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The “2-dimensional” analog of the unit tangent bundle with the geodesic flow is a “space of pointed k-surfaces”, which can be considered as the space of germs of complete k-surfaces passi

Annals of Mathematics

Random k-surfaces

By Franc¸ois Labourie

Random k-surfaces

By Franc ¸ ois Labourie *

Abstract

Invariant measures for the geodesic flow on the unit tangent bundle of

a negatively curved Riemannian manifold are a basic and well-studied ject This paper continues an investigation into a 2-dimensional analog of this

sub-flow for a 3-manifold N Namely, the article discusses 2-dimensional surfaces immersed into N whose product of principal curvature equals a constant k between 0 and 1, surfaces which are called k-surfaces The “2-dimensional”

analog of the unit tangent bundle with the geodesic flow is a “space of pointed

k-surfaces”, which can be considered as the space of germs of complete k-surfaces passing through points of N Analogous to the 1-dimensional lam-

ination given by the geodesic flow, this space has a 2-dimensional lamination

An earlier work [1] was concerned with some topological properties of chaotictype of this lamination, while this present paper concentrates on ergodic prop-erties of this object The main result is the construction of infinitely manymutually singular transversal measures, ergodic and of full support The novelfeature compared with the geodesic flow is that most of the leaves have expo-nential growth

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, and

will be 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

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

When dealing with ordinary differential solutions, one is led 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 mimic 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 cisely, we proved the space just described can be compactified by a space,

pre-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

byN , 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

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 us just

notice that since many leaves are hyperbolic (cf Theorem 2.4.1), one cannotproduce transversal measures by Plante’s argument Our strategy will be to

“code” by a combinatorial model on which it will be easier to build transversalmeasures

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, and state some of its properties

proved in [1]

3 Transversal measures We present our main result, Theorem 3.2.1,

dis-cuss other constructions and questions, and sketch the main construction

4 A combinatorial model In this section, we explain a combinatorial struction Starting from configuration data, we consider “configuration

con-spaces” These are spaces of mappings from QP1 to a space W We produce invariant and ergodic measures under the action of PSL(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 CP1

6 Convex surfaces and configuration data We prove here that the

combi-natorial model constructed in the previous section actually codes for the

space of k-surfaces.

7 Conclusion We summarise our constructions and prove our main result,

Theorem 3.2.1

I would like to thank W Goldman for references about CP1-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 detail 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 expounded 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 tangent 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 summarise 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 k-surface 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 ¯ S immersed by ¯ f satisfying f (S) = ¯ f ( ¯ S), there exists a covering π : ¯ S → Σ such that ¯ f = 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 tangent 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 Alternatively, we can think of it as the space of germs

of u-complete k-surfaces, or by analytic continuation as the space of ∞-jet of complete k-surfaces If we denote by J k (2, U N ) the finite dimensional manifold

of k-jets of surfaces in U N , N , can be seen as a subset of the projective limit

J ∞ (2, U N ); this point of view is interesting, but one should stress it seems hard to detect from the germ (or the jet) if a k-surface is complete or not.

The space N inherits a topology coming from the topology of pointed

immersed 2-manifolds in the unit tangent bundle (cf Section 2.3 of [1]);

alter-natively, this topology coincides with the topology induced by the embedding

in J ∞ (2, U N ).

We describe now the structure of a lamination of N First notice that each k-surface (or tube) S0 determines a leaf L S0 defined by

L S0 ={(S0, x)/x ∈ S0}.

We proved in [4] thatN is compact Furthermore, the partition of N into

leaves is a lamination, i.e admits a local product structure Notice thatN has

two parts:

(1) a dense set which turns out to be infinite dimensional, and which truly

consists of k-surfaces,

(2) a “boundary” consisting of the union of tubes; this “boundary” is closed,

finite dimensional, and is an S1 fibre bundle over the geodesic flow.Therefore, in some sense, N is an extension of the geodesic flow To enforce

this analogy, one should also notice that the 1-dimensional analogue, namely

the space of curves of curvature k in a hyperbolic surface, is precisely the

geodesic flow

2.3 Examples of k-surfaces In order to give a little more flesh to our discussion, we give some examples of k-surfaces.

Equidistant surfaces to totally geodesic planes in H3 If we suppose N

is of constant curvature, or equivalently that the universal cover of N is

H3, a surface equidistant to a geodesic plane is a k-surface It follows the

subset of N corresponding to such k-surfaces (with an orientation) in N

is identified with the unit tangent bundle of the hyperbolic space U N =

S1\PSL(2, C)/π1(N ) The lamination structure comes from the right action

homeomor-Σ homeomorphic by φ to S, such that φ ◦ ι = NΣ For instance, an uisdistant surface, as discussed in the previous paragraph, is the solution ofthe asymptotic Plateau problem given by the injection of a ”circular” disc in

eq-∂ ∞H3 = CP1 We proved in [1] that there exists at most one solution to agiven asymptotic problem Furthermore many asymptotic problems admit so-lutions, and in Section 6.3 we explain some of the results obtained in [1] The

general heuristic idea to keep in mind is that, most of the time, an totic Plateau problem has a solution, at least as often as a Riemann surface

asymp-is hyperbolic instead of being parabolic We give three examples from [1] In

all these examples M is assumed to be a negatively curved 3-manifold with

bounded geometry, for instance with a compact quotient

Theorem C If (S, ι) is an asymptotic Plateau problem such that

∂ ∞ M \ i(S) contains at least three points then (S, ι) admits a solution.

Theorem D Let Γ be a group acting on S, such that S/Γ is a compact surface of genus greater than 2 Let ρ be a representation of Γ in the isometry group of M If ι satisfies

∀γ ∈ Γ, ι ◦ γ = ρ(γ) ◦ ι, then (S, ι) admits a solution.

Theorem E Let (U, ι) be an asymptotic Plateau problem Let S be a relatively compact open subset of U , then (S, ι) admits a solution.

2.4 Dynamics of the space of k-surfaces The main Theorem of [1] which

we quote now shows that N , with is lamination considered as a dynamical

system, enjoys the chaotic properties of the geodesic flow:

Theorem 2.4.1 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-surfaces 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 ¯ h is close to h, then there exists a homeomorphism from N h to N¯h

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 thisspace, 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

k-surfaces is just the unit tangent 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 S1-bundle over the unit tangent bundle, where the fibre over u is

the set of unit vectors orthogonal to 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 now complete 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 minus 1 Let

k ∈ ]0, 1[ be a real number Let N be the space of k-surfaces of N.

3.1 First examples Let us first show some simple examples of natural

transversal measures on N The first three are ergodic They all come from

the existence of natural finite dimensional subspaces in N

- Dirac measures supported on closed leaves By Theorem 2.4.1(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

curva-ture Then, the space of oriented totally geodesic planes carries a

trans-verse invariant measure Indeed, the Haar measure for SL(2, C)/π1(N ) is invariant under the SL(2,R) action But every oriented totally geodesic

plane gives rise to a k-surface, namely the one equidistant to the geodesic

plane This way, we can construct an ergodic transversal measure onN , when N has constant curvature Its support is finite dimensional.

- Measures on spaces of ramified coverings We sketch briefly here a

con-struction yielding transversal, but nonergodic, measures onN 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 topological ramified covering of

Σ into ∂ ∞ M Let S π be the set of singular points of π and s π its

car-dinal Let S be a set of extra marked points of cardinal s Assume 2g + s π + s ≥ 3, so that the surface with s π + s deleted points is hyper-

bolic 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 unique solution to the asymptotic Plateau

problem (as described in Paragraph 6.3) represented by (π, Σ \ (S π ∪ S)).

To be honest, this last result is not stated as such in [1] However, one

can prove it using the ideas contained in the article Let now [π] be

the space of ramified coverings equivalent up to homeomorphisms of the

target to π, modulo homeomorphisms of Σ More precisely, let H be the group of homeomorphism of ∂ ∞ M , let F be the group of homeomor- phism of Σ preserving the set S ∪ S π Notice that bothH and H act on

C0(Σ, ∂ ∞ M ) Then

[π] = H.π.F/F.

The group π1(N ) acts properly on [π], and explicit invariant measures

can be obtained using equivariant families of measures (cf Section 5.1.1)

and configuration spaces of finite points Since [π]/π1(N ) is a space of

leaves of N , this yields transversal measures on this latter space.

None of these examples has full support, and they all have finite sional support So far, apart from these and the construction I will present inthis article, I do not know of other examples of transversal measures which areeasy to construct

dimen-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 minus 1 Assume the metric on N can be deformed, through negatively curved metrics, to a constant curvature 1 Then the space of k-surfaces admits in- finitely many mutually singular, ergodic transversal finite measures of full sup- port.

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 2.4.1, in order to prove our main result, it suffices to show theexistence of transversal ergodic finite measures of full support in the case ofconstant 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

Here, ν(x, y, z) is assumed to be of full support, and to fall in the same sure class, independently of (x, y, z) Such maps are easily obtained through equivariant families of measures (also described in F Ledrappier’s article [5]

mea-as Gibbs current, crossratios etc.) and a barycentric construction mea-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,

nonrigorous, 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 (§6) is to associate to (almost) every k-surface a locally convexpleated 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 QP1

to CP1 This is the aim of Sections 5and 6 IdentifyingQP1 with the space of connected components ofH2 minus atrivalent tree, we build invariant measures on this space of maps as projectivelimits of measures on finite configuration spaces of points onCP1 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

construction only works in the case of constant curvature manifolds, extending

to other cases through the stability Of course, it would be more pleasant toobtain transversal measures without any restriction on the metric Some parts

of the construction do not require any hypothesis on the metric, and we tried

to keep, sometimes at the price of slightly longer proof, the proof as general aspossible

3.5.2 Equidistribution of closed leaves Keeping in mind the analogy with

the geodesic flow and the construction of the Bowen-Margulis measure, wehave 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 nof 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 π1 surjects onto a given group This is a whole range ofquestions on which I am afraid to say I have no hint of answer However, theconstructions in this article should be related to equidistribution of ramifiedcoverings 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

(4.0.3), which are, roughly speaking, spaces of infinite sets of points on a

topological space W , associated to configuration data (4.1) Our main result

is Theorem 4.2.1 which defines invariant ergodic measures of full support onthese spaces, starting from measures defined on configuration data as in 4.1.2.One may think of these restricted infinite configuration spaces as analogues

of subshifts of finite type, where the analogue of the Bernoulli shift is thespace of maps ofQP1 (instead ofZ) into a space W with the induced action of PSL(2,Z) We call this latter space the infinite configuration space as described

in the first paragraph, as well as related notions The role of the configuration data is that of local transition rules.

4.0.1 The trivalent tree We consider the infinite trivalent tree T , with a

fixed cyclic ordering on the set of edges stemming from any vertex tively we can think of this ordering as defining a proper embedding of the tree

Alterna-in the real planeR2, such that the cyclic ordering agrees with the orientation.Another useful picture to keep in mind is to consider the periodic tiling of thehyperbolic plane H2 by ideal triangles, 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 =Z2∗ Z3 = PSL(2,Z)

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

We now consider the set B of connected components of H2\ T In our tiling picture this set B is in one-to-one correspondence 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 QP1 and this identification agrees

with the action of PSL(2,Z)

4.0.2 Quadribones, tribones Every edge of T defines a set of four points

in B, namely the connected components of H2\ T that touch this edge; 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 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

ori-ented sets When our quadribone is given by (a, b, c, d) the two corresponding tribones are (a, b, c) and (d, c, b).

a

b

c

d a

c

b

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

4.0.3 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 fromB ∞ to W3 (resp W4) given respectively by f

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, O3, O4) defines

(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 nth-symmetric group σ n Let σ+

n be the subgroup of σ n

of signature +1 Let λ3 = σ+3, and λ4 ⊂ σ+

4 be the subgroup generated by

(a, b, c, d) n = {(x1, , x n)|∃i = j, xi = x j } Assume

furthermore that:

(c) O n are open λ n ×Γ-invariant subsets of W n \∆ n, on which Γ acts properly

(d) p(O4) = O3, where p is the projection from W4 to W3 defined by

In 4.3.4, this property will have a geometric consequence

4.1.2 Measured configuration data Our next goal is to associate sures to this situation We shall say (W, Γ, O3, O4, µ3, µ4) is a (3,4)-measured configuration data if:

mea-(f) µ n are λ n × Γ-invariant measures, such that p ∗ µ4 = µ3

(g) The pushforward measures on O n /Γ are probability measures.

We shall say that the measured configuration data are regular if they

satisfy the following extra condition:

(h) The measure µ4 is in the measure class of IO4m ⊗ m ⊗ m ⊗ m where

m is of full support in W It follows that µ3 is in the measure class of

IO3m ⊗ m ⊗ m.

We also say two regular measured configuration data (W, Γ, O3, O4, µ3, µ4)

and (W, Γ, O3, O4, ¯ µ3, ¯ µ4), defined on the same configuration data, are mutually singular if µ3 and ¯µ3 are mutually singular

4.1.3 Remarks (i) From disintegration of measures, it follows from

the hypotheses (f) and (g) that for µ3-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 W4:

(ii) Conversely, there is a way to build regular measured configuration

data starting from configuration data (W, Γ, O3, O4), if we assume that O4 is invariant under σ4+.

Assume we have:

- a Γ-invariant measure ¯µ3 on W3 in the measure class of IO3m ⊗

m ⊗ m where m has full support, such that the pushforward on O3/Γ is

a probability measure;

- a Γ-equivariant map ¯ν:



O3 → P m (W ) (a, b, c) (a,b,c)

where P m (W ) is the set of finite Radon measures on W in the measure class of m.

Then, we can build µ3and µ4which will fulfil the requirements of the definition.Let us describe the procedure:

Firstly, we define a probability measure ¯µ4 on O4 to be proportional to

Secondly, we average ¯µ4 using the group σ4+ and obtain a finite measure

µ4 on O4/Γ, and we ultimately take µ3 = p ∗ µ4

It is routine now that µ3 and µ4 defined this way satisfy our needs.Furthermore, if ¯µ3 has full support in O3 as well as ν (a,b,c) for ¯µ3-almost

every (a, b, c) in W3, then µ3 and µ4 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 PSL(2,C);

- W = CP1 with the canonical action of Γ; it is a well-known fact that Γacts properly on

U n={(x1, , x n)∈ (CP1)n | x i = x j if i = j}.

Actually Γ acts properly on U3

- O3= U3,

- O4 is the set of points whose crossratios have a nonzero imaginary part;

it will satisfy hypothesis (e) for N = 1000 (cf 5.2).

This is Markov configuration data and furthermore in this specific

situa-tion O4 is invariant under σ4+ We will explain in subsection 5.1 how to attachmeasures to this situation, and discuss the case of general negatively curved3-manifolds

4.1.5 Final remark Even though we only wish to study this specific class

of examples, it is a little more comfortable to work in a more general setting,since very little of the geometry is used at this stage

4.2 Restricted infinite configuration spaces and the main result Let now (W, Γ, O i) be (3,4)-configuration data (cf 4.1)

We define the restricted infinite configuration space of W to be the subset

¯

B ∞ of B ∞, consisting of those maps such that the image of every tribone lies

in O3, and the image of every quadribone is in O4

¯

B ∞={f ∈ B ∞ | for all tribone t, quadribone q, f(t) ∈ O3, f (q) ∈ O4}.

Let alsoB0

∞be the open set of the infinite configuration space such that the

image of at least one tribone lies in O3 Let us call this subset the nondegenerate 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 (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 ¯ B ∞ is of full measure and µ has full support on it provided the data are regular ;

(ii) The pushforward of µ on B0

∞ /Γ is finite, where B0

∞ is the nondegenerate

infinite configuration space;

(iii) Given any tribone or quadribone, the pushforward of µ by the associated maps on W3 and W4 is our original µ3, µ4;

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

(v) If the configuration data are Markov and regular, then the pushforward

of µ on B0

∞ /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 (3,4)-configurationdata We shall use the notation and definitions of the preceding sections

Also in our constructions, for every (x, y, z) ∈ O3, we shall denote by

ν (x,y,z) the probability measure coming from the disintegration of µ4 over µ3

it will be called v-connected if furthermore e(A) contains v In other words

a connected subset of B is the union of the connected components of H2\ T 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 union of fewer 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 denote:

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

- ¯W(A) the set of maps such that the image of every tribone of A lies in

O3, and the image of every quadribone is in O4; if A is finite, ¯ W(A) is

an open set on which Γ acts properly Again, ¯W(B) = ¯ B ∞

4.3.3 Finite construction We can now prove:

Proposition 4.3.1 Let A be a finite v0-connected subset of B Then, there exists a Radon measure µ A,v0 on ¯ W(A) enjoying the following properties: (i) The pushforward of µ A,v0 on ¯ W(A)/Γ is finite; it is of full support if the data are regular ;

(ii) Let t0 be the tribone corresponding to the vertex v0; also let t0 be the associated map from A to W3; then t ∗ µ A,v0 = µ3.

(iii) Let q be a v0-connected quadribone; assume q ⊂ A; let q be the associated map from A to W4; then q ∗ µ A,v0 = µ4.

(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

cor-One should notice that the listed properties define µ A,v0 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 ourmeasure

Proposition 4.3.3 Assume the configuration data are Markov and ular There exists an integer P , such that if A0 and A1 are two P -disconnected subsets of a finite set C ⊂ B, then (p0, p1) µ C and p0

reg-∗ µ C ⊗p1

∗ 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 section

4.3.4 Proof of Proposition 4.3.1 We introduce first some notation with spect to a vertex v By definition B n (v) will denote the union of all v-connected n-bones; also, for any subset A of B, we put A n (v) = B n (v) ∩ A.

re-For the moment, we will work with a fixed v0and will omit the dependence

in v0 in the notation for the sake of simplicity; in particular A n = A n (v0) Wewill construct this measure by an induction procedure

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

Let us 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={a1, , a q }.

In particular,W(A n+1 \A n ) is identified with W q Let T A

n =∪ i=q i=1 t ai We have

a natural restriction map

Next, notice the following fact Let f ∈ ¯ W(A n) and ¯W f (A n+1) be the

fibre, over f , of the restriction map We use the identification

W(A n+1) =W(A n+1 \ A n)× W(A n ).

Then, ¯W f (A n+1 ) has full measure for ν f A,n ⊗ δ f

We can now define our measure on ¯W(A n+1) by an induction procedure:

- ¯W(A0) is identified with O3 using t0; we define µ A0 = (t −10 ) µ3;

- Assuming by induction that µ An is defined on W(A n) such that ¯W(A n)has full measure, we set

From the previous observation, we deduce that ¯W(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

By construction, using the obvious identifications, we have

(∗) µ C,p = µ A,p , for p < p0,

µ C,p =

W(Ap)(δ f ⊗ ν f (q \a) )dµ A,p (f ), for p = p0.

To conclude the proof of (iv), it remains to prove (∗) for p > p0 By induction,

this follows from the fact that, for p > p0, T p A = T p C We check this step bystep By definition,

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 this ever 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

when-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 µ4 under

the permutation (a, b, c, d)

4.3.6 A consequence of hypothesis (e) of 4.1 Using the previous notation,

we have:

Proposition 4.3.4 Assume the configuration data are Markov Then, there exists an integer P , such that if A0 and A1 are connected and P -discon- nected, and if C is a connected set that contains both, then

(p0, p1)( ¯W(C)) = ¯ W(A0)× ¯ W(A1).