature. Our first step (section 6) is to associate to (almost) every k-surface a locally convex pleated surface, analogous to a “convex core boundary”. It turns out that this way we can describe a dense subset of k-surfaces, by locally convex pleated surfaces, and in particular by their pleating loci at infinity. Such pleating loci are described as special maps from ℚℙ^{1} to ℂℙ^{1}. This is the aim of sections 5 and 6. Identifying ℚℙ^{1} with the space of connected components of ℍ^{2} minus a trivalent tree, we build invariant measures on this space of maps as projective limit of measures on finite configuration spaces of points on ℂℙ^{1}. This is done in section 4.

3.5 Comments and questions

3.5.1 General negatively curved 3-manifolds

As we have seen before, the proof only works in the case of constant curvature manifolds, extending to other cases through the stability poperty. Of course, it would be more pleasant to obtain transversal measures without any restriction on the metric. Some parts of the construction do not require any hypothesis on the metric, and we have tried to keep, at the price of slightly longer proof sometimes, the proof as general as possible.

3.5.2 Equirepartition of closed leaves

Keeping in mind the analogy with the geodesic flow and the construction of the Bowen-Margulis measure, here is a completely different attempt to exhibit transversal measures, without any initial assumption on the metric. Define the H-area of a k-surface to be the integral of its mean curvature. It is not difficult to show that for any real number A, the number N(A) of k-surfaces in N of H-area less than A is bounded. Starting from this fact, one would like to know if closed leaves are equidistributed in some sense, i.e. that some average μ_{n} of measures supported on closed leaves of area less than n weakly converges as n goes to infinity. We can be more specific and ask about closed leaves of a given genus, or closed leaves whose π_{1} surjects onto a given group. This is a whole range of questions on which I am afraid to say I have no hint of answer. However, the constructions in this article should be related to equirepartition of ramified coverings of the boundary at infinity by spheres.

4 A combinatorial model

In general, P(X) will denote the space of probability radon measures on the topological space X, δ_{x} ∈ P(X) will be the Dirac measure concentrated at x ∈ X, and 𝕀_{S} will be the characteristic function of the set S.

In this section, we shall describe restricted infinite configuration spaces (4.0.5), which are roughly speaking spaces of infinite sets of points on a topological space W, associated to a configuration data (4.1). Our main result is theorem 4.2.1 which defines invariant ergodic measures of full support on these spaces, starting from measures defined on configuration data as in paragraph 4.1.2. One may think of these restricted infinite configuration spaces as analogue of subshifts of finite type, where the analogue of the Bernouilli shift is the space of maps of ℚℙ^{1} (instead of ℤ) into a space W with the induced action of P SL(2, ℤ). We call this latter space infinite configuration space and describe in the first paragraph, as well as related notions. The role of the configuration data is that of local transition rules.

4.0.3 Trivalent tree

We consider the infinite trivalent tree T, with a fixed cyclic ordering on the set of edges stemming from any vertex. Alternatively we can think of this ordering as defining a proper embedding of the tree in the real plane ℝ^{2}, such that the cyclic ordering agrees with the orientation. Another useful picture to keep in mind is to consider the periodic tiling of the hyperbolic plane ℍ^{2} by ideal triangle, and our tree is the dual to this picture (Figure 1). The group F of symmetries of that picture, which we abusively call the ideal triangle group, acts transitively on the set of vertices. It is isomorphic to F = ℤ_{2} ∗ ℤ_{3} = P SL(2, ℤ).

Figure 1: The infinite trivalent tree dual to the ideal triangulation

We now consider the set B of connected conponents of ℍ^{2}\T. In our tiling picture this set B is in one-to one correspondance with the set of vertices of triangles, and it follows that the ideal triangle group F acts also transitively on B. Actually B can be identified with ℚℙ^{1} and this identification agrees with the action of P SL(2, ℤ).

4.0.4 Quadribones, tribones

Every edge of T defines a set of four points in B, namely the connected components of ℍ^{2}\T that touches this edge, and we shall call these particular sets quadribones. We consider this set as an oriented set, i.e. up to signature 1 permutations, as labelled in the figure 2. Also, every vertex of the tree defines special subsets of three points in B, that we shall call tribones. Obviously every quadribone contains two tribones corresponding to the extremities of the corresponding edge, and again these quadribones are oriented sets. When our quadribone is given by (a, b, c, d) the two corresponding tribones are (a, b, c) and (d, c, b).

Figure 2: tribone (a, b, c) and quadribone (a, b, c, d)

4.0.5 Infinite configuration spaces

We define the infinite configuration space of W to be the space, denoted B_{∞}, of maps from B to W.

Notice that every tribone t (resp. quadribone q) of B defines a natural map from B_{∞} to W^{3} (resp. W^{4}) given respectively by f ↦ f(t) and f ↦ f(q); we call these maps associated maps to the tribone t (resp. to the quadribone q).

4.1 Local rules

For the construction of our combinatorial model, we need the following definitions.

4.1.1 Configuration data

We shall say that (W, Γ, O_{3}, O_{4}) defines a (3,4)-configuration data if:

(a) W is a metrisable topological space;

(b) Γ is a discrete group acting continuously on W.

We deduce from that a (diagonal) action of Γ on W^{n} which commutes with the action of the n^{th}-symmetric group σ_{n}. Let σ_{n}^{+} be the subgroup of σ_{n} of signature +1. Let λ_{3} = σ_{3}^{+}, and λ_{4} ⊂ σ_{4}^{+} be the subgroup generated by (a, b, c, d) ↦ (d, c, b, d). Let Δ_{n} = {(x_{1}, …, x_{n})/∃i ≠ j, x_{i} = x_{j}}. We assume furthermore that:

(c) O_{n} are open λ_{n} × Γ-invariant subsets of W^{n} \ Δ_{n}, on which Γ acts properly.

(d) p(O_{4}) = O_{3}, where p is the projection from W^{4} to W^{3} defined by

(a, b, c, d)) ↦ (a, b, c).

We shall also say a configuration data is Markov if it satisfies the following extra hypothesis:

(e) There exist some constant p ∈ ℕ, such that if (a, b, c) and (d, e, f) both belong to O_{3}, then there exists a sequence (q_{1}, …, q_{j}) of elements of O_{4}, where j ≤ p and q_{n} = (q_{n}^{1}, q_{n}^{2}, q_{n}^{3}, q_{n}^{4}), satisfying:

– (q_{1}^{1}, q_{1}^{2}, q_{1}^{3}) = (a, b, c) ;

– (q_{j}^{2}, q_{j}^{3}, q_{j}^{4}) = (d, e, f) ;

– (q_{n}^{2}, q_{n}^{4}, q_{n}^{3}) = (q_{n+1}^{1}, q_{n+1}^{2}, q_{n+1}^{3}) or (q_{n}^{3}, q_{n}^{2}, q_{n}^{4}) = (q_{n+1}^{1}, q_{n+1}^{2}, q_{n+1}^{3}).

In paragraph 4.3.4, this property will have a geometric consequence.

4.1.2 Measured configuration data

Our next goal is to associate measures to this situation. We shall say (W, Γ, O_{3}, O_{4}, μ^{3}, μ^{4}) is a (3,4)- measured configuration data if:

(f) μ^{n} are λ_{n} × Γ-invariant measures, such that p_{∗}μ^{4} = μ^{3}.

(g) The pushforward measures on O_{n}/Γ are probability measures.

We shall say that the measured configuration data is regular if it satisfies the following extra condition:

(h) the measure μ_{4} is in the measure class of 𝕀_{O_{4}}m ⊗ m ⊗ m ⊗ m where m is of full support in W. It follows that μ_{3} is in the measure class of 𝕀_{O_{3}}m ⊗ m ⊗ m.

We also say two regular measured configuration data (W, Γ, O_{3}, O_{4}, μ^{3}, μ^{4}) and (W, Γ, O_{3}, O_{4}, μ̄^{3}, μ̄^{4}), defined on the same configuration data, are mutually singular if μ^{3} and μ̄^{3} are mutually singular.

4.1.3 Remarks

(i) From desintegration of measures, it follows from the hypothesis (f) and (g) that for μ^{3}-almost every triple of points (a, b, c) in W, we have a probability measure ν_{(a,b,c)} on W such that for every positive measurable function f on W^{4}:

∫_{W^{4}} f(a, b, c, d)dμ^{4}(a, b, c, d) = ∫_{W^{3}} ( ∫_{W} f(a, b, c, d)dν_{(a,b,c)}(d))dμ^{3}(a, b, c)

which we can rewrite as

dμ^{4} = ∫_{W^{3}} (δ_{(a,b,c)} ⊗ dν_{(a,b,c)})dμ^{3}(a, b, c),

(ii) Conversely, here is a way to build a regular measured configuration data starting from a configuration data (W, Γ, O_{3}, O_{4}), if we assume that O_{4} is invariant under σ_{4}^{+}.

Assume we have:

- a Γ-invariant measure μ̄^{3} on W^{3} in the measure class of 𝕀_{O_{3}}m ⊗ m ⊗ m where m has full support, such that the push forward on O_{3}/Γ is a probability measure;

- a Γ-equivariant map ν̄:

\{ \begin{matrix}O_{3} & → P_{m}(W) \\ (a, b, c) & ↦ ν̄_{(a,b,c)} \end{matrix}

where P_{m}(W) is the set of finite radon measures on W in the measure class of m.

Then, we can build μ^{3} and μ^{4} which will fulfill the requirements of the definition. Let’s describe the procedure:

Firstly, we define a probability measure μ̄^{4} on O_{4} to be proportionnal to

𝕀_{O_{4}} ∫_{W^{3}} (δ_{(a,b,c)} ⊗ ν̄_{(a,b,c)})dμ̄^{3}(a, b, c).

Secondly, we average μ̄^{4} using the group σ_{4}^{+} and obtain a finite measure μ^{4} on O_{4}/Γ, and we utimately take μ^{3} = p_{∗}μ^{4}.

It is a routine check now that μ^{3} and μ^{4} defined this way satisfy our needs. Futhermore, if μ̄^{3} has full support in O_{3} as well as ν_{(a,b,c)} for μ̄^{3}-almost

every (a, b, c) in W_{3}, then μ^{3} and μ^{4} have full support.

4.1.4 Example

In the sequel, we only wish to study one example that we describe briefly now and more precisely in section 5. Our specific interest lies in the following situation.

- Γ is a cocompact discrete subgroup of P SL(2, ℂ);

- W = ℂℙ^{1} with the canonical action of Γ; it is a well known fact that Γ acts properly on

U_{n} = {(x_{1}, …, x_{n}) ∈ (ℂℙ^{1})^{n}/ x_{i} ≠ x_{j} if i ≠ j}.

- O_{3} = U_{3},

- O_{4} is the set of points whose crossratios have a non zero imaginary part; it will satisfy hypothesis (e) for N = 1000 (cf paragraph 5.2).

This is a Markov configuration data and furthermore in this specific situation O_{4} is is invariant under σ_{4}^{+}. We will explain in paragraph 5.1 how to attach measures to this situation, and discuss the case of general negatively curved 3-manifolds.

4.1.5 Final remark

Even though we only wish to study that specific class of examples, it is a little more comfortable to work in our more general setting, since very little of the geometry is used at this stage.

4.2 Restricted infinite configuration spaces and main result

Let now (W, Γ, O_{i}) be a (3,4)-configuration data (cf 4.1),

We define the restricted infinite configuration space of W to be the subset B̄_{∞} of B_{∞}, consisting of those maps such that the image of every tribone lies in O_{3}, and the image of every quadribone is in O_{4}.

B̄_{∞} = {f ∈ B_{∞}/for all tribone t, quadribone q, f(t) ∈ O_{3}, f(q) ∈ O_{4}}.

Let also B_{∞}^{0} be the open set of the infinite configuration space such the image of at least one tribone lies in O_{3}. Let’s call this subset the non degenerate configuration space, and notice that Γ acts properly on this open subset of B_{∞}.

Now we can state the theorem we wish to prove:

Theorem 4.2.1 Let (W, Γ, O_{i}, μ^{i}) be a (3,4)-measured configuration data. Then there exists a Γ-invariant measure μ on the infinite configuration space of W, which is invariant by the action of the ideal triangle group, such that:

(i) the restricted infinite configuration space B̄_{∞} is of full measure and μ has full support on it provided the data is regular;

(ii) the pushforward of μ on B_{∞}^{0}/Γ is finite, where B_{∞}^{0} is the non degenerate infinite configuration space;

(iii) given any tribone or quadribone, the pushforward of μ by the associated maps on W^{3} and W^{4} are our original μ^{3}, μ^{4};

(iv) two regular mutually singular measured configuration data give rise to mutually singular measures ;

(v) if the configuration data is Markov and regular, then the pushforward of μ on B_{∞}^{0}/Γ is ergodic with respect to the action of the infinite triangle group.

Essentially, this measure is built by a Markov type procedure.

4.3 Construction of the measure

Let (W, Γ, O_{i}, μ^{i}) be a (3,4)-configuration data. We shall use the notations and definitions of the preceding sections.

Also in our constructions, for every (x, y, z) ∈ O_{3}, we shall denote by ν_{(x,y,z)} the probability measure coming from the desintegration of μ^{4} over μ^{3} as defined in paragraph 4.1.3.

4.3.1 Connected sets, P -bones, P -disconnected sets

For our constructions, we need a terminlogy for some subsets of B which roughly correspond to certain subtrees of T.

A subset A of B will be called connected if it is a reunion of quadribones such that the reunion e(A) of the associated edges is connected; if v is a vertex, it will be called v-connected if furthermore e(A) contains v. In other words a connected subset of B is the reunion of the connected components touching the edges of a connected subtree of T.

A subset A of B will be called a P -bone if it is connected and the reunion of less than P quadribones; two subsets A and C will be called P -disconnected if there is no P -bone which intersects both A and C.

4.3.2 Relative configuration spaces

If A is a subset of B, we shall note:

- W(A) the set of maps from A to W; in particular, W(B) = B_{∞}.

- W̄(A) the set of maps such that the image of every tribone of A lies in O_{3}, and the image of every quadribone is in O_{4}; if A is finite, W̄(A) is an open set on which Γ acts properly. Again, W̄(B) = B̄_{∞}.

4.3.3 Finite construction

We can now prove:

Proposition 4.3.1 Let A be a finite v_{0}-connected subset of B. Then, there exists a radon measure μ^{A,v_{0}} on W̄(A) enjoying the following properties:

(i) the pushforward of μ^{A,v_{0}} on W̄(A)/Γ is finite ; it is of full support if the data is regular;

(ii) let t_{0} be the tribone corresponding to the vertex v_{0}; let also t_{0} be the associated map from A to W^{3}; then t_{∗}μ^{A,v_{0}} = μ^{3};

(iii) let q be a v_{0}-connected quadribone ; assume q ⊂ A; let q be the associated map from A to W^{4}; then q_{∗}μ^{A,v_{0}} = μ^{4} (iv) assume there exist a tribone t ⊂ A, some element a ∈ B \ A, such that

q = t ∪ {a} is a quadribone ; let now C = A ∪ {a} and identify W(C) with W(A) × W then

μ^{C,v_{0}} = ∫_{W(A)} (δ_{f} ⊗ ν_{f(t)})dμ^{A,v_{0}}(f).

(v) let A ⊂ C ; let p be the natural restriction from W̄(C) to W̄(A). Then p_{∗}μ^{C,v_{0}} = μ^{A,v_{0}} ;

(vi) if (μ^{3}, μ^{4}) and (μ̄^{3}, μ̄^{4}) are regular and mutually singular, then the corresponding measures μ^{A,v_{0}} and μ̄^{A,v_{0}} are mutually singular.

One should notice that the listed properties defines μ^{A,v_{0}} uniquely.

We shall also say in the sequel that if C and A are as in (iv), that C is obtained from A by gluing a quadribone along a tribone, as in figure 3.

Figure 3: Gluing a quadribone (a, b, c, d) along a tribone (a, b, c)

We have a useful consequence of the previous proposition:

Corollary 4.3.2 Let A be a finite set and let v and w such that A is both v-connected and w-connected, then μ^{A,v} = μ^{A,w}.

Now of course, we may write μ^{A} = μ^{A,v}.

Our last proposition exhibits some kind of “Markovian” property of our measure.

Proposition 4.3.3 Assume the configuration data is Markov and regular. There exists an integer P, such that if A_{0} and A_{1} are two P -disconnected subsets of a finite set C ⊂ B, then (p^{0}, p^{1})_{∗}μ^{C} and p_{∗}^{0}μ^{C} ⊗ p_{∗}^{1}μ^{C} are in the same measure class. Here, p^{i} : W(C) → W(A_{i}) are the natural restriction maps.

We will now prove the results stated in this paragraph.

4.3.4 Proof of proposition 4.3.1

Introduce first some notations with respect to a vertex v. By definition B_{n}(v) will denote the union of all v-connected n-bones; also, for any subst A of B, we pose A_{n}(v) = B_{n}(v) ∩ A.

For the moment, we will work with a fixed v_{0} and will omit the dependance in v_{0} in the notations for the sake of simplicity, in particular A_{n} = A_{n}(v_{0}). We will construct this measure by an induction procedure.

Our first task is to build for every n ∈ ℕ, a map:

ν^{A,n} : \{ \begin{matrix}W̄(A_{n}) & → & P(W(A_{n+1} \ A_{n})) \\ f & ↦ & ν_{f}^{A,n}. \end{matrix}

Let’s do it. If a ∈ A_{n+1}\A_{n}, it belongs to a unique quadribone q_{a} ⊂ A_{n+1}. Let t_{a} = q_{a} \ {a}; notice that t_{a} is a subset of A_{n}. Let A_{n+1} \ A_{n} = {a_{1}, …, a_{q}}. In particular, W(A_{n+1} \ A_{n}) is identified with W^{q}. Let T_{n}^{A} = ∪_{i=1}^{i=q}t_{a_{i}}. We have a natural restriction map

i^{A,n} : W̄(A_{n}) ⟶ W̄(T_{n}^{A}).

We define

ν̄^{A,n} : \{ \begin{matrix}W̄(T_{n}^{A}) & → & P(W^{q}) = P(W(A_{n+1} \ A_{n})) \\ f & ↦ & ⨂_{i} ν_{f(t_{i})}. \end{matrix}

Finally, we set: ν^{A,n} = ν̄^{A,n} ∘ i^{A,n}.

Next, notice the following fact. Let f ∈ W̄(A_{n}). Let W̄_{f}(A_{n+1}) be the fiber, over f, of the restriction map. Let’s use the identification

W(A_{n+1}) = W(A_{n+1} \ A_{n}) × W(A_{n}).

Then, W̄_{f}(A_{n+1}) has full measure for ν_{f}^{A,n} ⊗ δ_{f}.

We can now define our measure on W̄(A_{n+1}) by an induction procedure:

- W̄(A_{0}) is identified with O_{3} using t_{0}; we define μ^{A_{0}} = (t_{0}^{−1})_{∗}μ^{3};

- assuming by induction that μ^{A_{n}} is defined on W(A_{n}) such that W̄(A_{n}) has full measure, we set

μ^{A,n+1} = ∫_{W̄(A_{n})} (ν_{f}^{A,n} ⊗ δ_{f})dμ^{A,n}(f).

From the previous observation, we deduce that W̄(A_{n+1}) has full measure. Furthermore, if the μ^{i} have full support, then μ^{A,n+1} has full support.

Finally, there exists p ∈ ℕ such that A = A_{p}, and we define

μ^{A,v_{0}} = μ^{A,p}.

Properties (i), (ii), (iii), and (vi) are immediately checked. Let’s prove property (iv).

Notice first that a lies in exactly one quadribone q of C. Let d be the unique tribone of C that contains a. Then, there exists p_{0} such that

C_{p} = A_{p} for p < p_{0},
C_{p} = A_{p} ∪ {a} for p ≥ p_{0}.

By construction, using the obvious identifications, we have

μ^{C,p} = μ^{A,p}, for p < p_{0},
μ^{C,p} = ∫_{W(A_{p})}(δ_{f} ⊗ ν_{f(q\a)})dμ^{A,p}(f), for p = p_{0}. (∗)

To conclude the proof of (iv), it remains to prove (∗) for p > p_{0}. By induction, this follows from the fact that, for p > p_{0}, T_{p}^{A} = T_{p}^{C}.Let’s check that step by step. By definition,

μ^{C,p+1} = ∫_{W(C_{p})} (ν_{f(T_{p})} ⊗ δ_{f})dμ^{C,p}(f).

But, by induction

μ^{C,p} = ∫_{W(A_{p})}(δ_{g} ⊗ ν_{g(q\a)})dμ^{A,p}(g).

Combining the two last equalities, and using T_{p}^{A} = T_{p}^{C}, we get

μ^{C,p+1} = ∫_{W(A_{p})}(δ_{g} ⊗ ν_{g(q\a)} ⊗ ν_{g(T_{p}^{A})})dμ^{A,p}(g)
= ∫_{W(A_{p+1})}(δ_{g} ⊗ ν_{g(q\a)})dμ^{A,p+1}(g).

This is what we wanted to prove.

Property (v) is an immediate consequence of (iv). Indeed, if C contains A, it is obtained inductively from A by “gluing quadribones along tribones” as in (v). ⋄ 4.3.5 Proof of corollary 4.3.2

Obviously, it suffices to prove it whenever v and w are the extremities of a common edge e. Let q be the associated quadribone. Since we can build A from q by successively “gluing quadribones along tribones”, using property (v) of proposition 4.3.1, it suffices to show that

μ^{q,v} = μ^{q,w}.

Thanks to proposition 4.3.1 (iii), this follows from the invariance of μ^{4} under the permutation (a, b, c, d) ↦ (d, c, b, a).⋄

4.3.6 A consequence of hypothesis (e) of 4.1

Using the previous notations, we have:

Proposition 4.3.4 Assume the configuration data is Markov. Then, there exists an integer P, such that if A_{0} and A_{1} are connected and P -disconnected, if C is a connected set that contains both then

(p^{0}, p^{1})(W̄(C)) = W̄(A_{0}) × W̄(A_{1}).

Proof: Let A_{0} and A_{1} be two P -disconnected subsets. Then there exists a N-bone K, where N > P, such that K intersects each A_{i} exactly along one tribone t_{i} as in figure 4. Let D = A_{0} ∪ K ∪ A_{1}.

Figure 4: t_{0} = (a, b, c), t_{1} = (r, s, t)

Let f_{0} (resp. f_{1}) be an element of W̄(A_{0}) (resp. W̄(A_{1})). Hypothesis (e) of 4.1 implies there exists some element g of W̄(K) such that g coincides with f_{0} (resp. f_{1}) on t_{0} (resp t_{1}) provided N is greater than p. Gluing together g and the f_{i}, we obtain an element h of W̄(D), whose restriction to A_{i} is f_{i}.

In other words, the restriction from W̄(D) to W̄(A_{0}) × W̄(A_{1}) is surjective. To conclude, its suffices to notice that since D is connected, the restriction from W̄(C) to W̄(D) is surjective.⋄

4.3.7 Proof of proposition 4.3.3

The first point to notice is that if μ^{3} and μ^{4} are in the measure class of 𝕀_{O_{3}}m ⊗ m ⊗ m and 𝕀_{O_{4}}m ⊗ m ⊗ m ⊗ m respectively, then for m-almost every tribone t, ν_{t} is also in the measure class of 𝕀_{O_{t}}m where O_{t} is such that {t} × O_{t} = p^{−1}(t) ∩ O_{4}. It follows that if A is connected then μ^{A} is actually in the measure class of

𝕀_{W̄(A)}m^{⊗#A}.

Let’s prove this last assertion by induction: let C_{n}, 1 ≤ n ≤ p be an increasing sequence of sets such that C_{1} is quadribone, C_{p} = A and C_{n+1} is obtained from C_{n} by gluing a quadribone along a tribone t_{n} as in proposition 4.3.1 (iv). Then

W̄(C_{n+1}) = ⋃_{f∈W̄(C_{n})}{f} × O_{f(t_{n})},

where we identified W(C_{n+1}) et W(C_{n}) × W. An inductive use of 4.3.1 (iv) implies our statement.

Assume now the configuration data is Markov. Then according to proposition 4.3.4, we have

(p^{0}, p^{1})(W̄(C)) = W̄(A_{0}) × W̄(A_{1}).

Hence our proposition.⋄

4.3.8 Infinite construction, and proof of properties (i), …, (iv) of theorem 4.2.1

We first define a measure μ on B_{∞}. We consider as before the set B_{n} = B_{n}(v_{0}), and pose μ_{n} = μ^{B_{n}}.

The set B_{∞} equipped with the product topology is the projective limit of the sequence (W(B_{n}). We define μ as the projective limit of the sequence {μ_{n}}_{n∈ℕ}. If μ^{3}, μ^{4} have full support on O_{3} and O_{4} respectively, then the measure μ_{n} has full support on W̄(B_{n}). It follows that μ has full support on B̄_{∞}.

The only non immediate property of μ is the invariance under the ideal triangle group P SL(2, ℤ).

Notice first that if g belongs to the stabilizer of the vertex v_{0}, we have that g_{∗}μ_{n} = μ_{n}: this follows from the invariance of μ^{3} under cyclic permutations, and from property (iv) of proposition 4.3.1.

Then, because of the symetries of T and the uniqueness of our construction, we have that if g ∈ F, g_{∗}μ^{A,v} = μ^{g(A),g(v)}, and therefore g_{∗}μ^{A} = μ^{g(A)}, because of corollary 4.3.2.

It follows that g_{∗}μ is the projective limit measure of the projective limit of {W(g(B_{n})}_{n∈ℕ}, which is also B_{∞}.

To conclude, we just have to remark that, thanks to (v) of proposition 4.3.1, for whatever sequence of finite v-connected set {D_{n}}_{n∈ℕ} in B, such that D_{n+1} ⊂ D_{n} and ∪_{n}D_{n} = B, the projective limit measure associated with the sequence of {μ^{D_{n}}}_{n∈ℕ} coincides with μ.⋄

4.4 Ergodicity

We shall now prove property (vi) of theorem 4.2.1. Let’s first introduce some definitions.

4.4.1 Hyperbolic elements, pseudo-markov measure

Let F = P SL(2, ℤ) be the ideal triangle group, which we consider emmbedded in the isometry group of the Poincaré disk. We shall say γ ∈ F is hyperbolic, if γ is an hyperbolic isometry. Notice that since F is Zariski dense, it contains many hyperbolic elements.

We also say a measure on B_{∞}^{0}/Γ is pseudo-markov if it satisfies the following property : there exists an integer P, such that for any P -disconnected and connected subsets A and C in B, if p_{A} and p_{B} are the associated projections, then p_{∗}^{A}μ ⊗ p_{∗}^{B}μ and (p^{A}, p^{B})_{∗}μ are in the same measure class. By proposition 4.3.3, the measure we constructed in the last section enjoys that property.

4.4.2 Main result

To conclude it suffices to prove:

Proposition 4.4.1 Let μ be a F -invariant finite measure on B_{∞}^{0}/Γ, which is the pushforward of a pseudo-Markov measure. Then μ is ergodic for the action of any hyperbolic element of F, hence ergodic for F itself.

The proof is closely related to the proof of the ergodicity of subshifts of finite type, and is an avatar of Hopf’s argument. We introduce stable and unstable leaves in paragraph 4.4.4, using vanishing sequences of sets defined in paragraph 4.4.3. We finally conclude using Birkhoff Ergodic Theorem.

4.4.3 Hyperbolic elements of F

Let X be a topological space and γ ∈ C^{0}(X, X), we shall say for short that a sequence of non empty subsets {V_{n}}_{n∈ℕ} is a vanishing sequence for γ if:

(i) V_{n+1} ⊂ V_{n};

(ii) ⋂_{n∈ℕ} V_{n} = ∅;

(ii) for all compact subset K of X, and n ∈ ℕ, there exists p ∈ ℕ, such that γ^{p}(K) ⊂ V_{n}.

Lemma 4.4.2 Let γ be an hyperbolic element in F. Then, there exist two families of connected subsets of B, {U_{n}^{+}}_{n∈ℕ} and {U_{n}^{−}}_{n∈ℕ}, , which are respectively vanishing sequences for γ and for γ^{−1}, such that furthermore U_{0}^{+} ∩ U_{0}^{−} = ∅.

Proof: this is a consequence of elementary hyperbolic geometry. Indeed, if we consider F as a subgroup of the hyperbolic plane, the fixed points of γ on the boundary at infinity are not vertices of the tiling by ideal triangles, and the lemma follows. ⋄

4.4.4 Contractions

Let now γ, {U_{n}^{±}}_{n∈ℕ} be as in lemma 4.4.2. We first introduce equivalence relations amongst elements of B_{∞}^{0}. We say f ∼_{n}^{+} g, if f|_{U_{n}^{+}} = g|_{U_{n}^{+}}. If f ∈ B_{∞}^{0}, let F_{n}^{+}(f) be the equivalence class of f. Finally define f ∼^{+} g, if there exists n such that f ∼_{n}^{+} g and note F^{+}(f) the equivalence class of f. Observe that

F^{+}(f) = ⋃_{n∈ℕ}F_{n}^{+}(f),

and let’s define ∼^{−}, and F^{−}(f) in a symmetric way. These equivalence classes are going to play the role of the stable and unstable leaves of hyperbolic systems.

We shall prove:

Proposition 4.4.3 There exist a Γ-invariant metric on B_{∞}^{0} inducing the natural topology, such that for all f ∈ B_{∞}^{0} and g ∈ F^{+}(f) then

lim_{p→+∞} d(γ^{p}(f), γ^{p}(g)) = 0,

and similarly if g ∈ F^{−}(f) then

lim_{p→+∞} d(γ^{−p}(f), γ^{−p}(g)) = 0.

Proof: We first define a metric on B_{∞}^{0} depending on the choice of a vertex v_{0} of the tree T. Let B_{n} ⊂ B be defined as in paragraph 4.3.4. Let T_{n} be the set of tribones of B_{n}. Let t be a tribone, then

- let B_{∞}^{t} to be the set of maps from B to W, such that the image of t lies in O^{3};

- if t ∈ T_{n}, let B_{n}^{t} be the set of maps from B_{n} to W, such that the image of t lies in O^{3}; notice that Γ acts properly on B_{n}^{t}.

Next,

- let δ_{n}^{t} be a Γ-invariant distance of diameter less than 1 on B_{n}^{t} which induces the product topology;

- let d_{n}^{t} the semidistance on B_{∞}^{t}, induced from δ_{n}^{t} by the canonical projection; notice that the product topology of B_{∞}^{t} is induced by the family of semidistances {d_{n}^{t}}_{n∈ℕ,t∈T_{n}}.

By definition, recall that

B_{∞}^{0} = ⋃_{tribones t} B_{∞}^{t}.

If t ∈ T_{n}, we extend d_{n}^{t} to B^{0} in the following way:

\{ \begin{matrix}d_{n}^{t}(x, y) & = 0, if x, y ∈ B_{∞}^{0} \ B_{∞}^{t}, \\ d_{n}^{t}(x, y) & = 1, if y ∉ B_{∞}^{t}, x ∈ B_{∞}^{t}. \end{matrix}

Utimately, we define a Γ-invariant metric d on B_{∞}^{0}, by the formula

d(x, y) = ∑_{n∈⋉} \frac{1}{2^{n}} ∑_{t∈T_{n}} \frac{1}{#T_{n}} d_{n}^{t}(x, y).

By construction of this distance, it follows that is f and g coincides on B_{n} then d(f, g) ≤ (\frac{1}{2})^{n−1}. In particular,since

∀q, n ∈ ℕ , ∃p ∈ ℕ such that γ_{p_{n}}(B_{n}) ⊂ U_{q},

it follows that for every n, if f ∼_{q}^{+} g, then there exists p ∈ ℕ, such that

d(γ^{p}(f), γ^{p}(g)) ≤ (\frac{1}{2})^{n−1}.

This ends the proof of the proposition .⋄

4.4.5 Preliminary steps for the proof of ergodicity

Define for every bounded function ϕ on B_{∞}^{0}

ϕ^{+} = lim sup_{n→+∞}(ϕ ∘ γ^{n}),

and

ϕ^{−} = lim sup_{n→+∞}(ϕ ∘ γ^{−n}).

Let’s first prove:

Proposition 4.4.4 Let ϕ be a continuous Γ-invariant function on B_{∞}^{0}, such that the quotient function on B_{∞}^{0}/Γ is compactly supported. Then f ∼^{+} g implies ϕ^{+}(f) = ϕ^{+}(g) and, f ∼^{−} g implies ϕ^{−}(f) = ϕ^{−}(g).

Proof: Notice first that ϕ is bounded and uniformly continuous. Hence, the proposition follows at once from proposition 4.4.3.⋄

A second preliminary step is:

Proposition 4.4.5 Let μ be a locally finite pseudo-Markov measure of full support on B_{∞}^{0}. Let E be a set of μ-full measure. Then for μ-almost every f, there exists a set F_{f} of μ-full measure such that

∀g ∈ F_{f}, ∃h ∈ E, s.t. f ∼^{+} h ∼^{−} g.

Proof: We should first notice that the set of equivalence classes of ∼_{n}^{+} is precisely W(U_{n}^{+}), the space of maps from U_{n}^{−} to W. A similar statement holds for ∼_{n}^{−}. Fix some integer n, for which U_{n}^{+} and U_{n}^{−} are P -disconnected. Let p^{+} be the natural continuous projection

B_{∞}^{0} ↦ W(U_{n}^{+}).

Define p^{−} a similar way. At last, let p = (p^{+}, p^{−}).

If E has full measure, then p(E) has full measure for p_{∗}μ. Hence by the pseudo-Markov property, it has full measure for p_{∗}^{+}μ ⊗ p_{∗}^{−}μ.

From Fubini Theorem, we deduce there is a set of full measure A in W(U_{n}^{+}), such that for every a ∈ A, the set

V_{a} = {c ∈ W(U_{n}^{−}), (a, c) ∈ p(E)}

has full measure for p_{∗}^{−}μ.

In particular, for every f ∈ (p^{+})^{−1}(A), the set F_{f} = (p^{−})^{−1}(V_{p^{+}(f)}) has full measure.

Now, by construction if f ∈ (p^{+})^{−1}(A) and g ∈ F_{f}, then p^{−}(g) = p^{−}(h), where h ∈ E and p^{+}(h) = p^{+}(f). This is exactly what we wanted to prove.⋄

4.4.6 End of the proof of ergodicity

In this paragraph, we will prove proposition 4.4.1. Let γ be some hyperbolic element in F. Let μ be the F -invariant measure on B_{∞}^{0}/Γ, constructed in 4.2.1. We first use the ergodic decomposition theorem and write

μ = ∫_{Z} ν_{z}dλ(z),

where for λ-almost every z in Z, ν_{z} is an ergodic measure for γ.

To conclude, it suffices to show that for any continuous and compactly supported function ψ on B_{∞}^{0}/Γ, and for every z and u in Z, we have ∫ ψdv_{z} = ∫ ψdv_{u}.

Let now ϕ = ψ ∘ π, where π is the natural projection from B_{∞}^{0} to B_{∞}^{0}/Γ. Let’s define as for proposition 4.4.4, the measurable functions ϕ^{+} and ϕ^{−}. From Birkhoff ergodic theorem, we deduce that for ν_{z}-almost every x, if π(y) = x, we have

(∗)  ϕ^{+}(y) = ϕ^{−}(y) = ∫_{B_{∞}^{0}/Γ} ψdν_{z}.

In particular, there exists a set of μ-full measure E on which ϕ^{+} = ϕ^{−}.

Now, we apply proposition 4.4.5, and deduce that for μ-almost every x, there exists a set of full measure F_{x} with the following property: if b ∈ F_{x} then there exists a ∈ E such that x ∼^{+} a ∼^{−} b.

From proposition 4.4.4, we deduce that ϕ^{+}(a) = ϕ^{+}(x) and ϕ^{−}(b) = ϕ^{−}(a). From the definition of E, we get that ϕ^{−} is constant and equal to ϕ^{−} on F_{x}, hence μ-almost everywhere.

Using (∗), we ultimately get that for almost every z, u ∈ Z

∫_{B_{∞}^{0}/Γ} ψdν_{z} = ∫_{B_{∞}^{0}/Γ} ψdν_{u},

which is what we wanted to prove.⋄

5 Configuration data and the boundary at infinity of a hyperbolic 3-manifold

We describe here our main, and actually unique useful example : the Markov configuration data associated to a hyperbolic 3-manifold.

Let in general ∂_{∞}M be the boundary at infinity of a negatively curved 3-manifold M. Let Γ be a discrete, torsion free and cocompact group of isometries of M.

Unless otherwise specified, we shall assume M is the hyperbolic 3-space ℍ^{3}. Then, ∂_{∞}M = ∂_{∞}ℍ^{3} is canonically identified with ℂℙ^{1}. In this identification, the action of the group of isometries of M on ∂_{∞}M coincides with the action of P SL(2, ℂ) on ℂℙ^{1}.

As we explained in 4.1.4, the (3,4)-configuration data we shall study is the following :

- W = ∂_{∞}M = ℂℙ^{1},

- O_{3} is the subset of ∂_{∞}M^{3} consisting of triples of different points :

O_{3} = {(x, y, z) ∈ ∂_{∞}M / x ≠ z ≠ y ≠ x}.

- O_{4} is the set of points whose crossratios have a non zero imaginary part;

We have Proposition 5.0.6 The quadruple (ℂℙ^{1}, Γ, O_{3}, O_{4}) is a Markov (3,4)-confi-guration data.

It is obvious. The only point that requires a check is hypothesis (e). In the last paragrpah 5.2, we will devise a fancy (and far too long) proof of this fact. Of course, a straightforward check would give that this configuration data satisfies (e) for N = 10, instead of N = 1000, provided by our proof. However, I hope the scheme of this proof might be useful in more general situations.

In the next paragraph we explain how to turn this example in a regular measured configuration data in many ways, using equivariant family of measures (cf 5.1.1).

5.1 Measured configuration data

In view of 4.1.3(ii) we need to produce μ̄^{3} in the Lebesgue class of m⊗m⊗m for some measure class m of full support, such that the pushforward of μ̄^{3} on O_{3}/Γ is finite. Then we have to build a Γ-equivariant map ν̄:

\{ \begin{matrix}O_{3} & → P_{m}(W) \\ (a, b, c) & ↦ ν̄_{(a,b,c)} \end{matrix}

where P_{m}(W) is the set of finite radon measures on W in the measure class of m.

We shall do that using the notion of equivariant family of measures described by F. Ledrappier in [5], and which is a generalisation of that of conformal densities due to D. Sullivan in [6].

5.1.1 Equivariant family of measures

An equivariant family of measures on the boundary a map μ which associates to every x ∈ M a finite measure μ_{x} on ∂_{∞}M such that:

(i) for all γ in Γ, μ_{γx} = γ_{∗}μ_{x},

(ii) For all x, y ∈ M, μ_{x} and μ_{y} are in the same Lebesgue class.

In particular we can write dμ_{x}(a) = e^{−γ_{a}(x,y)}dμ_{y}(a). Actually, the original definition requires some regularity of the function (a, x, y) ↦ γ_{a}(x, y), which we shall not need in the sequel.

A typical example arises when one associates to a point x the pushforward by the exponential map of the Liouville measure on the unit sphere at x.

When c_{η}(x, y) = δB_{η}(x, y), where B_{η}(x, y) is the the Buseman function defined by

B_{η}(x, y) = lim_{z→η}(d(x, z) − d(y, z))

the corresponding equivariant family of measures is called a conformal density of ratio δ. Among these is the Paterson-Sullivan measure.

In [5] which also contains many references to related results, F. Ledrappier discusses various ways of building equivariant families of measures, and in particular relates them to other notions like crossratios, Gibbs currents, transverse invariant measures to the horospherical foliations etc. As a conclusion, there exist numerous examples of equivariant families of measures, all mutually singular.

5.1.2 End of the construction

Let’s go back to our construction now. Let first β

(a, b, c) ↦ β_{a,b,c}

be a Γ-equivariant map from O_{3} to M. For instance, we can take the barycenter of the sum of the three Dirac measures concentrated at a, b, and c. Now define, if x ∈ M

dμ̄^{3}(a, b, c) = e^{c_{a}(x,β_{a,b,c})+c_{b}(x,β_{a,b,c})+c_{c}(x,β_{a,b,c})}dμ_{x} ⊗ dμ_{x} ⊗ dμ_{x}(a, b, c).

It follows from the definition of equivariant families of measure that this definition is independant on x. If Γ is a group of isometries then μ̄^{3} is Γinvariant. Furthermore, if Γ is cocompact then the corresponding measure is finite on O_{3}/Γ.

For ν̄, we can now just take the map (a, b, c) ↦ μ_{β_{a,b,c}}.

5.1.3 Negatively curved 3-manifolds

We do not have used previously the hyperbolic structure. Let’s take for a general negatively curved M and cocompact group of isometries Γ

- W = ∂_{∞}M,

- O_{3} = ∂_{∞}M^{3} \ Δ_{3},
- O_{4} any λ_{4}-invariant subset such that p(O_{4}) = O_{3}.

Then the previous construction works provided that O_{4} is invariant under all σ_{4}^{+}. For instance, we could take O_{4} = U_{4}, but the corresponding construction seems to be of no use for our problem. For the moment, I have not been able to construct a configuration data adapted to the problem, for general negatively curved 3-manifolds.

5.2 Complex crossratio

Let [a; b; c; d] be the complex crossratio of four points of ℂℙ^{1}, such that [0; 1; ∞; z] = z. Let ℑ(α) be the imaginary part of the complex number α.

We will single out the geometric properties of the crossratio which are actually used in proposition 5.0.6.

5.2.1 disks

We associate to every triple (a, b, c), a disk D(a, b, c), defined by

D(a, b, c) = {z ∈ ℂℙ^{1} / ℑ([a; b; c; z]) < 0}.

If we consider D as a map from O_{3} to the set of subsets of ∂_{∞}M, it enjoys the following properties:

(i) the map D is Γ-equivariant;

(ii) D(a, b, c) ∪ D(a, c, b) is dense;

(iii) for every (a, b, c) in O_{3}, a belongs to the closure of D(a, b, c);

(iv) Let O_{4} = {a, b, c, d / (a, b, c) ∈ O_{3} , d ∈ D(a, b, c)}, then O_{4} is an open set invariant under the oriented permutations (a, b, c, d) ↦ (d, c, b, d).

In our particular case, all these properties are easily checked from the invariance of the crossratio and it’s behaviour under permutations.

We will prove:

Proposition 5.2.1 Let D be satisfying properties (i) to (iv) of 5.2.1. Then the quadruple (∂_{∞}M, Γ, O_{3}, O_{4}) is a Markov (3,4)-configuration data.

In our very precise situation, we could devise a quick proof of that fact. However, we will give a somewhat longer proof: the idea is to stress the importance of properties (i) to (v) of 5.2.1, and forget a while the complex structure on ∂_{∞}M.

Proof of the proposition: it only remains to prove (e) of definition 4.1 which caracterizes Markov configuration data. Let’s recall it:

(e) There exist some constant p ∈ ℕ, such that if (a, b, c) and (d, e, f) both belong to O_{3}, then there exists a sequence (q_{1}, …, q_{j}) of elements of O_{4}, where j ≤ p, such that if q_{n} = (q_{n}^{1}, q_{n}^{2}, q_{n}^{3}, q_{n}^{4}) then (q_{1}^{1}, q_{1}^{2}, q_{1}^{3}) = (a, b, c), (q_{j}^{3}, q_{j}^{2}, q_{j}^{4}) = (d, e, f) and at last (q_{n}^{2}, q_{n}^{4}, q_{n}^{3}) = (q_{n+1}^{1}, q_{n+1}^{2}, q_{n+1}^{3}) or (q_{n}^{3}, q_{n}^{2}, q_{n}^{4}) = (q_{n+1}^{1}, q_{n+1}^{2}, q_{n+1}^{3}) ,

In order to proceed towards a proof we shall write that (a, b, c) ⇝^{p} (d, e, f) if (a, b, c) and (d, e, f) satisfies this condition (e). With these notations at hands, one immediately checks:

- (a, b, c) ⇝^{p} (u, v, w) implies (w, u, v) ⇝^{p} (b, c, a);

- composition rule: if t_{1} ⇝^{p} (a, b, c), and (a, b, c) ⇝^{q} t_{3} or (b, c, a) ⇝^{q} t_{3}

then t_{1} ⇝^{p+q} t_{3};

- (a, b, c, d) ∈ O_{4} exactly means that (a, b, c) ⇝^{1} (c, b, d).

We are going to proceed through various steps.

Step 1: for any (a, b, c) there exists (a_{1}, b_{1}, c_{1}) arbitrarily close to (a, b, c)

such that (a, b, c) ⇝^{3} (a_{1}, c_{1}, b_{1}).

We shall prove that using property (iii) of the definition 5.2.1. First, using (iii) of 5.2.1, we choose b_{1} arbitrarily close to b such that (b, c, a, b_{1}) ∈ O_{4}.

Next, using (iii) again, we choose c_{1} arbitrarily close to c such that (c, b, a, c_{1}) ∈ O_{4}, and still, because O_{4} is open, (b, c_{1}, a, b_{1}) ∈ O_{4}.

At last, using (iii) again, we choose a_{1} arbitrarily close to a such that (a, b, c, a_{1}) ∈ O_{4} and still (b, c_{1}, a_{1}, b_{1}) and (c, b, a_{1}, c_{1}) in O_{4}.

It follows that we have

(a, b, c) ⇝ (c, b, a_{1})
(c, b, a_{1}) ⇝ (a_{1}, b, c_{1})
(b, c_{1}, a_{1}) ⇝ (a_{1}, c_{1}, b_{1}).

The composition rule implies now that (a, b, c) ⇝^{3} (a_{1}, c_{1}, b_{1}).

Step 2: for any (a, b, c) there exists (a_{1}, b_{1}, c_{1}) arbitrarily close to (a, b, c)

such that (a, b, c) ⇝^{3} (b_{1}, a_{1}, c_{1}).

The proof is symmetric: first we notice, using (iii), that we can find c_{1} arbitrarily close to c such that

(c, a, b) ⇝ (b, a, c_{1}).

We choose b_{1}, arbitrarily close to b, such that

(b, a, c) ⇝ (c, a, b_{1})
(c, a, b_{1}) ⇝ (b_{1}, a, c_{1}).

Lastly, we choose a_{1}, arbitrarily close to a, such that

(a, b, c) ⇝ (c, b, a_{1})
(b, a_{1}, c) ⇝ (c, a_{1}, b_{1})
(c, a_{1}, b_{1}) ⇝ (b_{1}, a_{1}, c_{1}).

The composition rule implies the desired statement.

Step 3: for any (a_{1}, b_{1}, c_{1}) close enough to (a, b, c) then (a, b, c) ⇝^{36} (a_{1}, b_{1}, c_{1}).

It suffices to prove that (a, b, c) ⇝^{36} (a, b, c). From the first step, we have that given (a, b, c) there exists (a_{4}, b_{4}, c_{4}) arbitrarily close to (a, b, c) such that

(a, b, c) ⇝^{3} (a_{4}, c_{4}, b_{4}).

Next applying the first step one more time, we can choose (a_{3}, b_{3}, c_{3}) arbitrarily close to (a_{4}, b_{4}, c_{4}) such that

(a_{4}, c_{4}, b_{4}) ⇝^{3} (a_{3}, b_{3}, c_{3}),

and this leads to

(a, b, c) ⇝^{6} (a_{3}, b_{3}, c_{3}).

Actually we can choose (a_{3}, b_{3}, c_{3}) close enough to (a_{4}, b_{4}, c_{4}) such that still

(a, b, c) ⇝^{3} (a_{3}, c_{3}, b_{3}).

At last, we choose thanks to step 2, (a_{2}, b_{2}, c_{2}) arbitrarily close to (a_{3}, b_{3}, c_{3}) such that

(a_{3}, c_{3}, b_{3}) ⇝^{3} (c_{2}, a_{2}, b_{2}),

and this implies

(a, b, c) ⇝^{6} (c_{2}, a_{2}, b_{2}),

As before, we can choose (a_{2}, b_{2}, c_{2}) close enough so that we still have

(a, b, c) ⇝^{6} (a_{2}, b_{2}, c_{2}).

From this last relation, we obtain

(c_{2}, a_{2}, b_{2}) ⇝^{6} (b, c, a).

It follows

(a, b, c) ⇝^{12} (b, c, a),

Hence

(a, b, c) ⇝^{36} (a, b, c).

Step 4: for any a, b, c and any permutation σ we have

(a, b, c) ⇝^{100} (σ(a), σ(b), σ(c)).

This follows easily from the previous steps. Step 5: for any a, b, c there exist an open dense set of d such that

(a, b, c) ⇝^{300} (b, c, d).

Indeed, from hypothesis (ii) of 5.2.1, there exist an open dense set of d such that either (a, b, c) ⇝^{1} (c, b, d) - in which case we are done - or (a, c, b) ⇝^{1} (b, c, d) and we obtain our assertion using step 4 twice.

Final step: for any (a, b, c, d, e, f), we have (a, b, c) ⇝^{1000} (d, e, f). Using step 5 three times, we obtain there exists an open dense set of (u, v, w) such that (a, b, c) ⇝^{900} (u, v, w), hence our conclusion thanks to step 3.

The proof is complete although, obviously, 1000 is not the optimal constant. Also this proof is far too complicated in our case, but one of my hope is to build a map D satisfying (i), (ii), (iii) (iv) and (v) of 5.2.1 for a general negatively curved 3-manifold.⋄

6 Convex surfaces and configuration data

Let’s again N = M/Γ be a compact hyperbolic 3-manifold. In the last section we have built a configuration data associated to that situation, and we can extend that to a measured configuration data in many ways (cf 4.1.4)

We consider now the restricted configuration space B̄_{∞}, associated to that situation with it’s measure μ, invariant under Γ and ergodic under the action of the ideal triangle group F as constructed in 4.2.1.

We turn B̄_{∞} into an ergodic riemannian lamination by a suspension procedure, namely we consider

F = (B̄_{∞} × ℍ^{2})/F,

where F acts as an isometry group on ℍ^{2} and diagonally on B̄_{∞} × ℍ^{2}.

The ergodic and Γ × F -invariant measure μ gives rise to a transversal Γ-invariant and ergodic measure on F that we shall also call μ.

Our aim is now to prove:

Proposition 6.0.2 There exist a continuous leaf preserving map Φ with dense image from F/Γ to N, the space of k-surfaces in N.

This proposition, loosely speaking, explains that our combinatorial construction codes for convex surfaces. As a corollary, we obtain our main theorem

Theorem 6.0.3 Let N = M/Γ be a compact negatively curved 3-manifold whose metric can be deformed through negatively curved metrics to a hyperbolic one. Then there there exist infinitely many mutually singular ergodic transversal measures of full support on N, the space of k-surfaces of N.

6.1 Bent and pleated surfaces

Recall that a ℂℙ^{1}-surface is a surface locally modelled on ℂℙ^{1}.

We shall have to recall facts about (locally convex) pleated surfaces, mostly without demonstrations, especially when dealing with the relation between meausred geodesic laminations and ℂℙ^{1}-structures which has been described by W. Thurston. A useful reference is [7], where H. Tanigawa gives a description and some results about this relation.

The main fact about this construction is the following: to every hyperbolic surface S (maybe non complete) and every measured geodesic lamination μ we can associate • a pleated locally convex surface in the hyperbolic space,

• a 3-manifold B(S, μ), the end of (S, μ),

• a ℂℙ^{1}-surface Σ which is going to be the boundary at infinity of the end.

The map which asssociates to (S, μ) the ℂℙ^{1}-surface Σ is called the Thurston map, we shall denote it by Θ. Notice that since S is not assumed to be complete, this map has no reason to be injective.

6.1.1 An example

For the sake of completeness, we briefly recall the Thurston’s construction in a special case, which will be the one we shall actually need.

Let S be a open subset of ℍ^{2} (maybe non complete) which is the union of totally geodesic ideal polygons. To every edge e of this tiling, we associate a positive number θ_{e} less than π. The data μ consisting of the edges of the tilings and of the assigned positive numbers is a specific example of a geodesic lamination.

We may think of every polygon T as totally geodesically emebedded in ℍ^{3}. Let n_{T} be the exterior normal field along T. Let p_{T} be the map from T ×]0, ∞[ defined by

p_{T} : (x, s) ↦ exp(sn_{T} (x)).

Let P_{T} be the prism over T, i.e. the image of p_{T}.

Let e be an edge of the tiling of S, intersection of two polygons T_{0}^{e} and T_{1}^{e} and considered as a geodesic in ℍ^{3}. The θ_{e}-wedge over e is the closed set delimited by the two half-planes whose boundary is e and forming an angle θ_{e}.

Finally, the end B(S, μ) of (S, μ) is the reunion of all prisms and edges. Notice that there is a canonical isometric local homeomorphism from M_{Σ} to ℍ^{3}.

In this special case, S is isometrically immersed in ℍ^{3} as a pleated surface.

6.1.2 Facts

The following propositions, whose proof follows from results explained in [7], summarizes the property of Thurston’s construction we shall need in the sequel. All these properties relie on the following observation:

Observation 6.1.1 Let S be a hyperbolic surface (maybe non complete) and μ a geodesic lamination, let D be an embedded ℂℙ^{1}-disk in Θ(S, μ), then there is an embedding of the (hyperbolic) half space P in B(S, μ) such that, the boundary at infinity of this embedded P is precisely D.

From this observation, we deduce easily the following results.

Proposition 6.1.2 Let S be a (maybe non complete) hyperbolic surface and μ_{1} a measured geodesic lamination on S. Let μ_{2} be a measured geodesic lamination on ℍ^{2} supported on finitely many geodesics. Assume Θ(ℍ^{2}, μ_{2}) injects by f (as a ℂℙ^{1}-surface) in Θ(S, μ_{1}). Then B(ℍ^{2}, μ_{2}) injects in B(S, μ_{1}) in such a way the associated injection between the boundary at infinity is f.

Proposition 6.1.3 Let M be a ℂℙ^{1}-surface. Then there exists an exhaustion of M by relatively compact ℂℙ^{1}-surfaces M_{i} such that M_{i} = Θ(ℍ^{2}, μ_{i}) where μ_{i} is supported on finitely many geodesics.

Let S be a locally convex immersed surface in ℍ^{3}. Let n be its exterior normal field. We define the end, to be the 3-manifold B_{S} diffeomorphic to S×]0, ∞[ equiped with the hyperbolic metric induced by the immersion (s, t) ↦ exp(tn(s)). In particular, the boundary at infinity S_{∞} of B_{S} is a ℂℙ^{1}-surface.

Proposition 6.1.4 Let S be a locally convex immersed surface in ℍ^{3} such that B_{∞} injects as a ℂℙ^{1}-surface in Θ(ℍ^{2}, μ). Then B_{S} injects in B(ℍ^{2}, μ).

6.2 Tilings and related definitions

We shall denote by T (a, b, c) the ideal triangle in ℍ^{2} whose vertices are a, b and c in ∂_{∞}ℍ^{2} = ℝℙ^{1}.

Recall that we consider ℍ^{2} periodically tiled by ideal triangles. Let’s denote T^{0} the collection of ideal triangles of this triangulation. The set B = ℚℙ^{1} is the set of vertices at infinity of this triangulation (cf 4.0.3).

Notice now that every monotone map g from B = ℚℙ^{1} to ∂_{∞}ℍ^{2} = ℝℙ^{1} defines a tiling by ideal triangles of an open set U_{g} of ℍ^{2}. This triangulation is given by the collection T^{g} of triangles defined by

T^{g} = {T (g(a), g(b), g(c))/T (a, b, c) ∈ T^{0}}.

With these notations we have:

U_{g} = ⋃_{T ∈T^{g}} T.

6.2.1 Pleated surfaces and tilings

We will prove the following immediate proposition:

Proposition 6.2.1 For every f ∈ B̄_{∞}, there exists a unique monotone map g(f) from B to ∂_{∞}ℍ^{2}, a unique map ψ_{f} from U_{g(f)} to ℍ^{3}, such that its restriction to every tile T (a, b, c) is totally geodesic, and the ideal triangle ψ_{f}(T (a, b, c)) has f(a), f(b) and f(c) as vertices at infinity. Furthermore, S_{f}^{0} = ψ_{f} is locally convex and ψ_{f} depends continously on f.

Using this proposition, we introduce the following notations: we shall denote by μ_{f} the geodesic lamination on U_{g(f)} whose support is the set of edges of T^{g}, each edge being labelled with the angle of the two corresponding triangles in ℍ^{3}. Then we shall write B_{f} for B(U_{g(f)}, μ_{f}) and S_{f}^{∞} for Θ(U_{g(f)}, μ_{f}).

Proof: The construction of ψ_{f} is described in the statement of the proposition. The only point to check is the local convexity of S_{f}^{0}. This follows at once from the following observation: three points (a, b, c) at infinity in ℍ^{3} determine an oriented totally geodesic plane P in ℍ^{3}, and the points in ℂℙ^{1} “below” P are precisely those points d such that ℑ(a, b, c, d) < 0. ⋄

6.2.2 Tiling map

Later on, we shall need a technical device, called a tiling map, associated to every element of B̄_{∞}.

Let C^{0}(ℍ^{2}, ℂℙ^{1}) be the space of continuous maps from ℍ^{2} to ℂℙ^{1} with the topology of uniform convergence on every compact set. We use the notations of the previous section 6.1.

The following proposition is obvious.

Proposition 6.2.2 There exists a continuous map ξ

\{ \begin{matrix}B̄_{∞} & → & C^{0}(ℍ^{2}, ℂℙ^{1}) \\ f & ↦ & ξ_{f}. \end{matrix}

which satisfies the following properties

(i) there exists an homeomorphism h_{f} from ℍ^{2} to S_{f}^{∞}, such that ξ_{f} = i_{f} ∘ h_{f}, (ii) for every T (a_{1}, a_{2}, a_{3}) in T^{0}, the map ξ_{f}|_{T (a_{1},a_{2},a_{3})} extends continously

to {a_{1}, a_{2}, a_{3}} in such a way that ξ_{f}(a_{i}) = f(a_{i}),

(iii) for every element γ in F, ξ_{f∘γ} = ξ_{f} ∘ γ.

By definition, ξ_{f} is a tiling map associated to f.

6.3 k-surfaces and asymptotic Plateau problems

We recall definitions and results from [1] that we specialize in the case of ℍ^{3}.

Let S be a locally convex surface immersed in ℍ^{3}. Let ν_{S} be the exterior normal vector field to S. The Gauss-Minkowski (Figure 5) map from S to ∂_{∞}ℍ^{3} is the local homeomorphism n_{S}:

\{ \begin{matrix}S & → & ∂_{∞}ℍ^{3} \\ x & ↦ & n_{S}(x) = exp(∞ν_{S}(x)). \end{matrix}

Figure 5: Gauss-Minkowski map

An asymptotic Plateau problem is a pair (i, U) where U is a surface, and i is a local homeomorphism from U to ∂_{∞}ℍ^{3}. A k-solution to an asymptotic Plateau problem (i, U), is a k-surface S immersed in ℍ^{3}, such that there exists a homeomorphism g from U to S such that i = n_{s} ∘ g.

We proved (theorem A of [1]) that there exists at most one solution of a given asymptotic Plateau problem. We also proved (theorem E of [1]) that if (i, U) is an asymptotic Plateau problem, and if O is a relatively compact open set of U, then (i, O) admits a solution.

We need the following proposition which uses the notations of section 6.1 Proposition 6.3.1 Let f be an element of B̄_{∞}, then the asymptotic Plateau problem (i_{f}, S_{f}^{∞}) admits a k-solution.

Proof: Using proposition 6.1.3, we have an exhaustion of S_{f}^{∞}, by ℂℙ^{1}- surfaces M_{i}, such that M_{i} = Θ(ℍ^{2}, μ_{i}) where μ_{i} is supported on finitely many geodesics. Let B_{i} = B(ℍ^{2}, μ_{i}). According to 6.1.2, we have

B_{i} ⊂ B_{i+1} ⊂ B_{f}.

Since M_{i} is relatively compact in S_{f}^{∞}, there exists, according to theorem E of [1], a k-solution Σ_{i} to the asymptotic Plateau problem defined by M_{i}. According to proposition 6.1.4,

B_{Σ_{i}} ⊂ B_{i} ⊂ B_{f}.

Let now

W = ⋃_{i∈ℕ}B_{Σ_{i}}.

We wish now to prove that ∂W the boundary of W is a k-surface solution of the asymptotic Plateau problem defined by S_{f}^{∞}.

First we should notice that since B_{Σ_{i}} ⊂ B_{f}, there exists a constant A just depending on f, such that evey ball Σ_{i}(x, A) of ecenter x and radius A in Σ_{i}, when considered immersed in ℍ^{3}, is a subset of the boundary of a convex set. Hence, according to lemma 5.4.(iii) of [2], there exists a constant C such that we have, if H is the mean curvature of Σ_{i} and dσ the area element:

∫_{Σ_{i}(x,A)} Hdσ ≤ C.

Let now x_{i} be a point in Σ_{i}. Assume the sequence {x_{i}}_{i∈ℕ} converges in the metric completion of B_{f} to a point x_{0}. We conclude from theorem D of [2] that {(Σ_{i}, x_{i})}_{i∈ℕ} converges smoothly to a pointed k-surface (Σ_{∞}, x_{0}). It follows that x_{0} is in the interior of B_{f}. Indeed, let ∂B_{f} be the boundary of the metric completion of B_{f}. Notice that every point of ∂B_{f} is included in an open geodesic segment drawn on ∂B_{f}. If x_{0} belongs to ∂B_{f}, it would follow that the corresponding open segment is actually drawn on Σ_{∞} and this is impossible.

This argument finally shows that ∂W is a k-surface, and that every half infinite geodesic joining a point of ∂B_{f} to a point of S_{f}^{∞} intersects ∂W. The conclusion follows. ⋄

6.4 Construction of the map Φ

In this section, we summarize the previous sections and build a continous map Φ from F to N.

Let f ∈ B̄_{∞}. Let ξ_{f} the tiling map of f ( cf 6.2.2 ). Let Σ_{f} be the k-solution of the asymptotic Plateau problem (i_{f}, S_{f}^{∞}) ( cf 6.3.1 ). Let n_{f} be the Gauss-Minkowski map of Σ_{f}. We define Φ by

Φ([f, x]) = (Σ_{f}, n_{f}^{−1}(ξ_{f}(x))).

Continuity follows from the uniqueness of the solution of an asymptotic Plateau problem.

6.5 Density of the image of Φ

The only point left to be proved in proposition 6.0.2 is the density of the image of Φ.

We start with an observation. Let S be a compact surface, S̃ its universal cover. Let μ_{1} be a measured lamination on S supported on finitely many geodesics. Assume the weight of every geodesic is strictly less than π. Then, from the construction explained above we deduce that Θ(S̃, μ_{1}) lies in the image of Φ.

According to 2.2.1, the union of compact leaves of N is dense. It therefore suffices to prove that every compact k-surface belongs to the closure of the image of Φ.

Let S be such a compact k-surface in N. The underlying surface admits a ℂℙ^{1}-structure induced by the Minkowski-Gauss map. According to Thurston’s parametrisation theorem [7], such a surface is of the form Θ(S, μ_{0}) for a certain measured lamination μ_{0}. We proved, using different words (Corollary 1 of [3]) that the map which associates to every ℂℙ^{1}-structure on a compact surface, the k-surface solution of the corresponding asymptotic Plateau problem, is continuous. To complete our proof, we just have to remark that the set of measured geodesic laminations with finite support and such that the weight of every geodesic is strictly less than π is dense in the space of all measured geodesic laminations.

7 Conclusion

It remains to glue altogether the main propositions of the previous sections to obtain the proof of our main result.

From the stability property, it suffices to build a transverse invariant measure of full support on N, whenever N has constant curvature. Let Γ = π_{1}(N), and F = P SL(2, ℤ).

We consider the restricted configuration space B̄_{∞}, subset of the space of maps from ℚℙ^{1} to ℂℙ^{1}, as defined in paragraph 4.0.5, and associated to the Markov configuration data (as defined in 4.1) coming from the complex crossratio on ∂_{∞}Mℍ^{3}, according to section 5 and proposition 5.2.1.

We can turn now this configuration data into a measured one as shown in paragraph 5.1.

Thanks now to the main result of section 4, theorem 4.2.1, we obtain a finite F -invariant ergodic measure of full support on B̄_{∞}/Γ. Here, the action of F is by right composition.

Furthermore, choices of mutually singular equivariant families of measures lead to mutually singular transversal measure.

Next, we suspend the action of F on B̄_{∞}. Namely, namely we consider the riemaniann lamination.

F = (B̄_{∞} × ℍ^{2})/F, ,

where F acts as an isometry group on ℍ^{2} and diagonally on B̄_{∞} × ℍ^{2}.

The finite ergodic and F -invariant measure on B̄_{∞}/Γ gives rise to a transversal Γ-invariant and ergodic measure on F that we call μ.

Finally proposition 6.0.2 defines a map Φ from F/Γ to N, which is leaf preserving, continuous with a dense image. Therefore, we can pushforward μ using Φ to obtain a transversal ergodic finite measure of full support.

References

[1] F. Labourie Un lemme de Morse pour les surfaces convexes. A paraître à Invent. Math.

[2] F. Labourie Immersions isométriques elliptiques et courbes pseudoholomorphes. J. Diff. Geom. 30 (1989) pp 395-44.

[3] F. Labourie Surfaces convexes dans l’espace hyperbolique et ℂℙ^{1}- structures. J. Lond. Mat. Soc. 45 (1992)pp 549-565.

[4] F. Labourie Problèmes de Monge-Ampères, courbes pseudo-holomorphes et laminations G.A.F.A. 7 (1997), pp 496-534.

[5] F. Ledrappier Structure au bord à des variétés à courbure négative. Séminaire de théorie spectrale et géometrie (1995) pp 97-122

[6] D. Sullivan The Density at Infinity of a Discrete Group of Hyperbolic Motions. Public. Math. I.H.E.S. 50 (1979) pp 171-202

[7] H. Tanigawa Grafting, Harmonic Maps and Projective Structures on Surfaces. J. of Diff. Geom. 47 (1997) pp 399-419

François Labourie Topologie et Dynamique Université Paris-Sud F-91405 Orsay (Cedex)