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Measured flat geodesic laminations

Since their introduction by Thurston, measured geodesic laminations on hyperbolic surfaces occur in many contexts. In [Mor], we have introduced a notion of flat laminations on surfaces endowed with a half-translation structure (that is a singular flat surface with holonomy {+/-Id}, similar to geodesic laminations on hyperbolic surfaces. Here is a sequel to this article that aims at defining transverse measures on flat laminations similar to transverse measures on hyperbolic laminations, taking into account that two different leaves of a flat lamination may no longer be disjoint. One aim of this paper is to construct a tool that could allow a fine description of the space of degenerations of half-translation structures on a surface. In this paper, we define a nicer topology than the Hausdorff topology on the set of measured flat laminations and a natural continuous projection of the space of measured flat laminations onto the space of measured hyperbolic laminations, for some arbitrary half-translation structure and hyperbolic metric on a surface. We prove in particular that the space of measured flat laminations is projectively compact.

The main aim of this article is to propose a definition of transverse measure on the (geodesic) flat laminations, introduced in [Mor], on a surface endowed with a half-translation structure, that is a flat metric with conical singular points and with holonomies in {± Id}. Although the definition is inspired of transverse measures on the geodesic laminations on hyperbolic surfaces (see for instance [Bon1]), the extension is non trivial, notably since the images of two leaves of a flat lamination are not necessarly disjoint. We will call measured flat lamination a flat lamination endowed with a transverse measure. We will define a sufficiently fine topology on the set of measured flat laminations and we will construct a (non injective) natural continuous projection of the space of measured flat laminations onto the space of measured hyperbolic laminations, for any choice of a half-translation structure and of a (complete) hyperbolic metric on a surface, and we will describe its lack of injectivity. This allows to consider the measured flat laminations that are the limits of some sequences of periodic local geodesics, in the projectivized space of measured flat laminations. This in turn could yield a better understanding of the degenerations of half-translation structures on a surface, as initiated in [DLR]. In particular, as spaces of measures are suitable for analysis tools (distributions as in [Bon1]), this could allow a finer study of the boundary of the space of half-translation structures that we will develop in a subsequent work. We refer to [?] for a survey of [Mor] and of this work.
We use the same notations as in [Mor] : let Σ be a compact, connected, orientable surface, without boundary (to simplify in the introduction). A half-translation structure (or flat structure with conical singularities and holonomies in {± Id}) on Σ is the data consisting in a (possibly empty) discrete set of points Z of Σ and of a Euclidean metric on Σ − Z with conical singular points of angles of the form kπ, with k ∈ N and k 3 at each point of Z, such that the holonomy of every piecewise C 1 loop of Σ − Z is contained in {± Id}. We refer to Section 2.1 notably when the boundary is not empty.
The surface Σ endowed with a half-translation structure is a complete and locally CAT(0) metric space (Σ, d). Let p : ( Σ, d) → (Σ, d) be a locally isometric universal cover. Two local geodesics , of (Σ, d), defined up to changing the origins, are said to be interlaced if they have some lifts , in Σ such that the image of intersects both complementary components of (R) in Σ, and conversely. A local geodesic is said to be self-interlaced if it is interlaced with itself. We endow the set of oriented, but non parametrized, local geodesics of (Σ, d) with the quotient topology of the compact-open topology for the action by translations on the parametrizations, of R on the parametrized local geodesics, that is called the geodesic topology.
Definition 1 [Mor,Déf. 2.2] A (geodesic) flat lamination on (Σ, d) is a non empty set Λ of complete local geodesics of (Σ, d), defined up to changing origin, whose elements are called leaves, such that : • the leaves of Λ are non self-interlaced and pairwise non interlaced ; • Λ is invariant by changing the orientations of the leaves ; • Λ is closed for the geodesic topology. We will call support of Λ the union of the images of the leaves of Λ.
New phenomenons appear in flat laminations compared with hyperbolic ones : the images of two leaves are generally not disjoint, the flat laminations are not determined by their supports (uncountably many flat laminations can have the same support), the cylinder components (see Section 2.2) may have uncontably many leaves. Finally, there are three types of minimal components of a flat lamination on a compact surface (periodic leaf travelled in both orientations, minimal component of recurrent type or of finite graph type, see Theorem 4 for a complete statement). Compared with hyperbolic laminations, the main difficulty to define transverse measures on flat laminations is that the images of the leaves are not necessarly disjoint and that the support does not determine the lamination. Hence, we no longer define the transverse measure as a family of measures on the images of the arcs transverse to the lamination, but as a family of measures on the sets of local geodesics that intersect them transversally, and we have to refine the notion of invariance by holonomy of these families of measures.
In the first section, we define the measured flat laminations and we endow their set with a topology. In the second one, we define the preimage of a measured flat lamination in a cover. In the third one, we define a homeomorphism between the space of measured flat laminations on a compact surface endowed with a fixed half-translation structure and the space of Radon measures on the set of geodesics (defined up to changing origin) of a locally isometric universal cover, that are invariant by the covering group action and whose supports are some flat laminations. In the fourth one, we define a proper, surjective, continuous map from the space of measured flat laminations to the space of measured hyperbolic laminations, for a fixed complete hyperbolic metric with totally geodesic boundary, and we characterize its lack of injectivity. In the fifth one, we define the intersection number between a measured flat lamination and a free homotopy class of closed curves. In the last one, we define the tree associated to a measured flat lamination together with the universal covering group action on it. These tools should be useful to do analysis on the space of degenerations of half-translation structures on surfaces, and we plan to develop this in future work (see [?]). Acknoledgement : I want to thank Frederic Paulin for many advices and corrections that have deeply improved the redaction of this paper.

Definitions.
In this section, we recall the definition and some properties of half-translation structures on surfaces and the definition of flat laminations introduced in [Mor]. Then, we define measured flat laminations and we endow their set with a topology.

Half-translation structures.
As in [Mor], in the whole paper, we will use the definitions and notation of [BH] for a surface endowed with a distance (Σ, d) : (locally) CAT(0), δ-hyperbolic,... Notably, a geodesic (resp. a local geodesic) of (Σ, d) is an isometric (resp. locally isometric) map : I → Σ, where I is an interval of R. It will be called a segment, a ray or a geodesic line of (Σ, d) if I is respectively a compact interval, a closed half line (generally [0, +∞[) or R. If there is no precision, a geodesic is a geodesic line. A germ of geodesic ray, or simply a germ, is an equivalence class of locally geodesic rays for the equivalence relation r 1 ∼ 0 r 2 if r 1 and r 2 coïncide on a non empty initial segment that is not reduced to a point. Similarly, the relation r ∼ ∞ r if there exist T, T > 0 such that r(t + T ) = r (t + T ) for all t 0, is an equivalence relation on the set of subrays of a local geodesic. An equivalence class for this equivalence relation is called an end (in the sense of Freudhental) of a local geodesic. A local geodesic has two ends. We call geodesic topology the compact-open topology on the set G d of local geodesics for the distance d or the quotient topology of this topology by the action by translations of R at the source, on the set [G d ] of local geodesics defined up to changing origin. The quotient map from G d to [G d ] will be denoted by g → [g], and if F is a subset of G d , we will denote by Let Σ be a connected, orientable surface, with (possibly empty) boundary. Assume that Σ is endowed with a Euclidean metric on Σ − Z, where Z is a discrete subset of Σ. If the holonomy of every piecewise C 1 loop in Σ − Z is contained in {± Id}, two vectors v 1 and v 2 tangent to Σ are said to have the same direction if v 2 is the image of ±v 1 by holonomy along a piecewise C 1 path in Σ − Z between the basepoints of v 1 and v 2 . This definition does not depend on the choice of this path. A piecewise C 1 path or union of paths is said to have constant direction, if all its tangent vectors, at the points in Σ − Z, have the same direction.
Definition 2 A half-translation structure (or flat structure with conical singularities and holonomies in {± Id}) on a surface Σ is the data of a (possibly empty) discrete subset Z of Σ and a Euclidean metric on Σ − Z with conical singularity of angle k z π at each z ∈ Z, with k z ∈ N, k z 3 if z ∈ Z − Z ∩ ∂Σ and k z 2 if z ∈ Z ∩ ∂Σ, such that the holonomy of every piecewise C 1 loop in Σ − Z is contained in {± Id} and such that the union of the boundary components has constant direction.
We will denote by [q] a half-translation structure on Σ, with q a holomorphic quadratic differential on Σ (see [Mor,§ 2.5] for an explanation of the notation and to [Str,Def. 1.2 p. 2] for a definition of a holomorphic quadratic differential in the case where the boundary is non empty). A half-translation structure defines a geodesic distance d on Σ that is locally CAT(0). We will call local flat geodesics the local geodesics of a half-translation structure. A continuous map : R → Σ is a local flat geodesic if and only if it satisfies (see [Str,Th. 5.4 p.24] and [Str,Th. 8.1 p. 35]) : for every t ∈ R, • if (t) does not belong to Z, there exists a neighborhood V of t in R such that |V is a Euclidean segment (hence, |V has constant direction) ; • if (t) belongs to Z −Z ∩∂Σ, then the two angles defined by the germs of ([t, t+ε[) and (]t−ε, t]), with ε > 0 small enough, measured in both connected components of U − (]t − ε, t + ε[), with U a small enough neighborhood of (t), are at least π.
Let Σ be a connected, orientable surface with (possibly empty) boundary. Let [q] be a halftranslation structure and let m be a hyperbolic metric with totally geodesic boundary on Σ. Let p : Σ → Σ be a universal cover of covering group Γ Σ , let [ q] be the unique half-translation structure and let m be the unique hyperbolic metric on Σ such that p : ( Σ, [ q]) → (Σ, [q]) and p : ( Σ, m) → (Σ, m) are locally isometric. To be consistent with [Mor], and notably to be allowed to use [Mor,Rem. 2.9], we will always assume that Σ is a cover (possibly trivial) of a compact surface whose Euler characteristic is negative. In particular, if d is the distance defined by [ q] or m, according to the theorem of Cartan-Hadamard, the metric space ( Σ, d) is complete, CAT(0), and δ-hyperbolic, with δ > 0. Furthermore, there exists a unique Γ Σ -equivariant homeomorphism between the boundaries at infinity of Σ for the two metrics, thank to which we identify them. Let ∂ ∞ Σ denote this boundary at infinity and ∂ 2 [Mor,§ 2.3], we have given a very global definition of interlaced local geodesics, in a locally CAT(0), complete, connected metric space, whose boundary at infinity of a universal cover is endowed with a (total) cyclic order. Here, we only recall the specific definition in the case of connected, orientable surfaces, endowed with a complete locally CAT(0) metric. Since ( Σ, d) is CAT(0), the intersection of the images of two geodesics of ( Σ, d) is connected, possibly empty. If is a geodesic of ( Σ, d), since (R) is not necessarly disjoint from ∂ Σ, the complementary Σ − (R) may have more than two connected components. However, the orientation of Σ allows do distinguish the two sides of (R) (except if (R) is a boundary component of Σ, in which case (R) has only one side). Two geodesics , of ( Σ, d), defined up to changing the origins, are interlaced if neither (R) nor (R) is a boundary component of Σ, and intersects two connected components of Σ − (R) corresponding to one and the other of its sides (or the same after exchanging and , which is equivalent). Two local geodesics of (Σ, d) are interlaced if they admit some lifts in ( Σ, d) which are interlaced, and a local geodesic is self-interlaced if it is interlaced with itself. We easily convinced that this definition is equivalent to [Mor,§ 2.3] in the case of surfaces.
If d is the distance defined by m, two local geodesics are non interlaced if and only if they are disjoint and a local geodesic is non self-interlaced if and only if it is simple. We refer to [Mor,§ 3.1] for a characterization of the local geodesics for [q] that are non interlaced.
Definition 3 A geodesic lamination (or simply a lamination) of (Σ, d), with d the distance defined by m or [q], is a non-empty set Λ of (complete) local geodesics of (Σ, d), defined up to changing origin, whose elements are called leaves, such that : • leaves are non self-interlaced ; • leaves are pairwise non interlaced ; • if belongs to Λ then so does − , with − (t) = (−t) ; • Λ is closed for the geodesic topology.
We say that Λ is a flat lamination if d is defined by [q] and that Λ is a hyperbolic lamination if d is defined by m. Usually, a hyperbolic lamination of (Σ, m) is defined as a non-empty closed subset of Σ, that is a union of images of simple and pairwise disjoint local geodesics of (Σ, m). The definitions are equivalent in the case of hyperbolic laminations but not in the case of flat laminations (see [Mor,§ 4.1]). We recall the two main results of [Mor] about flat laminations. In Theorem 4, a cylinder component is a maximal set of leaves of Λ whose images are contained in a non degenerated flat cylinder (hence, these leaves are periodic), a minimal component is a sublamination which is the closure, for the geodesic topology, of a leaf and its opposite − . The minimal component is of recurrent type if is regular (i.e. does not meet any singular point) and is not periodic, all the images of its leaves are then dense in a domain of Σ, i.e. the closure of a connected open subset bounded by some periodic local geodesics, if it is not equal to Σ. The minimal component is of finite graph type if the image of is a finite graph, and if neither nor its opposite are eventually periodic. All the images of its leaves are then equal, and no leaf is eventually periodic. An end of a local geodesic terminates in a minimal component or in a cylinder component if there exists a ray in the equivalence class of this end which is the ray of a leaf of the minimal component or a ray of a boundary component of the corresponding flat cylinder.
Theorem 4 [Mor, § 6] Let Λ be a flat lamination on a compact, connected, orientable surface endowed with a half-translation structure. Then Λ is a finite union of cylinder components, of minimal components (of recurrent type, finite graph type and periodic leaf travelled in both senses) and of isolated leaves (for the geodesic topology) both of whose ends terminate in a minimal component or a cylinder component.
A cyclic orientation on a finite metric graph X is the data of cyclic orders (see [Wol,§ 2.3.1] for the definition) on the sets of germs of locally geodesic rays issued from each vertice of X.
Theorem 5 Every cyclically oriented, connected, finite, metric graph X, without extremal point, may be the support of an uncountable minimal flat lamination with no eventually periodic leaf, on a compact and connected surface endowed with a half-translation structure, except if X is homeomorphic to a circle, a dumbbell pair, a flat height or a flat theta, by a homeomorphism preserving the cyclic orientation (i.e , where the orientations are given by the plan).

Definition of measured flat laminations.
Let (Σ, [q]) be a connected, orientable surface with (possibly empty) boundary, endowed with a half-translation structure. An arc is a piecewise C 1 map α : [0, 1] → Σ which is a homeomorphism onto its image. Let Λ be a flat lamination of (Σ, [q]). An arc α is transverse to a leaf or to a segment of leaf of Λ if • α is transverse to outside the singular points of [q] and the singular points of α ; • for every singular point x of [q] or of α in Image( )∩α(]0, 1[)−Image( )∩α(]0, 1[)∩∂Σ, there exists a neighborhood U of x that is a topological disk, and a segment S of such that U − Image(S) ∩ U has two connected components and the connected components of U ∩(α([0, 1])−{x}) are contained in different components of U − Image(S) ∩ U ; • for every singular point x of [q] or of α in Image( ) ∩ α(]0, 1[) ∩ ∂Σ, there exists a neighborhood U of x that is a topological quarter of disk, and a segment S of such that U − Image(S) ∩ U has two or three connected components and the connected components of U ∩ (α([0, 1]) − {x}) are contained in two different components of U − Image(S) ∩ U , with only one which is not disjoint from ∂Σ. The arc α may intersect ∂Σ along a segment ; x U α U U x α x α • α is tangent to neither in 0 nor in 1. However, α(0) and α(1) may belong to (R).
An arc α is transverse to a set F of leaves or of segments of leaves of Λ if it is transverse to every element of F , and F is transverse to α if α is transverse to F . In particular, an arc is transverse to Λ if it is transverse to every leaf of Λ.
Definition 6 A transverse measure on Λ is a family µ = (µ α ) α of Radon measures µ α defined on G(α), for every arc α transverse to Λ, such that : We will denote by (Λ, µ) a flat lamination endowed with a transverse measure, that we will call a measured flat lamination, and we will denote by M L p (Σ) the set of measured flat laminations on Σ. We endow M L p (Σ) with the topology such that a sequence (Λ n , µ n ) n∈N converges to (Λ, µ) if and only if for every arc α, if α is transverse to Λ, then α is transverse to Λ n for n large enough and µ n,α * µ α in the space of Radon measures on G(α).
A leaf of Λ is positively recurrent if there exists an arc α transverse to such that intersects α([0, 1]) at an infinite number of positive times. For example, if Σ is compact, the leaves terminating in some minimal components are positively recurrent.
Lemma 7 If Λ is endowed with a transverse measure µ, then the only leaves of Λ which are isolated and positively recurrent are the periodic leaves.
Proof. Assume there exists an isolated leaf of Λ which is positively recurrent, and let α be an arc as above. Then, there exists an increasing sequence (t n ) n∈N of successive positive times such that (t n ) ∈ α([0, 1]). For every n ∈ N, let n : t → (t+t n ). Let n ∈ N, and let t n = s 0 · · · s k = t n+1 be the finite sequence of times between t n and t n+1 such that (s i ) is a singular point for every i ∈ [1, k − 1] ∩ N, and choose some arcs α i and β i transverse to at (s i ) and at (s i+1 ) if i k − 1.
We always can choose α i and β i such that there exists a homotopy between α i and β i that allows to define a holonomy between αi : t → (t + s i ) ∈ G(α i ) and βi : t → (t + s i+1 ) ∈ G(β i ), and such that the arcs α 0 and β k−1 are some subarcs of α (If (s i ) or (s i+1 ) belong to ∂Σ, in order to have a holonomy between α i and β i , we may choose α i or β i such that they intersect ∂Σ along a segment or such one of their extremal points belong to (R)).
By iteration, we have µ α ({ n }) = µ α ({ 0 }) for all n ∈ N. Since is isolated, the leaf 0 is isolated in G(α), and since 0 belongs to the support of µ α , we have µ α ( 0 ) > 0. Finally, the set { n } n∈N is contained in G(α), which is relatively compact according to the theorem of Ascoli. And if were not periodic, the set { n } n∈N would be infinite and µ α ({ n } n 0 ) n 0 µ α ( 0 ) would be infinite. Hence µ α would not be locally finite, which is a contradiction.

Preimage of a measured flat lamination.
Let (Σ, [q]) be a connected, orientable surface, with (possibly empty) boundary, endowed with a half-translation structure and let (Λ, µ) be a measured flat lamination on Σ. Let p : (Σ , [q ]) → (Σ, [q]) be a locally isometric cover of (Σ, [q]) with covering group Γ Σ and let Λ be the preimage of Λ in Σ (see the remark before [Mor,Lem. 2.3]). Since p is a local diffeomorphism, if α is an arc of Σ which is transverse to Λ, and if α is a lift of α in Σ , then α is transverse to Λ , and p induces a homeomorphism f α : G(α ) → G(α). We set µ α = (f −1 α ) * µ α . More generally, if α is any arc transverse to Λ , then its image is a union of images of some lifts of some arcs transverse to Λ, say α ([0, 1]) = α 1 ([0, 1]) ∪ · · · ∪ α n ([0, 1]) such that, for every 2 k n, the intersection is the image of a lift of an arc transverse to Λ, and if p ∈ {k, k + 1, k − 1}, where T is the set of arcs transverse to Λ , as the family of measures just constructed. By naturality, Λ is Γ Σ -invariant and µ is invariant by the action by Lemma 8 The family µ is the unique transverse measure on Λ such that if α is the lift of an arc α transverse to Λ, Proof. The properties (1), (3) and (4) of Definition 6 are clearly satisfied by µ , and if h : Otherwise, we see h as a concatenation of compositions of holonomies that are lifts of some holonomies of Λ, and we show similarly that which is Γ Σ -invariant, then the set of projections of the leaves of Λ by p : Σ → Σ is a flat lamination Λ of (Σ, [q]), and if α is an arc transverse to Λ and α is a lift of α in Σ , then (f α ) * µ α is a measure on G(α), that does not depend on the choice of the lift, by Γ Σ -invariance. The family of measures defined in this way satisfies clearly the properties (1), (3) and (4) of Definition 6. Moreover if h : F 1 → F 2 is a holonomy between two Borel sets of G(α 1 ) and G(α 2 ), with α 1 , α 2 two disjoint arcs transverse to Λ, then it lifts to a holonomy between two Borel sets of G(α 1 ) and G(α 2 ), with α 1 , α 2 two lifts of α 1 , α 2 , and we have µ α2|F2 = h * (µ α1|F1 ). Hence (Λ, µ) is a measured flat lamination, and its image by the previous map is (Λ , µ ).
Hence, this map is a bijection between M L p (Σ) and the set of measured flat laminations of (Σ , [q ]) that are Γ Σ -invariant. Finally, if (Λ n , µ n ) n∈N is a sequence of M L p (Σ) that converges to (Λ, µ), and if (Λ n , µ n ) n∈N and (Λ , µ ) are their images in M L p (Σ ), then if α is an arc transverse to Λ which is the lift of an arc α transverse to Λ, the arc α is transverse to Λ n for n large enough, and since p is a local diffeomorphism, α is transverse to Λ n for n large enough. Moreover, we have If α is an arc transverse to Λ which is not the lift of an arc transverse to Λ, then by finite decomposition, we prove similarly that if n is large enough, Λ n is transverse to α and µ n,α * µ α . Hence, the map (Λ, µ) → (Λ , µ ) is continuous. The same property holds for its inverse, hence it is a homeomorphism.
4 Measured flat laminations and Radon measures on the space of flat geodesics of the universal cover.
Let (Σ, [q]) be a connected, orientable surface with (possibly empty) boundary, endowed with a half-translation structure and let p : ( Σ, [ q]) → (Σ, [q]) be a locally isometric universal cover, with covering groupe Γ Σ . The local geodesics of ( Σ, [ q]) are geodesics, and if Λ is a flat lamination of ( Σ, [ q]) and if α is an arc transverse to Λ such that every geodesic of G(α) intersects α([0, 1]) only at its origin, then the map g α : We denote by M Γ Σ ([G [ q] ]) the space of Radon measures on the space [G [ q] ] endowed with the geodesic topology, which are Γ Σ and ι-invariant (where ι( ) = − : t → (−t)) and whose supports are Γ Σ -invariant flat laminations, endowed with the weak-star topology. Let ν be an element of M Γ Σ ([G [ q] ]) whose support is Λ and let α be an arc transverse to Λ such that the geodesics of . If α is an arc transverse to Λ, but if some geodesic of G(α) possibly intersects α([0, 1]) at several points, then we define the measure µ α by a finite gluing process as in Section 3, which is always possible since the geodesics of G [ q] are proper. Let µ ν be the family of transverse measures defined in this way.
Proposition 9 The family of measures µ ν is a transverse measure on Λ, and the map ν → µ ν is a homeomorphism between M Γ Σ ([G [ q] ]) and the space of measured flat laminations on ( Σ, Proof. Unless the opposite is specified, until the end of the proof, if α is an arc transverse to a flat lamination Λ, we assume that the geodesics of G(α) only intersect α([0, 1]) at their origins. We may always assume this up to shortening α, since the geodesics of G(α) are proper and transverse to α.
The properties (1), (3), and (4) of Definition 6 are clearly satisfied by µ ν . If h : F 1 → F 2 is a holonomy of Λ between two Borel sets F 1 of G(α 1 ) and F 2 of G(α 2 ), then, by the definition of the holonomies, the sets [ ) and µ ν is invariant by holonomy, so it is a transverse measure on Λ, which is Γ Σinvariant by naturality.
Assume that two measures ν 1 and ν 2 of M Γ Σ ([G [ q] ]) define the same measured flat lamination ( Λ, µ) by this construction. Then, if U is a relatively compact open set of [G [ q] ], the set U is contained in a finite union of open sets (U i ) 1 i n such that for every U i there exists an arc α i transverse to Λ, such that U i ⊂ [G(α i )]. We have ν 1 (U i ) = ν 2 (U i ) for all 1 i n, and then ν 1 (U ) = ν 2 (U ). Since the relatively compact open sets span the Borel tribute, we have ν 1 = ν 2 . Hence, the map ν → µ ν is injective.
Let us now prove it is surjective. Let ( Λ, µ) be a measured flat lamination and let R be the sets of subsets A ⊂ [G [ q] ] such that there exists a finite family of arcs α 1 , . . . , α n , with n ∈ N, transverse to Λ (and such that the elements of G(α i ) intersect α i ([0, 1]) only at their origins), for all 1 i n, and some Borel subsets Then R is non empty and ∅ ∈ R. If A, B are two elements of R, let (α i ) 1 i n and (β j ) 1 j k be some finite families of arcs transverse to Λ and (A i ⊆ G(α i )) 1 i n and (B j ⊆ G(β j )) 1 j k be some families of Borel subsets of G [ q] such that A = [A i ] and B = [B j ]. For all 1 i n and 1 j k, we replace A i by Hence, the set R is a ring. Let A ∈ R and let (A i ⊆ G(α i )) 1 i n , be as above. We define ν(A) = n i=1 µ αi (A i ). Then ν(A) 0 for all A ∈ R and ν(∅) = 0. Let (A n ) n∈N be a sequence of pairwise disjoint elements of R such that n∈N A n belongs to R. Let {α i } 1 i k be a family of arcs transverse to Λ associated to n∈N A n as above. For every n ∈ N, there exists a family {A n,i ⊂ G(α i )} 1 i k of Borel subsets such Hence, the constructed map is σ-additive on R, and since ν(∅) = 0 it is a premeasure on R. For every arc α transverse to Λ, the measure µ α is locally finite, hence if A ∈ R has a compact closure, then ν(A) < +∞. Moreover, according to the theorem of Ascoli, the set of geodesics whose origins belong to a compact subset of Σ is compact, and since Σ is a countable union of compact sets, the space [G [ q] ] is a countable union of compact sets, hence ν is σ-finite. According to the theorem of extension of Caratheodory, the premeasure ν can be extended to a unique measure, still denoted by ν, on the σ-algebra generated by R, which is equal to the Borel tribute of [G [ q] ]. According to the properties (1) and (3) of Definition 6, the support of the measure ν is Λ and ν is ι and Γ Σ -invariant by naturality. By construction, the measured flat lamination associated to ν by the previous map is ( Λ, µ). Hence, the map ν → µ ν is bijective.
Finally q] ]) which converges to ν, and if ( Λ n , µ n ) n∈N and ( Λ, µ) are their images in M L p ( Σ), then if α is an arc transverse to Λ (such that the geodesics of G(α) intersect α([0, 1]) only at their origins), for n large enough, the lamination Λ n is transverse to α and µ n, Hence, the map ν → µ ν is continuous. Similarly, if ( Λ n , µ n ) n∈N is a sequence of measured flat laminations of Σ which are Γ Σ -invariant, that converges to ( Λ, µ), and if (ν n ) n∈N and ν are their preimages in M Γ Σ ([G [ q] ]), then if f is a continuous map from [G [ q] ] to R whose support is compact, there exists a finite family {α 1 , . . . , α p } of arcs transverse to Λ and to Λ n for n large enough, such that Supp(f ) ⊂ 1 i p [G(α i )], and for all 1 i p and n large enough, we have ν . Hence ν n * ν and the inverse map is continuous. Hence, the map ν → µ ν is a homeomorphism. q] ]) be the composition of the map µ → µ defined at Lemma 8, with Σ = Σ, with the inverse of the map ν → µ ν defined at Lemma 9.
Corollary 10 The map φ is a homeomorphism.
5 Links between measured flat laminations and measured hyperbolic laminations.
In this Section 5, we still denote by (Σ, [q]) a connected, orientable surface with (possibly empty) boundary, endowed with a half-translation structure, and by p : ( Σ, [ q]) → (Σ, [q]) a locally isometric universal cover whose covering group is Γ Σ . We assume that Σ is compact and that χ(Σ) < 0, and we denote by m a hyperbolic metric with totally geodesic boundary on Σ and by m the unique hyperbolic metric on Σ such that p : ( Σ, m) → (Σ, m) is locally isometric. For every geodesic g of [G [ q] ] or [G m ], we denote by E(g) ∈ ∂ 2 ∞ Σ its ordered pair of points at infinity, and if F is a set of geodesics, then E(F ) = {E(g) : g ∈ F }. The space M L h (Σ) of measured hyperbolic laminations on (Σ, m), endowed with the topology defined in [Bon1,p. 19], is homeomorphic to the space of Radon measures on [G m ] that are Γ Σ et ι-invariant, whose supports are hyperbolic laminations, denoted by M Γ Σ ([G m ]) (see [Bon2,Prop. 17 p. 154]). Here, we use this fact and the homeomorphism between M Γ Σ ([G [ q] ]) and M L p (Σ) defined in Corollary 10 to investigate the links between M L p (Σ) and M L h (Σ). We denote by ϕ : [G [ q] ] → [G m ] the map associating to the geodesic ∈ [G [ q] ] the geodesic ϕ( ) ∈ [G m ] that corresponds to , i.e. such that E(ϕ( )) = E( ) (see [Mor,§4.2]). Then, ϕ is surjective and continuous, and a closed subset F of [G [ q] ] is a flat lamination if and only if ϕ(F ) is a hyperbolic lamination. Moreover ϕ is proper. Indeed, if K is a compact set of [G m ] and if ( n ) n∈N is a sequence of ϕ −1 (K), then by definition of ϕ, up to taking a subsequence, the sequence (E( n )) n∈N converges in ∂ 2 ∞ Σ. Since ( Σ, [ q]) is δ-hyperbolic (as Σ is compact and χ(Σ) < 0, see [Mor,Rem. 2.10]), according to [Mor,Lem. 2.6], up to taking a subsequence, the sequence ( n ) n∈N converges to a geodesic such that E( ) ∈ E(K). Hence belongs to ϕ −1 (K), therefore ϕ −1 (K) is compact.
However, the map ϕ is not injective. By definition, two different geodesics , of ( Σ, [ q]) have the same image by ϕ if and only if they have the same ordered pair of points at infinity, and according to the flat strip theorem (see for example [BH,Th. 2.13 P. 182]), their images are parallel and contained in a maximal flat strip of ( Σ, [ q]). Then, their projections in (Σ, [q]) are freely homotopic periodic local geodesics, and hence their images are contained in a maximal flat cylinder (see for example [MS1,Th. 2.(c)]). The points at infinity of , and of their images ϕ( ) = ϕ( ) are therefore the attractive and repulsive fixed points of an element of the covering group, and the projection of ϕ( ) is a closed local geodesic of (Σ, m). Hence, the restriction of ϕ to the set of geodesics of ( Σ, [ q]) whose projections are not periodic is an injective map whose image is the set of geodesics of ( Σ, m) whose projections are not closed geodesics. Moreover, the preimage of a geodesic λ of ( Σ, m) whose projection is a closed local geodesic is the set of geodesics of ( Σ, [ q]) having the same ordered pair of points at infinity than λ, which may be a unique geodesic, or may be a set of parallel geodesics whose images are contained in a flat strip, and whose projections are periodic.

Lemma 11
The map ϕ defines a continuous, surjective and proper map ϕ * from the space of Radon measures on [G [ q] ] to the space of Radon measures on [G m ]. Moreover, ϕ * ν belongs to Proof. The map ϕ is continuous and proper, hence it defines a continuous map ϕ * from the space of Radon measures on [G [ q] ] to the space of Radon measures on [G m ]. Let us first prove that ϕ * is surjective. The map s : [G m ] → [G [ q] ] which associates, to a hyperbolic geodesic, the flat geodesic to which it corresponds (if it is unique) and the "middle" geodesic of the set of geodesics (contained in a flat strip) to which it corresponds otherwise, is a measurable section (that is not continuous) of ϕ. Since ϕ is continuous, the preimage of a compact set by s is relatively compact. Hence s defines a map s * from the set of Radon measures on [G m ] to the space of Radon measures on [G [ q] ], and ϕ * • s * = Id. Therefore ϕ * is surjective.
Let us now prove that ϕ * is proper. The space [G [ q] ] is countable at infinity. Thus, there exists a sequence (K n ) n∈N of compact sets such that, for all n ∈ N, K n is contained in the interior of K n+1 and n∈N K n = [G [ q] ]. If C is a compact set of the space of Radon measures on [G m ] and if K is a compact set of [G [ q] ], then the set {ν(K), ν ∈ (ϕ * ) −1 (C)} is bounded by the maximum of {ν(ϕ(K)), ν ∈ C}, which is finite since C is compact. Hence, for all n ∈ N, the set {ν |Kn , ν ∈ (ϕ * ) −1 (C)} is compact and if (ν k ) k∈N is a sequence of (ϕ * ) −1 (C), proceeding by diagonal extraction, we show that there exists a subsequence, still denoted by (ν k ) k∈N , and a Radon measure ν on [G [ q] ] such that for all n ∈ N, ν k|Kn * ν |Kn . And, by the choice of the , there exists n ∈ N such that Supp(f ) ⊂ K n , and then (ν k (f )) k∈N = (ν k|Kn (f )) k∈N converges to ν |Kn (f ) = ν(f ). Hence, we have ν k * ν and ϕ * is proper on the space of Radon measures. Finally, by definition of ϕ, the measure ϕ * ν belongs to is closed (see [Bon2,Prop. 3 et 17]), and since ϕ * is continuous, its preimage is closed. Hence, the restriction of ϕ * to these spaces is continuous, proper and surjective.
The map ϕ * is not injective. Considering the lack of injectivity of ϕ, the preimage by ϕ * of a Radon measure on [G m ] whose support contains no geodesic whose projection in (Σ, m) is a closed geodesic, consists in a unique Radon measure on [G [ q] ] whose support contains no geodesic whose projection in (Σ, [q]) is a periodic local geodesic. However, the preimage by ϕ * of the Γ Σ -orbit of a Dirac measure, whose support is the Γ Σ -orbit of a geodesic λ, of mass δ, whose projection in (Σ, m) is a closed geodesic, is the set of Radon measures on [G [ q] ], whose support is the Γ Σ -orbit of a closed subset F of the set of geodesics of ( Σ, [ q]) having the same ordered pair of points at infinity that λ, such that the mass of F is δ. If there exist at least two such geodesics, then this set is the set of parallel geodesics contained in a flat strip, of width L > 0, hence it is homeomorphic to [0, L]. Hence, the set F is homeomorphic to a closed subset of [0, L]. Consequently, the preimage of ν by ϕ * is homeomorphic to the set of Borel measures on [0, L] of mass δ.
The spaces M L h (Σ) and M L p (Σ) are respectively homeomorphic to M Γ Σ ([G m ]) and to M Γ Σ ([G [ q] ]), hence ϕ * defines a continuous map ψ : M L p (Σ) → M L h (Σ) which is proper and surjective. The group R + * acts on these two spaces by multiplication of the measures. We denote by PM L p (Σ) and PM L h (Σ) the quotient spaces for these actions. Since ψ is equivariant by these actions, it defines a continuous map ψ : PM L p (Σ) → PM L h (Σ) which is surjective and proper. We deduce from this the following lemmas.
Lemma 12 The space PM L p (Σ) is compact.
If Σ is compact, a measured cylinder lamination is a measured flat lamination having a unique component which is a cylinder component (see [Mor,§6]).
Lemma 13 If Σ is compact, the set of measured cylinder laminations having finitely many leaves is dense in M L p (Σ). In particular, M L p (Σ) is separable.
Proof. The set of simple closed local geodesics endowed with a transverse measure which is a Dirac measure of positive mass is dense in M L h (Σ) (see [Bon1,Prop. 15]), and its preimage by ϕ * is the set of measured cylinder laminations. Since ϕ * is continuous, this set is dense in M L p (Σ). If (Λ, µ) is a measured cylinder lamination whose support is not reduced to a single leaf, we denote by α : [0, T ] → C (T > 0) a geodesic arc such that the interior of the maximal flat cylinder C that contains the support of Λ, endowed with the induced distance, is isometric to α(]0, T [) × S 1 . Then, the set of local geodesics contained in C and parallel to the boundary of C is homeomorphic to [0, T ], and since the set of Radon measures with finite support on [0, T ] is dense in the space of Radon measures on [0, T ], there exists a sequence (µ n,α ) n∈N of Radon measures with finite support on G(α) such that, for all n, every leaf of the support of µ n,α is parallel to the boundary of C and µ n,α * µ α . Moreover, Λ n = [Supp(µ n,α )] is a flat lamination and µ n,α defines a transverse measure on Λ n such that the sequence (Λ n , µ n ) n∈N converges to (Λ, µ).
The maps ψ and ψ are not injective. Assume that Σ is compact with genus g ∈ N and b ∈ N boundary components. Considering the lack of injectivity of ϕ * , the preimage of a measured hyperbolic lamination having no closed leaf consists in a unique measured flat lamination having no periodic leaf. However, if (Λ m , µ m ) is a closed leaf λ endowed with a transverse measure which, for every arc α such that G(α) contains λ, is a Dirac measure at λ of mass δ > 0, the preimage of (Λ m , µ m ) by ψ is the set of measured cylinder laminations whose supports are closed sets F of leaves that are freely homotopic to λ. If the set of local geodesics of (Σ, [q]) that are freely homotopic to λ contains at least two elements, it foliates a maximal flat cylinder. Then, this set is homeomorphic to [0, L], with L is the height of the flat cylinder, so F is homeomorphic to a closed subset of [0, L]. Hence, the preimage of (Λ m , µ m ) by ψ is homeomorphic to the set of Borel measures on [0, L] of mass δ. Since ϕ * is equivariant for the addition of measures, we see that if a measured hyperbolic lamination has some closed leaves λ 1 , . . . , λ p , of respective masses δ 1 , . . . , δ p (p is always at most 3g − 3 + b), then the preimage of (Λ m , µ m ) by ψ is homeomorphic to the Cartesian product of the sets of Borel measures on [0, L i ], 1 i p, where L i is the height of the maximal flat cylinder, union of the local geodesics of (Σ, [q]) freely homotopic to λ i , whose total mass is δ i .
Since Σ is compact, the projectified space PM L h (Σ) is homeomorphic to the sphere S 6g−6+2b (see [Bon1,Th. 17]). If the support of the measured hyperbolic laminations in the equivalence class of x ∈ PM L h (Σ) has no closed leaf, then the preimage of {x} by ψ is a single point. However, if the closed leaves λ 1 , . . . λ p belong to the support of the measured hyperbolic laminations in the equivalence class of x, and if L 1 , . . . , L p are the heights of the maximal flat cylinders, unions of the images of the periodic local geodesics of (Σ, [q]) that are freely homotopic to λ 1 , . . . , λ p (L i = 0 if there is only one periodic local geodesic that is freely homotopic to λ i ), then the preimage of {x} by ψ is homeomorphic to the Cartesian product with p terms of the sets of Borel measures on [0, L i ] of total masses at most 1. In particular, since the set of measured hyperbolic laminations whose support contains a closed leaf is projectively dense in the space of measured hyperbolic laminations, the subset of PM L h (Σ) of points whose preimages by ψ are not a single point is dense in PM L h (Σ).

Intersection number.
If g : R → Σ is a geodesic of ( Σ, [ q]) or of ( Σ, m), the orientation of Σ allows to define the two sides (arbitrarily) + and − of g(R) (except if g(R) is a boundary component of ∂ Σ, in which case g(R) has one side). We denote by C + (g) and C − (g) the two complementary sides which are the unions of g(R) with the connected components of Σ−g(R) corresponding to one and the other sides of g(R) ( Σ − g(R) may have more than two connected components if g(R) is not disjoint from ∂ Σ). If Λ is a geodesic lamination of ( Σ, [ q]) or ( Σ, m), and if g is a leaf of Λ, then the image of another leaf of Λ is contained in C + (g) or C − (g), since the leaf is not interlaced with g. A leaf g separates two other leaves if the image of one is contained in C + (g), and the image of the otherone in C − (g). Let g 0 and g 1 be two leaves of Λ. We denote by C i the complementary side of g i (R) that contains g i+1 (R) (with i ∈ Z/2Z). We define C(g 0 , g 1 ) = C 0 ∩ C 1 . Let c be a geodesic segment joining their images (if the images are not disjoint, the segment c may be a point). A leaf of Λ intersects c non trivially if it is contained in C(g 0 , g 1 ) and if it intersects both complementary components of the image of c in C(g 0 , g 1 ), and we denote by B(g 0 , g 1 ) = B Λ (g 0 , g 1 ) the set of leaves of Λ intersecting c non trivially.
Lemma 14 The compact set B(g 0 , g 1 ) does not depend on the choice of c.
Proof. The set of leaves whose images are contained in C 0 ∩ C 1 is compact and the requirement to intersect c non trivially is closed in this set, hence B(g 0 , g 1 ) is compact. Let c be another geodesic segment joining the images of g 0 and g 1 . Since both c and c separate C 0 ∩ C 1 in two connected components and since the intersection of the images of two geodesic segments of ( Σ, [ q]) (or between the image of a geodesic and a point) is connected, every leaf that intersects c non trivially also intersects c non trivially, and conversely.
The set of free homotopy classes of simple closed curves on Σ endowed with transverse measures which are Dirac measures of positive masses, embeds into M L h (Σ), and the intersection number on this set can be extended, in a unique way, to a continuous map i : [Bon2,Prop. 3]). According to Lemma 11, the map ϕ * defines a map ψ : M L p (Σ) → M L h (Σ). Let α be a non trivial free homotopy class of closed curves, let (Λ [q] , µ [q] ) be a measured flat lamination and let ν µ ] ] (see Corollary 10). We define the intersection number between (Λ [q] , µ [q] ) and α by , α). Hence, we assume that α is primitive (i.e. if there exists a free homotopy class α 0 such that α = α k 0 , then k = ±1). We denote by α [q] a periodic local flat geodesic in the class of α, and by α [q] a lift of α [q] in Σ. Let γ ∈ Γ Σ − {e} be one of the two primitive hyperbolic elements of Γ Σ whose translation axis is α [q] (R), and let ( Λ [q] , µ [q] ) be the preimage of (Λ [q] , µ [q] ) in Σ.
Lemma 15 Let be a leaf of Λ [ q] which is interlaced with α [q] . The number i [q] (µ [q] , α) is equal to where F m is the set of leaves of the measured hyperbolic lamination ( Λ m , µ m ) defined by ϕ * ν µ [q] which intersect a segment I = [a, γa[ transversally, with a ∈ α m (R), where α m (R) is the translation axis of γ in ( Σ, m), which is a fundamental domain of α m (R) for the action by translations of γ Z (see [Bon2,Prop. 3]). Since the choice of a is arbitrary, if F m is not empty, we may assume that a is an intersection point between a leaf λ of Λ m and α m (R). Hence, is γ-invariant, if several leaves belong to ϕ −1 (λ), we can choose arbitrarily in this set). Finally, if Remark. We could define the intersection number between a free homotopy class of closed curves with (Λ [q] , µ [q] ) by the infimum of the masses given by the measured flat lamination to the closed curves that are piecewise transverse to the lamination, similarly to the intersection number with a measured foliation, but this infimum would not be necessarly attained since the periodic local geodesics are generally not piecewise transverse to the lamination.
Furthermore, in the case of compact surfaces endowed with a half-translation structure, whose boundary is empty, contrarily to measured hyperbolic lamination (see [Ota,Th. 2]), the intersection numbers with the free homotopy classes of closed curves of Σ do not separate the measured flat laminations, but only their images in M L h (Σ). In particular, the topology defined after Definition 6 is not equivalent to the one induced by the product topology on R H , with H the set of free homotopy classes of closed curves, on the image of M L p (Σ) by the map (Λ [q] , µ [q] ) → (i(µ [q] , α)) α∈H .
6 Tree associated to a measured flat lamination.
If { 1 , 2 } is a pair of leaves of Λ, we define d Λ ( 1 , 2 ) = 1 2 ν µ (B( 1 , 2 )) (see Section 5.1 for the definition of B( 1 , 2 ))). Then d Λ ( 1 , 2 ) 0 and d Λ ( 1 , 2 ) = d Λ ( 2 , 1 ). Moreover, if 1 , 2 and 3 are three leaves of Λ and c 1 , c 2 and c 3 are some geodesic segments joining respectively the images of 1 and 2 , 2 and 3 , and 1 and 3 (denoted by c 1,2,3 on the figure, cases 2, 3 and 4), then either none of the three leaves separates the other ones (case 1, see Section 5.1 for the definition of a leaf separating to other leaves) or one separates the other ones (case 2, 3 and 4). In case 1, every leaf of B( 1 , 3 ) intersects c 1 or c 2 non trivially, hence We denote by (T, d T ) the quotient metric space ( Λ, d Λ )/ ∼, where ∼ if and only if d Λ ( , ) = 0, and if F is a set of leaves of Λ, we denote by F T its image by the quotient map.
Remark 16 Let 1 and 2 be two distinct leaves of Λ. Since ν µ has no atom and its support is Λ, we have ν µ (B( 1 , 2 )) = 0 if and only if B( 1 , 2 ) = { 1 , 2 }, and the topology defined by the distance d T is equivalent to the quotient topology of the topology induced by the geodesic topology on Λ, by the equivalence relation 1 R 2 if and only if B( 1 , 2 ) = { 1 , 2 }.
Finally, since the image of a leaf has two complementary sides (except if its image is a boundary component), and the image of each of the other leaves is contained in one of them, the relation defined on B( 1 , 2 ) by if belongs to B( 1 , ) is a total order which is compatible with R, hence defines a total order on B( 1 , 2 ) T .

Lemma 17
The metric space (T, d T ) is an R-tree.
Proof. Let 1 and 2 be two leaves of Λ. The map f : B( 1 , 2 ) → R + defined by f ( ) = d Λ ( 1 , ) is nondecreasing (with respect to ) and continuous since ν µ has no atom. Moreover, it is compatible with the equivalence relation R and defines an increasing continuous map f : B( 1 , 2 ) T → R + . Since B( 1 , 2 ) T is compact, it is a homeomorphism onto its image. Assume that its image is not an interval. Since it is a compact subset of R, if U is a bounded complementary component of f (B( 1 , 2 )) in R + , then its closure is an interval [a, b] with a < b. Let a and b be some leaves of f −1 (a) and f −1 (b). If there exists a leaf ∈ B( a , b ) − { a , b }, since ν µ has no atom and its support is T is a geodesic segment between T and T . Moreover, up to reparametrization, it is the unique arc joining T to T . Indeed, let g : [0, 1] → T be another arc joining T and T . If a leaf 0 belongs to B( , ) then it separates and (in the sense of Section 5.1) and since g is continuous for the quotient topology of the geodesic topology by the equivalence relation R, the point T 0 belongs to the image of g. Hence B( , ) T is contained in the image of g. Assume that there exists an element x = g(t), with t ∈]0, 1[, in the image of g that does not belong to B( , ) T and let x be a leaf representing x. Assume for a contradiction that x (R) is contained in C( , ) (case 1 below, see Section 5.1 for the definition of C( , )). Then B( , x ) is the union of the compact sets B( , x ) ∩ B( , ) and B( , x ) ∩ B( x , ), whose intersection is { x }. Let y be the closest element to in the compact set (B( , x )−B( , )∩B( , x ))∪{ x } (for ). By assumption on x , we see that y is also the closest element to in B( x , ) − B( , ) ∩ B( , x ). If y = x , the leaf y separates from x and x from . Since g is continuous, the element T y belongs to g([0, t]) and to g([t, 1]). If B( y , x ) = { y , x }, then T x = T y , and T y would belong to g([0, t[) and to g(]t, 1]), thus g would not be injective. If B( y , x ) = { y , x }, we denote by z the closest element to y in the compact set B( , y )∩B( , ). By definition of y , we have B( z , y ) = { z , y }, thus T y = T z , and since T x = T y , we have T x ∈ B( , ) T , which is a contradiction.
Hence x (R) is contained in the complementary side of (R) that does not contain (R), or the opposite (cases 2 and 3 below). However, since g is continuous and since separates x and (or separates x and ), there would then exist t ∈ ]0, 1[ such that (or ) represents g(t), et g would not be injective. Hence, if g is an arc between T and T , it has the same image as f −1 . Thus, up to reparametrization, the unique arc between T and T is f −1 which is isometric to [0, d T ( T , T )] by construction. Since it is true for all pair of leaves, the metric space (T, d T ) is an R-tree.
Assume that the measure ν µ defined by ( Λ, µ) has an atom . We replace (R) by a flat strip of width ν µ ( ) by gluing isometrically each of the complementary sides of (R) on the boundary components of FS( ) = R × [0, ν µ ( )], endowed with the Euclidean distance. By doing the same thing for every atom of ν µ , we get a surface Σ endowed with a complete CAT(0) metric d and the isometric action of the covering group Γ Σ on Σ minus the images of the atoms of ν µ extends in a unique way to an isometric action on ( Σ , d ). Note that ( Σ, d ) is a complete CAT(0) metric space whose boundary at infinity is endowed with a (total) cyclic order, compatible with the covering group action, hence it enters in the generalized framework of the geodesic laminations introduced in [Mor]. However, the distance d does not necessarily come from a half-translation structure, since the angles of the conical singular points belonging to the boundary of an added flat strip may not be multiples of π.
Let be an atom of ν µ and let F be the maximal set of geodesics whose images are contained in the corresponding flat strip FS( ) of ( Σ , d ), that are parallel to its boundary. Let α be a geodesic segment of FS( ) that orthogonally joins its boundary components. Then, the map r : F → Image(α) defined by r(g) = g(R) ∩ Image(α) is a homeomorphism. Hence, we can endow F with the measure ν = (r −1 ) * dx α , where dx α is the Lebesgue measure on Image(α), of total mass ν µ ( ). The canonical isometric embedding of every connected component of Σ − ν µ ( )>0 (R) into Σ induces an embedding of Λ − ν µ ( )>0 into the space [G d ] of geodesics of ( Σ , d ) defined up to changing origin. We denote by ν c the push forward of ν µ| Λ− ν µ ( )>0 by this embedding.
Then, we define ν µ = ν c + ν µ ( )>0 ν . The measure ν µ is a Γ Σ -invariant Radon measure on the set of geodesics (defined up to changing origin) of ( Σ , d ). Its support is the union of the image of Λ − ν µ ( )>0 by the canonical embedding, and of the sets F , with ν( ) > 0, defined above, which is a geodesic lamination Λ of ( Σ , d ).
Since, by construction, the Radon measure ν µ has no atom, we can define the tree associated to the measure ν µ , exactly as above, even if its support is not a flat lamination as defined in this paper, since it only matters than ( Σ , d ) is complete, CAT(0), and the support Λ of ν µ is a geodesic lamination in the generalized sense of [Mor]. We will call tree associated to (Λ, µ) the tree associated to ν µ defined in this way.
7 Covering group action on the tree associated to a measured flat lamination.
In this section, we use the definitions and notations of Section 6 and we consider the canonical action of the covering group Γ Σ on the tree (T, d T ) associated to the preimage ( Λ, µ) of the measured flat lamination (Λ, µ) in Σ. We may assume that ν µ has no atom, up to proceeding as in the last paragraph of Section 6. Since Λ is fixed, we will use the notation B( , ) instead of B Λ ( , ) for every pair of leaves , of Λ.
The covering group Γ Σ acts on Σ by isometries. Hence, it defines an action on the set [G [ q] ] of geodesics of ( Σ, [ q]) that are defined up to changing origin. Since Λ is Γ Σ -invariant, this action defines an action on Λ. Since for every γ ∈ Γ Σ we have γ * ν µ = ν µ and γB( , ) = B(γ , γ ), for every pair of leaves , of Λ, it defines an isometric action of Γ Σ on the tree (T, d T ) associated to ( Λ, µ) defined in Lemma 17.
Lemma 18 For every primitive element γ ∈ Γ Σ − {e}, if α γ (R) is a tranlation axis of γ in ( Σ, [ q]) and α γ is the projection of α γ in Σ, then the translation distance T (γ) of γ in (T, d T ) is equal to i [q] (µ, α γ ). Moreover, if T (γ) > 0, the translation axis of γ is the image in T of the set of leaves of Λ which are interlaced with α γ .
In the first possibility, the intersection equal to B( , γ ). Thus, no leaf of F is contained in the complementary side A of 0 (R) that contains α γ (R), if it is non interlaced with α γ (there is no leaf as represented by a dotted line points on the above picture). Consequently, there exists " ∈ B( , γ ) such that the set B( 0 , ") is reduced to { 0 , "} and d( ", 0 ) = 0. Thus T 0 = "' T ∈ B( , γ ) T . In the second possibility, there is a minimum, denoted by m in B( 0 , ), for defined in Remark 16, different from 0 . Then d(m, 0 ) = 0 and since m belongs to B( 0 , ) ⊂ F ∩ (B( −1 , 0 ), it is interlaced with α γ by assumption. Hence, the image of 0 in T belongs to the translation axis of γ in T and, by γ-invariance, so does the image of '. Hence, if i [q] (µ, α γ ) > 0, the translation distance T (γ) of γ is equal to i [q] (µ, α γ ) and the translation axis of γ is the image in T of the set of leaves of Λ which are interlaced with a translation axis of γ in ( Σ, [ q]).
Let be a leaf of Λ and (a, b) = ( (−∞), (+∞)). Since no leaf of Λ is interlaced with α γ , the union Λ ∪ { α γ , α −1 γ } is a flat lamination, and we can define B( , α γ ). We replace by the maximum of (the compact set) B( , α γ ) − α γ , for . Then B( , α γ ) = { , α γ }. However, the action of γ Z on ∂ ∞ Σ has a South-North dynamic, whose fixed points are N and S, with N attractive and S repulsive. Up to taking the opposite of the cyclic order on ∂ ∞ Σ, we can assume that the action of γ Z on the γ Z -orbits of a and b is increasing.

γb
Isometry between the tree associated to a measured flat lamination and the dual tree to the corresponding measured hyperbolic lamination.
In this section 8, we use the same notation as in Section 6. We denote by m a hyperbolic metric with totally geodesic boundary on Σ and by m the lifted hyperbolic metric on Σ. We begin by recalling the definition of the dual tree to a measured hyperbolic lamination (see for example [MS2,§1]), with a new presentation that allows to construct a Γ Σ -equivariant isometry between the tree associated to a measured flat lamination and the dual tree to the corresponding measured hyperbolic lamination. Let (Λ m , µ m ) be a measured hyperbolic lamination of (Σ, m) and let ( Λ m , µ m ) be its preimage in Σ. Then ( Λ m , µ m ) is Γ Σ -invariant and defines a measure ν µm ∈ M Γ Σ ([G m ]) (see [Bon2,Prop. 17 p. 154]). If λ is an atom of ν µm , we replace λ by a flat strip FS( λ) of width ν µm ( λ) foliated by geodesic lines parallel to its boundary. Proceeding similarly for every atom of ν µm , we get a geodesic lamination Λ on a surface endowed with a distance ( Σ , d ), which is CAT(0), and the isometric action of Γ Σ on ( Σ, m) extends uniquely to an isometric action on ( Σ , d ). Then, it is the same framework as at the end of Section 6. We can similarly defines the measure ν λ of total mass ν µm ( λ) on the parallel leaves foliating the flat strip FS( λ) associated to an atom λ of ν µm . The measure ν which is equal to ν µm outside of the atoms and equal to ν λ on the set of leaves foliating the flat stip associated to λ, for every atom λ of ν µm , is a Radon measure on [G d ] which is Γ Σ -invariant, atomless, and whose support is equal to Λ .
Let ( Λ [ q] , µ [ q] ) be a measured flat lamination on ( Σ, [ q]), let ν µ [q] be its associated measure on [G [ q] ] and let ν µm be its image by ϕ * (see Lemma 11). We denote by ( Λ m , µ m ) the measured hyperbolic lamination defined by ν µm , and by Λ and ν the geodesic lamination on ( Σ , d ) and the Radon measure on [G d ] defined by ( Λ m , µ m ) as above. We assume (up to proceeding as in the last paragraph of Section 6) that ν µ [q] has no atom.