MMeasured flat geodesic laminations.
Thomas Morzadec18 septembre 2018
Département de mathématique, UMR 8628 CNRS, Université Paris-Sud, Bât. 430, F-91405 Orsay Cedex, France Bureau : 16. [email protected]
Abstract :
Since their introduction by Thurston, measured geodesic laminations onhyperbolic surfaces occur in many contexts. In [Mor], we have introduced a notion of flatlaminations on surfaces endowed with a half-translation structure (that is a singular flatsurface 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 lamina-tions similar to transverse measures on hyperbolic laminations, taking into account thattwo different leaves of a flat lamination may no longer be disjoint. One aim of this paperis to construct a tool that could allow a fine description of the space of degenerationsof half-translation structures on a surface. In this paper, we define a nicer topology thanthe Hausdorff topology on the set of measured flat laminations and a natural continuousprojection of the space of measured flat laminations onto the space of measured hyperbo-lic laminations, for some arbitrary half-translation structure and hyperbolic metric on asurface. We prove in particular that the space of measured flat laminations is projectivelycompact. The main aim of this article is to propose a definition of transverse measure on the (geo-desic) flat laminations, introduced in [Mor], on a surface endowed with a half-translationstructure, 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 onhyperbolic surfaces (see for instance [Bon1]), the extension is non trivial, notably since theimages of two leaves of a flat lamination are not necessarly disjoint. We will call measuredflat lamination a flat lamination endowed with a transverse measure. We will define a suf-ficiently fine topology on the set of measured flat laminations and we will construct a (noninjective) natural continuous projection of the space of measured flat laminations onto thespace of measured hyperbolic laminations, for any choice of a half-translation structure andof 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 sequencesof periodic local geodesics, in the projectivized space of measured flat laminations. This inturn could yield a better understanding of the degenerations of half-translation structureson a surface, as initiated in [DLR]. In particular, as spaces of measures are suitable for
1. Keywords : Measured geodesic lamination, flat surface, half-translation structure, holomorphic qua-dratic differential, measured foliation, hyperbolic surface, dual tree. AMS codes 30F30, 53C12, 53C22. a r X i v : . [ m a t h . M G ] D ec nalysis tools (distributions as in [Bon1]), this could allow a finer study of the boundaryof the space of half-translation structures that we will develop in a subsequent work. Werefer to [ ? ] for a survey of [Mor] and of this work.We use the same notations as in [Mor] : let Σ be a compact, connected, orientablesurface, 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 dataconsisting in a (possibly empty) discrete set of points Z of Σ and of a Euclidean metricon Σ − Z with conical singular points of angles of the form kπ , with k ∈ N and k (cid:62) ateach point of Z , such that the holonomy of every piecewise C loop of Σ − Z is containedin {± 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 : ( (cid:101) Σ , (cid:101) d ) → (Σ , d ) be a locally isometric universal cover.Two local geodesics (cid:96), (cid:96) (cid:48) of (Σ , d ) , defined up to changing the origins, are said to be interlaced if they have some lifts (cid:101) (cid:96), (cid:101) (cid:96) (cid:48) in (cid:101) Σ such that the image of (cid:101) (cid:96) intersects both complementarycomponents of (cid:101) (cid:96) (cid:48) ( R ) in (cid:101) 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, localgeodesics of (Σ , d ) with the quotient topology of the compact-open topology for the actionby translations on the parametrizations, of R on the parametrized local geodesics, that iscalled 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 arecalled 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 : theimages of two leaves are generally not disjoint, the flat laminations are not determinedby their supports (uncountably many flat laminations can have the same support), thecylinder components (see Section 2.2) may have uncontably many leaves. Finally, thereare three types of minimal components of a flat lamination on a compact surface (periodicleaf travelled in both orientations, minimal component of recurrent type or of finite graphtype, see Theorem 4 for a complete statement). Compared with hyperbolic laminations, themain difficulty to define transverse measures on flat laminations is that the images of theleaves 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 imagesof the arcs transverse to the lamination, but as a family of measures on the sets of localgeodesics that intersect them transversally, and we have to refine the notion of invarianceby holonomy of these families of measures.In the first section, we define the measured flat laminations and we endow their setwith a topology. In the second one, we define the preimage of a measured flat laminationin a cover. In the third one, we define a homeomorphism between the space of measuredflat laminations on a compact surface endowed with a fixed half-translation structure andthe space of Radon measures on the set of geodesics (defined up to changing origin) ofa locally isometric universal cover, that are invariant by the covering group action and2hose 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 measuredhyperbolic laminations, for a fixed complete hyperbolic metric with totally geodesic boun-dary, and we characterize its lack of injectivity. In the fifth one, we define the intersectionnumber between a measured flat lamination and a free homotopy class of closed curves. Inthe last one, we define the tree associated to a measured flat lamination together with theuniversal covering group action on it. These tools should be useful to do analysis on thespace of degenerations of half-translation structures on surfaces, and we plan to developthis in future work (see [ ? ]). Acknoledgement : I want to thank Frederic Paulin for many advices and corrections that have deeplyimproved the redaction of this paper.
In this section, we recall the definition and some properties of half-translation structures onsurfaces and the definition of flat laminations introduced in [Mor]. Then, we define measured flatlaminations and we endow their set with a topology.
As in [Mor], in the whole paper, we will use the definitions and notation of [BH] for a surfaceendowed 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 (cid:96) : I → Σ , where I is aninterval of R . It will be called a segment , a ray or a geodesic line of (Σ , d ) if I is respectively acompact interval, a closed half line (generally [0 , + ∞ [ ) or R . If there is no precision, a geodesic is ageodesic line. A germ of geodesic ray , or simply a germ , is an equivalence class of locally geodesicrays for the equivalence relation r ∼ r if r and r coïncide on a non empty initial segmentthat is not reduced to a point. Similarly, the relation r ∼ ∞ r (cid:48) if there exist T, T (cid:48) > such that r ( t + T ) = r (cid:48) ( t + T (cid:48) ) for all t (cid:62) , 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 alocal geodesic. A local geodesic has two ends. We call geodesic topology the compact-open topologyon the set G d of local geodesics for the distance d or the quotient topology of this topology by theaction by translations of R at the source, on the set [ G d ] of local geodesics defined up to changingorigin. The quotient map from G d to [ G d ] will be denoted by g (cid:55)→ [ g ] , and if F is a subset of G d ,we will denote by [ F ] its image in [ G d ] .Let Σ be a connected, orientable surface, with (possibly empty) boundary. Assume that Σ isendowed with a Euclidean metric on Σ − Z , where Z is a discrete subset of Σ . If the holonomy ofevery piecewise C loop in Σ − Z is contained in {± Id } , two vectors v and v tangent to Σ aresaid to have the same direction if v is the image of ± v by holonomy along a piecewise C path in Σ − Z between the basepoints of v and v . This definition does not depend on the choice of thispath. A piecewise C path or union of paths is said to have constant direction , if all its tangentvectors, at the points in Σ − Z , have the same direction. Definition 2
A half-translation structure (or flat structure with conical singularities and holono-mies in {± Id } ) on a surface Σ is the data of a (possibly empty) discrete subset Z of Σ and aEuclidean metric on Σ − Z with conical singularity of angle k z π at each z ∈ Z , with k z ∈ N , k z (cid:62) if z ∈ Z − Z ∩ ∂ Σ and k z (cid:62) if z ∈ Z ∩ ∂ Σ , such that the holonomy of every piecewise C loopin Σ − Z is contained in {± Id } and such that the union of the boundary components has constantdirection. e will denote by [ q ] a half-translation structure on Σ , with q a holomorphic quadratic diffe-rential on Σ (see [Mor, § 2.5] for an explanation of the notation and to [Str, Def. 1.2 p. 2] for adefinition 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 (cid:96) : 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 (cid:96) ( t ) does not belong to Z , there exists a neighborhood V of t in R such that (cid:96) | V is a Euclideansegment (hence, (cid:96) | V has constant direction) ; • if (cid:96) ( t ) belongs to Z − Z ∩ ∂ Σ , then the two angles defined by the germs of (cid:96) ([ t, t + ε [) and (cid:96) (] t − ε, t ]) ,with ε > small enough, measured in both connected components of U − (cid:96) (] t − ε, t + ε [) , with U a small enough neighborhood of (cid:96) ( t ) , are at least π . • if (cid:96) ( t ) belongs to Z ∩ ∂ Σ , then the angle defined by the germs of (cid:96) ([ t, t + ε [) and (cid:96) (] t − ε, t ]) , with ε > small enough, measured in the connected component of U − (cid:96) (] t − ε, t + ε [) which is disjointfrom ∂ Σ , with U a small enough neighborhood of (cid:96) ( t ) , is at least π . (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1) U (cid:96) ( t ) (cid:62) π ∂ Σ ∂ Σ U (cid:96) ( t ) (cid:96) ( t ) π (cid:54) π (cid:54) Let Σ be a connected, orientable surface with (possibly empty) boundary. Let [ q ] be a half-translation structure and let m be a hyperbolic metric with totally geodesic boundary on Σ .Let p : (cid:101) Σ → Σ be a universal cover of covering group Γ (cid:101) Σ , let [ (cid:101) q ] be the unique half-translationstructure and let (cid:101) m be the unique hyperbolic metric on (cid:101) Σ such that p : ( (cid:101) Σ , [ (cid:101) q ]) → (Σ , [ q ]) and p : ( (cid:101) Σ , (cid:101) m ) → (Σ , m ) are locally isometric. To be consistent with [Mor], and notably to be allowedto use [Mor, Rem. 2.9], we will always assume that Σ is a cover (possibly trivial) of a compactsurface whose Euler characteristic is negative. In particular, if (cid:101) d is the distance defined by [ (cid:101) q ] or (cid:101) m , according to the theorem of Cartan-Hadamard, the metric space ( (cid:101) Σ , (cid:101) d ) is complete, CAT(0) ,and δ -hyperbolic, with δ > . Furthermore, there exists a unique Γ (cid:101) Σ -equivariant homeomorphismbetween the boundaries at infinity of (cid:101) Σ for the two metrics, thank to which we identify them. Let ∂ ∞ (cid:101) Σ denote this boundary at infinity and ∂ ∞ (cid:101) Σ = ∂ ∞ (cid:101) Σ × ∂ ∞ (cid:101) Σ − { ( x, x ) , x ∈ ∂ ∞ (cid:101) Σ } .In [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 isendowed with a (total) cyclic order. Here, we only recall the specific definition in the case ofconnected, orientable surfaces, endowed with a complete locally
CAT(0) metric. Since ( (cid:101) Σ , (cid:101) d ) is CAT(0) , the intersection of the images of two geodesics of ( (cid:101) Σ , (cid:101) d ) is connected, possibly empty. If (cid:101) (cid:96) is a geodesic of ( (cid:101) Σ , (cid:101) d ) , since (cid:101) (cid:96) ( R ) is not necessarly disjoint from ∂ (cid:101) Σ , the complementary (cid:101) Σ − (cid:101) (cid:96) ( R ) may have more than two connected components. However, the orientation of (cid:101) Σ allows do distinguishthe two sides of (cid:101) (cid:96) ( R ) (except if (cid:101) (cid:96) ( R ) is a boundary component of (cid:101) Σ , in which case (cid:101) (cid:96) ( R ) has only oneside). Two geodesics (cid:101) (cid:96), (cid:101) (cid:96) (cid:48) of ( (cid:101) Σ , (cid:101) d ) , defined up to changing the origins, are interlaced if neither (cid:101) (cid:96) ( R ) nor (cid:101) (cid:96) (cid:48) ( R ) is a boundary component of (cid:101) Σ , and (cid:101) (cid:96) intersects two connected components of (cid:101) Σ − (cid:101) (cid:96) (cid:48) ( R ) corresponding to one and the other of its sides (or the same after exchanging (cid:101) (cid:96) and (cid:101) (cid:96) (cid:48) , which is quivalent). Two local geodesics of (Σ , d ) are interlaced if they admit some lifts in ( (cid:101) Σ , (cid:101) d ) which areinterlaced, and a local geodesic is self-interlaced if it is interlaced with itself. We easily convincedthat 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 aredisjoint 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 definedby m or [ q ] , is a non-empty set Λ of (complete) local geodesics of (Σ , d ) , defined up to changingorigin, whose elements are called leaves, such that : • leaves are non self-interlaced ; • leaves are pairwise non interlaced ; • if (cid:96) belongs to Λ then so does (cid:96) − , with (cid:96) − ( t ) = (cid:96) ( − 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 closedsubset 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 flatlaminations (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 containedin a non degenerated flat cylinder (hence, these leaves are periodic), a minimal component is asublamination which is the closure, for the geodesic topology, of a leaf (cid:96) and its opposite (cid:96) − . Theminimal component is of recurrent type if (cid:96) is regular (i.e. does not meet any singular point) andis not periodic, all the images of its leaves are then dense in a domain of Σ , i.e. the closure ofa connected open subset bounded by some periodic local geodesics, if it is not equal to Σ . Theminimal component is of finite graph type if the image of (cid:96) is a finite graph, and if neither (cid:96) norits opposite are eventually periodic. All the images of its leaves are then equal, and no leaf iseventually periodic. An end of a local geodesic terminates in a minimal component or in a cylindercomponent if there exists a ray in the equivalence class of this end which is the ray of a leaf of theminimal 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 surfaceendowed with a half-translation structure. Then Λ is a finite union of cylinder components, of mi-nimal components (of recurrent type, finite graph type and periodic leaf travelled in both senses) andof isolated leaves (for the geodesic topology) both of whose ends terminate in a minimal componentor 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, ona compact and connected surface endowed with a half-translation structure, except if X is homeo-morphic to a circle, a dumbbell pair, a flat height or a flat theta, by a homeomorphism preservingthe cyclic orientation (i.e , where the orientations are given by the plan). Let (Σ , [ q ]) be a connected, orientable surface with (possibly empty) boundary, endowed with ahalf-translation structure. An arc is a piecewise C map α : [0 , → Σ which is a homeomorphismonto its image. Let Λ be a flat lamination of (Σ , [ q ]) . An arc α is transverse to a leaf or to a segmentof leaf (cid:96) of Λ if α is transverse to (cid:96) outside the singular points of [ q ] and the singular points of α ; • for every singular point x of [ q ] or of α in Image( (cid:96) ) ∩ α (]0 , − Image( (cid:96) ) ∩ α (]0 , ∩ ∂ Σ , there existsa neighborhood U of x that is a topological disk, and a segment S of (cid:96) such that U − Image( S ) ∩ U has two connected components and the connected components of U ∩ ( α ([0 , −{ x } ) are containedin different components of U − Image( S ) ∩ U ; • for every singular point x of [ q ] or of α in Image( (cid:96) ) ∩ α (]0 , ∩ ∂ Σ , there exists a neighborhood U of x that is a topological quarter of disk, and a segment S of (cid:96) such that U − Image( S ) ∩ U has two or three connected components and the connected components of U ∩ ( α ([0 , − { x } ) arecontained in two different components of U − Image( S ) ∩ U , with only one which is not disjointfrom ∂ Σ . The arc α may intersect ∂ Σ along a segment ; xU (cid:96)α U U xα (cid:96)x α (cid:96) • α is tangent to (cid:96) neither in nor in . However, α (0) and α (1) may belong to (cid:96) ( R ) .An arc α is transverse to a set F of leaves or of segments of leaves of Λ if it is transverseto every element of F , and F is transverse to α if α is transverse to F . In particular, an arc istransverse to Λ if it is transverse to every leaf of Λ .If α : [0 , → Σ is an arc of Σ , we denote by G ( α ) the subset of G [ q ] consisting in the localgeodesics of (Σ , [ q ]) which are transverse to α and whose origins belong to α ([0 , . By definition,if α (cid:48) ([0 , ⊆ α ([0 , , then G ( α (cid:48) ) ⊆ G ( α ) . Let F ⊆ G [ q ] be such that [ F ] ⊆ Λ , and let α and α be two disjoint arcs transverse to F , such that F ⊆ G ( α ) and every element of F intersects α ([0 , at a positive time. For every g ∈ F , we define t g = min { t > g ( t ) ∈ α ([0 , } . Let F be the subset of the elements g ∈ G ( α ) such that there exists g ∈ F with g ( t ) = g ( t + t g ) for all t ∈ R . A holonomy h : F → F of Λ is a homeomorphism between F and F defined by h ( g ) = g : t (cid:55)→ g ( t + t g ) such that there exists a homotopy H : [0 , × [0 , → Σ between α = H ( · , and α = H ( · , such that : • for every t ∈ [0 , , the map s (cid:55)→ H ( s, t ) is an arc transverse to every segment of leaf g | [0 , t g ] ,with g ∈ F ; • for every (cid:96) ∈ F , there exists s (cid:96) ∈ [0 , such that t (cid:55)→ H ( s (cid:96) , t ) is a segment of (cid:96) (up to changingthe parametrization) ; • the intersections H ([0 , × ]0 , ∩ α i ([0 , with i = 1 , are empty.Contrarily to the case of measured foliations, if the images of the geodesics are not pairwisedisjoint, the map H may not be injective. 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 : (1) the support of µ α is the set { (cid:96) ∈ G ( α ) : [ (cid:96) ] ∈ Λ } ; (2) if h : F → F is a holonomy of Λ , where α , α are two disjoint arcs transverse to F and F ⊂ G ( α ) and F ⊂ G ( α ) are some Borel sets, then h ∗ ( µ α | F ) = µ α | F ; (3) µ α is ι -invariant, with ι ( (cid:96) ) = (cid:96) − : t (cid:55)→ (cid:96) ( − t ) ; (4) if α (cid:48) ([0 , ⊆ α ([0 , , then µ α | G ( α (cid:48) ) = µ α (cid:48) . We will denote by (Λ , µ ) a flat lamination endowed with a transverse measure, that we will calla measured flat lamination , and we will denote by M L p (Σ) the set of measured flat laminationson 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 enoughand µ n,α ∗ (cid:42) µ α in the space of Radon measures on G ( α ) . leaf (cid:96) of Λ is positively recurrent if there exists an arc α transverse to (cid:96) such that (cid:96) intersects α ([0 , at an infinite number of positive times. For example, if Σ is compact, the leaves terminatingin some minimal components are positively recurrent. Lemma 7 If Λ is endowed with a transverse measure µ , then the only leaves of Λ which areisolated and positively recurrent are the periodic leaves. Proof.
Assume there exists an isolated leaf (cid:96) of Λ which is positively recurrent, and let α be an arcas above. Then, there exists an increasing sequence ( t n ) n ∈ N of successive positive times such that (cid:96) ( t n ) ∈ α ([0 , . For every n ∈ N , let (cid:96) n : t (cid:55)→ (cid:96) ( t + t n ) . Let n ∈ N , and let t n = s (cid:54) · · · (cid:54) s k = t n +1 be the finite sequence of times between t n and t n +1 such that (cid:96) ( s i ) is a singular point for every i ∈ [1 , k − ∩ N , and choose some arcs α i and β i transverse to (cid:96) at (cid:96) ( s i ) and at (cid:96) ( s i +1 ) if i (cid:54) k − .We always can choose α i and β i such that there exists a homotopy between α i and β i that allowsto define a holonomy between (cid:96) α i : t (cid:55)→ (cid:96) ( t + s i ) ∈ G ( α i ) and (cid:96) β i : t (cid:55)→ (cid:96) ( t + s i +1 ) ∈ G ( β i ) , andsuch that the arcs α and β k − are some subarcs of α (If (cid:96) ( s i ) or (cid:96) ( s i +1 ) belong to ∂ Σ , in order tohave a holonomy between α i and β i , we may choose α i or β i such that they intersect ∂ Σ along asegment or such one of their extremal points belong to (cid:96) ( R ) ). β(cid:96)(cid:96) α(cid:96)α β α β β α(cid:96) Then, according to the assertion (2) of Definition 6 we have µ α i ( (cid:96) α i ) = µ β i ( (cid:96) β i ) . We can assume,without loss of generality, that for every i ∈ [0 , k − ∩ N , the intersection α i +1 ([0 , ∩ β i ([0 , is the image of an arc transverse to (cid:96) , thus according to the assertion (4) of Definition 6, wehave µ α i +1 ( (cid:96) α i +1 ) = µ β i ( (cid:96) β i ) , thus µ α i +1 ( (cid:96) α i +1 ) = µ α i ( (cid:96) α i ) , and by iteration µ α ( (cid:96) n ) = µ α ( (cid:96) α ) = µ β k − ( (cid:96) β k − ) = µ α ( (cid:96) n +1 ) (the holonomies have to be built piece by piece instead of directly between (cid:96) n and (cid:96) n +1 , to include the case where (cid:96) ( R ) is not disjoint from ∂ Σ , and thus we cannot define thehomotopy globally).By iteration, we have µ α ( { (cid:96) n } ) = µ α ( { (cid:96) } ) for all n ∈ N . Since (cid:96) is isolated, the leaf (cid:96) isisolated in G ( α ) , and since (cid:96) belongs to the support of µ α , we have µ α ( (cid:96) ) > . Finally, the set { (cid:96) n } n ∈ N is contained in G ( α ) , which is relatively compact according to the theorem of Ascoli. Andif (cid:96) were not periodic, the set { (cid:96) n } n ∈ N would be infinite and µ α ( { (cid:96) n } n (cid:62) ) (cid:62) (cid:80) n (cid:62) µ α ( (cid:96) ) wouldbe infinite. Hence µ α would not be locally finite, which is a contradiction. (cid:3) Let (Σ , [ q ]) be a connected, orientable surface, with (possibly empty) boundary, endowed witha half-translation structure and let (Λ , µ ) be a measured flat lamination on Σ . Let p (cid:48) : (Σ (cid:48) , [ q (cid:48) ]) → (Σ , [ q ]) be a locally isometric cover of (Σ , [ q ]) with covering group Γ Σ (cid:48) and let Λ (cid:48) be the preimageof Λ in Σ (cid:48) (see the remark before [Mor, Lem. 2.3]). Since p (cid:48) is a local diffeomorphism, if α is anarc of Σ which is transverse to Λ , and if α (cid:48) is a lift of α in Σ (cid:48) , then α (cid:48) is transverse to Λ (cid:48) , and p (cid:48) induces a homeomorphism f α (cid:48) : G ( α (cid:48) ) → G ( α ) . We set µ α (cid:48) = ( f − α (cid:48) ) ∗ µ α . More generally, if α (cid:48) isany arc transverse to Λ (cid:48) , then its image is a union of images of some lifts of some arcs transverseto Λ , say α (cid:48) ([0 , α (cid:48) ([0 , ∪ · · · ∪ α (cid:48) n ([0 , such that, for every (cid:54) k (cid:54) n , the intersection α (cid:48) k − ([0 , ∩ α (cid:48) k ([0 , is the image of a lift of an arc transverse to Λ , and if p (cid:54)∈ { k, k + 1 , k − } ,then α (cid:48) k ([0 , ∩ α (cid:48) p ([0 , ∅ . Then, for every k ∈ [1 , n − ∩ N , we have α (cid:48) k | G ( α (cid:48) k ) ∩ G ( α (cid:48) k +1 ) = ( f − α (cid:48) k ) ∗ ( µ α k | G ( p (cid:48) ◦ α (cid:48) k ) ∩ G ( p (cid:48) ◦ α (cid:48) k +1 ) )= ( f − α (cid:48) k +1 ) ∗ ( µ α k +1 | G ( p (cid:48) ◦ α (cid:48) k ) ∩ G ( p (cid:48) ◦ α (cid:48) k +1 ) )= µ α (cid:48) k +1 | G ( α (cid:48) k ) ∩ G ( α (cid:48) k +1 ) Hence, there exists a unique measure µ α (cid:48) on G ( α (cid:48) ) such that µ α (cid:48) | G ( α (cid:48) k ) = µ α (cid:48) k for every k ∈ [1 , n ] ∩ N . We define µ (cid:48) = ( µ (cid:48) α (cid:48) ) α (cid:48) ∈ T (cid:48) , where T (cid:48) is the set of arcs transverse to Λ (cid:48) , as the family ofmeasures just constructed. By naturality, Λ (cid:48) is Γ Σ (cid:48) -invariant and µ (cid:48) is invariant by the action byhomeomorphisms of Γ Σ (cid:48) defined by γ ( µ (cid:48) α (cid:48) ) α (cid:48) ∈ T (cid:48) = ( γ ∗ µ (cid:48) γ − α (cid:48) ) α (cid:48) ∈ T (cid:48) , for every γ ∈ Γ Σ (cid:48) . Lemma 8
The family µ (cid:48) is the unique transverse measure on Λ (cid:48) such that if α (cid:48) is the lift of an arc α transverse to Λ , then µ α (cid:48) = ( f − α (cid:48) ) ∗ µ α . Moreover, the map µ (cid:55)→ µ (cid:48) from M L p (Σ) to M L p (Σ (cid:48) ) thus defined is a homeomorphism between M L p (Σ) and the space of measured flat laminations of Σ (cid:48) which are Γ Σ (cid:48) -invariant. Proof.
The properties (1) , (3) and (4) of Definition 6 are clearly satisfied by µ (cid:48) , and if h (cid:48) : F (cid:48) ⊂ G ( α (cid:48) ) → F (cid:48) ⊂ G ( α (cid:48) ) is a holonomy of Λ (cid:48) which is the lift of a holonomy h : F ⊂ G ( α ) → F ⊂ G ( α ) of Λ , with α , α two disjoint arcs transverse to Λ , then µ α (cid:48) | F (cid:48) = ( f − α (cid:48) ) ∗ ( µ α | F ) =( f − α (cid:48) ) ∗ h ∗ ( µ α | F ) = h (cid:48)∗ ( f − α (cid:48) ) ∗ ( µ α | F ) = h (cid:48)∗ ( µ α (cid:48) | F (cid:48) ) . Otherwise, we see h (cid:48) as a concatenation ofcompositions of holonomies that are lifts of some holonomies of Λ , and we show similarly that µ α (cid:48) | F (cid:48) = h (cid:48)∗ ( µ α (cid:48) | F (cid:48) ) . Hence µ (cid:48) is a transverse measure on Λ (cid:48) . The map from M L p (Σ) to M L p (Σ (cid:48) ) defined in this way is injective by construction. If (Λ (cid:48) , µ (cid:48) ) is a measured flat lamination of (Σ (cid:48) , [ q (cid:48) ]) which is Γ Σ (cid:48) -invariant, then the set of projections of the leaves of Λ (cid:48) by p : Σ (cid:48) → Σ is a flatlamination Λ of (Σ , [ q ]) , and if α is an arc transverse to Λ and α (cid:48) is a lift of α in Σ (cid:48) , then ( f α (cid:48) ) ∗ µ (cid:48) α (cid:48) is a measure on G ( α ) , that does not depend on the choice of the lift, by Γ Σ (cid:48) -invariance. The familyof measures defined in this way satisfies clearly the properties (1) , (3) and (4) of Definition 6.Moreover if h : F → F is a holonomy between two Borel sets of G ( α ) and G ( α ) , with α , α two disjoint arcs transverse to Λ , then it lifts to a holonomy between two Borel sets of G ( α (cid:48) ) and G ( α (cid:48) ) , with α (cid:48) , α (cid:48) two lifts of α , α , and we have µ α | F = h ∗ ( µ α | F ) . Hence (Λ , µ ) is a measuredflat lamination, and its image by the previous map is (Λ (cid:48) , µ (cid:48) ) .Hence, this map is a bijection between M L p (Σ) and the set of measured flat laminations of (Σ (cid:48) , [ q (cid:48) ]) that are Γ Σ (cid:48) -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 (cid:48) n , µ (cid:48) n ) n ∈ N and (Λ (cid:48) , µ (cid:48) ) are their images in M L p (Σ (cid:48) ) , then if α (cid:48) is an arc transverseto Λ (cid:48) 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 (cid:48) is transverse to Λ (cid:48) n for n large enough. Moreover, we have µ (cid:48) n,α (cid:48) = ( f − α (cid:48) ) ∗ µ n,α ∗ (cid:42) ( f − α (cid:48) ) ∗ µ α = µ (cid:48) α . If α (cid:48) is an arc transverse to Λ (cid:48) which is not the lift of anarc transverse to Λ , then by finite decomposition, we prove similarly that if n is large enough, Λ (cid:48) n is transverse to α (cid:48) and µ (cid:48) n,α (cid:48) ∗ (cid:42) µ (cid:48) α (cid:48) . Hence, the map (Λ , µ ) (cid:55)→ (Λ (cid:48) , µ (cid:48) ) is continuous. The sameproperty holds for its inverse, hence it is a homeomorphism. (cid:3) Let (Σ , [ q ]) be a connected, orientable surface with (possibly empty) boundary, endowed witha half-translation structure and let p : ( (cid:101) Σ , [ (cid:101) q ]) → (Σ , [ q ]) be a locally isometric universal cover,with covering groupe Γ (cid:101) Σ . The local geodesics of ( (cid:101) Σ , [ (cid:101) q ]) are geodesics, and if (cid:101) Λ is a flat laminationof ( (cid:101) Σ , [ (cid:101) q ]) and if α is an arc transverse to (cid:101) Λ such that every geodesic of G ( α ) intersects α ([0 , only at its origin, then the map g α : G ( α ) → [ G ( α )] defined by g α ( g ) = [ g ] is a homeomorphism.We denote by M Γ (cid:101) Σ ([ G [ (cid:101) q ] ]) the space of Radon measures on the space [ G [ (cid:101) q ] ] endowed with thegeodesic topology, which are Γ (cid:101) Σ and ι -invariant (where ι ( (cid:96) ) = (cid:96) − : t (cid:55)→ (cid:96) ( − t ) ) and whose supports re Γ (cid:101) Σ -invariant flat laminations, endowed with the weak-star topology. Let ν be an element of M Γ (cid:101) Σ ([ G [ (cid:101) q ] ]) whose support is (cid:101) Λ and let α be an arc transverse to (cid:101) Λ such that the geodesics of G ( α ) intersect α ([0 , only at their origins. Then (cid:101) µ α = ( g − α ) ∗ ( ν | [ G ( α )] ) is a Radon measure on G ( α ) whose support is g − α ( (cid:101) Λ ∩ [ G ( α )]) . If α is an arc transverse to (cid:101) Λ , but if some geodesic of G ( α ) possibly intersects α ([0 , at several points, then we define the measure (cid:101) µ α by a finite gluingprocess as in Section 3, which is always possible since the geodesics of G [ (cid:101) q ] are proper. Let (cid:101) µ ν bethe family of transverse measures defined in this way. Proposition 9
The family of measures (cid:101) µ ν is a transverse measure on (cid:101) Λ , and the map ν (cid:55)→ (cid:101) µ ν is a homeomorphism between M Γ (cid:101) Σ ([ G [ (cid:101) q ] ]) and the space of measured flat laminations on ( (cid:101) Σ , [ (cid:101) q ]) which are Γ (cid:101) Σ -invariant. Proof.
Unless the opposite is specified, until the end of the proof, if α is an arc transverse to aflat lamination (cid:101) Λ , we assume that the geodesics of G ( α ) only intersect α ([0 , at their origins. Wemay always assume this up to shortening α , since the geodesics of G ( α ) are proper and transverseto α .The properties (1) , (3) , and (4) of Definition 6 are clearly satisfied by (cid:101) µ ν . If h : F → F is aholonomy of (cid:101) Λ between two Borel sets F of G ( α ) and F of G ( α ) , then, by the definition of theholonomies, the sets [ F ] and [ F ] are equal. Hence (cid:101) µ ν,α | F = ( g − α ) ∗ ( ν | [ F ] ) = h ∗ ( g − α ) ∗ ( ν | [ F ] ) = h ∗ ( (cid:101) µ ν,α | F ) and (cid:101) µ ν is invariant by holonomy, so it is a transverse measure on (cid:101) Λ , which is Γ (cid:101) Σ -invariant by naturality.Assume that two measures ν and ν of M Γ (cid:101) Σ ([ G [ (cid:101) q ] ]) define the same measured flat lamination ( (cid:101) Λ , (cid:101) µ ) by this construction. Then, if U is a relatively compact open set of [ G [ (cid:101) q ] ] , the set U iscontained in a finite union of open sets ( U i ) (cid:54) i (cid:54) n such that for every U i there exists an arc α i transverse to (cid:101) Λ , such that U i ⊂ [ G ( α i )] . We have ν ( U i ) = ν ( U i ) for all (cid:54) i (cid:54) n , and then ν ( U ) = ν ( U ) . Since the relatively compact open sets span the Borel tribute, we have ν = ν .Hence, the map ν (cid:55)→ (cid:101) µ ν is injective.Let us now prove it is surjective. Let ( (cid:101) Λ , (cid:101) µ ) be a measured flat lamination and let R be thesets of subsets A ⊂ [ G [ (cid:101) q ] ] such that there exists a finite family of arcs α , . . . , α n , with n ∈ N ,transverse to (cid:101) Λ (and such that the elements of G ( α i ) intersect α i ([0 , only at their origins), forall (cid:54) i (cid:54) n , and some Borel subsets A i ⊂ G ( α i ) such that A = (cid:113) [ A i ] . Then R is non empty and ∅ ∈ R . If A, B are two elements of R , let ( α i ) (cid:54) i (cid:54) n and ( β j ) (cid:54) j (cid:54) k be some finite families of arcstransverse to (cid:101) Λ and ( A i ⊆ G ( α i )) (cid:54) i (cid:54) n and ( B j ⊆ G ( β j )) (cid:54) j (cid:54) k be some families of Borel subsetsof G [ (cid:101) q ] such that A = (cid:113) [ A i ] and B = (cid:113) [ B j ] . For all (cid:54) i (cid:54) n and (cid:54) j (cid:54) k , we replace A i by A i − A i ∩ g − α i ([ B j ] ∩ [ G ( α i )]) . Then A ∪ B = ( (cid:113) [ A i ]) (cid:113) ( (cid:113) [ B j ]) and A − B = (cid:113) [ A i ] . Hence, the set R is a ring. Let A ∈ R and let ( A i ⊆ G ( α i )) (cid:54) i (cid:54) n , be as above. We define ν ( A ) = n (cid:80) i =1 µ α i ( A i ) . Then ν ( A ) (cid:62) for all A ∈ R and ν ( ∅ ) = 0 . Let ( A n ) n ∈ N be a sequence of pairwise disjoint elements of R such that (cid:113) n ∈ N A n belongs to R . Let { α i } (cid:54) i (cid:54) k be a family of arcs transverse to (cid:101) Λ associated to (cid:113) n ∈ N A n as above. For every n ∈ N , there exists a family { A (cid:48) n,i ⊂ G ( α i ) } (cid:54) i (cid:54) k of Borel subsets suchthat A n = k (cid:113) i =1 [ A (cid:48) n,i ] . If n (cid:54) = m , we have A (cid:48) n,i ∩ A (cid:48) m,i = ∅ . Hence ν ( A ) = k (cid:88) i =1 (cid:101) µ α i ( (cid:113) n ∈ N A (cid:48) n,i )= (cid:88) n ∈ N k (cid:88) i =1 (cid:101) µ α i ( A (cid:48) n,i )= (cid:88) n ∈ N ν ( A n ) ence, the constructed map is σ -additive on R , and since ν ( ∅ ) = 0 it is a premeasure on R . Forevery arc α transverse to (cid:101) 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 originsbelong to a compact subset of (cid:101) Σ is compact, and since (cid:101) Σ is a countable union of compact sets, thespace [ G [ (cid:101) q ] ] is a countable union of compact sets, hence ν is σ -finite. According to the theorem ofextension of Caratheodory, the premeasure ν can be extended to a unique measure, still denotedby ν , on the σ -algebra generated by R , which is equal to the Borel tribute of [ G [ (cid:101) q ] ] . Accordingto the properties (1) and (3) of Definition 6, the support of the measure ν is (cid:101) Λ and ν is ι and Γ (cid:101) Σ -invariant by naturality. By construction, the measured flat lamination associated to ν by theprevious map is ( (cid:101) Λ , (cid:101) µ ) . Hence, the map ν (cid:55)→ (cid:101) µ ν is bijective.Finally, if ( ν n ) n ∈ N is a sequence in M Γ (cid:101) Σ ([ G [ (cid:101) q ] ]) which converges to ν , and if ( (cid:101) Λ n , (cid:101) µ n ) n ∈ N and ( (cid:101) Λ , (cid:101) µ ) are their images in M L p ( (cid:101) Σ) , then if α is an arc transverse to (cid:101) Λ (such that the geodesics of G ( α ) intersect α ([0 , only at their origins), for n large enough, the lamination (cid:101) Λ n is transverseto α and (cid:101) µ n,α = ( g − α ) ∗ ( ν n | [ G ( α )] ) ∗ (cid:42) ( g − α ) ∗ ( ν | [ G ( α )] ) = (cid:101) µ α . Hence, the map ν (cid:55)→ (cid:101) µ ν is continuous.Similarly, if ( (cid:101) Λ n , (cid:101) µ n ) n ∈ N is a sequence of measured flat laminations of (cid:101) Σ which are Γ (cid:101) Σ -invariant,that converges to ( (cid:101) Λ , (cid:101) µ ) , and if ( ν n ) n ∈ N and ν are their preimages in M Γ (cid:101) Σ ([ G [ (cid:101) q ] ]) , then if f is acontinuous map from [ G [ (cid:101) q ] ] to R whose support is compact, there exists a finite family { α , . . . , α p } of arcs transverse to (cid:101) Λ and to (cid:101) Λ n for n large enough, such that Supp( f ) ⊂ (cid:83) (cid:54) i (cid:54) p [ G ( α i )] , and for all (cid:54) i (cid:54) p and n large enough, we have ν n ( f | [ G ( α i )] ) = (cid:101) µ n,α i ( f ◦ g α i | G ( α i ) ) −→ (cid:101) µ α i ( f ◦ g α i | G ( α i ) ) = ν ( f | [ G ( α i )] ) . Hence ν n ∗ (cid:42) ν and the inverse map is continuous. Hence, the map ν (cid:55)→ (cid:101) µ ν is ahomeomorphism. (cid:3) Let φ : M L p (Σ) → M Γ (cid:101) Σ ([ G [ (cid:101) q ] ]) be the composition of the map µ (cid:55)→ µ (cid:48) defined at Lemma 8,with Σ (cid:48) = (cid:101) Σ , with the inverse of the map ν (cid:55)→ (cid:101) µ ν defined at Lemma 9. Corollary 10
The map φ is a homeomorphism. (cid:3) In this Section 5, we still denote by (Σ , [ q ]) a connected, orientable surface with (possiblyempty) boundary, endowed with a half-translation structure, and by p : ( (cid:101) Σ , [ (cid:101) q ]) → (Σ , [ q ]) a locallyisometric universal cover whose covering group is Γ (cid:101) Σ . We assume that Σ is compact and that χ (Σ) < , and we denote by m a hyperbolic metric with totally geodesic boundary on Σ and by (cid:101) m the unique hyperbolic metric on (cid:101) Σ such that p : ( (cid:101) Σ , (cid:101) m ) → (Σ , m ) is locally isometric. For everygeodesic g of [ G [ (cid:101) q ] ] or [ G (cid:101) m ] , we denote by E ( g ) ∈ ∂ ∞ (cid:101) 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 hyperboliclaminations on (Σ , m ) , endowed with the topology defined in [Bon1, p. 19], is homeomorphic tothe space of Radon measures on [ G (cid:101) m ] that are Γ (cid:101) Σ et ι -invariant, whose supports are hyperboliclaminations, denoted by M Γ (cid:101) Σ ([ G (cid:101) m ]) (see [Bon2, Prop. 17 p. 154]). Here, we use this fact andthe homeomorphism between M Γ (cid:101) Σ ([ G [ (cid:101) q ] ]) and M L p (Σ) defined in Corollary 10 to investigate thelinks between M L p (Σ) and M L h (Σ) . We denote by ϕ : [ G [ (cid:101) q ] ] → [ G (cid:101) m ] the map associating to thegeodesic (cid:101) (cid:96) ∈ [ G [ (cid:101) q ] ] the geodesic ϕ ( (cid:101) (cid:96) ) ∈ [ G (cid:101) m ] that corresponds to (cid:101) (cid:96) , i.e. such that E ( ϕ ( (cid:101) (cid:96) )) = E ( (cid:101) (cid:96) ) (see [Mor, §4.2]). Then, ϕ is surjective and continuous, and a closed subset F of [ G [ (cid:101) q ] ] is a flatlamination if and only if ϕ ( F ) is a hyperbolic lamination. Moreover ϕ is proper. Indeed, if K is a compact set of [ G (cid:101) m ] and if ( (cid:101) (cid:96) n ) n ∈ N is a sequence of ϕ − ( K ) , then by definition of ϕ , up totaking a subsequence, the sequence ( E ( (cid:101) (cid:96) n )) n ∈ N converges in ∂ ∞ (cid:101) Σ . Since ( (cid:101) Σ , [ (cid:101) q ]) is δ -hyperbolic (as Σ is compact and χ (Σ) < , see [Mor, Rem. 2.10]), according to [Mor, Lem. 2.6], up to takinga subsequence, the sequence ( (cid:101) (cid:96) n ) n ∈ N converges to a geodesic (cid:101) (cid:96) such that E ( (cid:101) (cid:96) ) ∈ E ( K ) . Hence (cid:101) (cid:96) belongs to ϕ − ( K ) , therefore ϕ − ( K ) is compact. owever, the map ϕ is not injective. By definition, two different geodesics (cid:101) (cid:96), (cid:101) (cid:96) (cid:48) of ( (cid:101) Σ , [ (cid:101) q ]) havethe same image by ϕ if and only if they have the same ordered pair of points at infinity, andaccording to the flat strip theorem (see for example [BH, Th. 2.13 P. 182]), their images areparallel and contained in a maximal flat strip of ( (cid:101) Σ , [ (cid:101) q ]) . Then, their projections in (Σ , [ q ]) arefreely homotopic periodic local geodesics, and hence their images are contained in a maximal flatcylinder (see for example [MS1, Th. 2.(c)]). The points at infinity of (cid:101) (cid:96) , (cid:101) (cid:96) (cid:48) and of their images ϕ ( (cid:101) (cid:96) ) = ϕ ( (cid:101) (cid:96) (cid:48) ) are therefore the attractive and repulsive fixed points of an element of the coveringgroup, and the projection of ϕ ( (cid:101) (cid:96) ) is a closed local geodesic of (Σ , m ) . Hence, the restriction of ϕ tothe set of geodesics of ( (cid:101) Σ , [ (cid:101) q ]) whose projections are not periodic is an injective map whose imageis the set of geodesics of ( (cid:101) Σ , (cid:101) m ) whose projections are not closed geodesics. Moreover, the preimageof a geodesic (cid:101) λ of ( (cid:101) Σ , (cid:101) m ) whose projection is a closed local geodesic is the set of geodesics of ( (cid:101) Σ , [ (cid:101) q ]) having the same ordered pair of points at infinity than (cid:101) λ , which may be a unique geodesic, or maybe a set of parallel geodesics whose images are contained in a flat strip, and whose projections areperiodic. Lemma 11
The map ϕ defines a continuous, surjective and proper map ϕ ∗ from the space ofRadon measures on [ G [ (cid:101) q ] ] to the space of Radon measures on [ G (cid:101) m ] . Moreover, ϕ ∗ ν belongs to M Γ (cid:101) Σ ([ G (cid:101) m ]) if and only if ν belongs to M Γ (cid:101) Σ ([ G [ (cid:101) q ] ]) and the restriction of ϕ ∗ to M Γ (cid:101) Σ ([ G [ (cid:101) q ] ]) is asurjective map ϕ ∗ : M Γ (cid:101) Σ ([ G [ (cid:101) q ] ]) → M Γ (cid:101) Σ ([ G (cid:101) m ]) . Proof.
The map ϕ is continuous and proper, hence it defines a continuous map ϕ ∗ from the spaceof Radon measures on [ G [ (cid:101) q ] ] to the space of Radon measures on [ G (cid:101) m ] . Let us first prove that ϕ ∗ issurjective. The map s : [ G (cid:101) m ] → [ G [ (cid:101) q ] ] which associates, to a hyperbolic geodesic, the flat geodesic towhich it corresponds (if it is unique) and the "middle" geodesic of the set of geodesics (containedin 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 (cid:101) m ] to the space of Radon measures on [ G [ (cid:101) q ] ] ,and ϕ ∗ ◦ s ∗ = Id . Therefore ϕ ∗ is surjective.Let us now prove that ϕ ∗ is proper. The space [ G [ (cid:101) q ] ] is countable at infinity. Thus, thereexists a sequence ( K n ) n ∈ N of compact sets such that, for all n ∈ N , K n is contained in theinterior of K n +1 and (cid:83) n ∈ N K n = [ G [ (cid:101) q ] ] . If C is a compact set of the space of Radon measureson [ G (cid:101) m ] and if K is a compact set of [ G [ (cid:101) q ] ] , then the set { ν ( K ) , ν ∈ ( ϕ ∗ ) − ( C ) } is bounded bythe maximum of { ν ( ϕ ( K )) , ν ∈ C } , which is finite since C is compact. Hence, for all n ∈ N ,the set { ν | K n , ν ∈ ( ϕ ∗ ) − ( C ) } is compact and if ( ν k ) k ∈ N is a sequence of ( ϕ ∗ ) − ( C ) , proceedingby diagonal extraction, we show that there exists a subsequence, still denoted by ( ν k ) k ∈ N , anda Radon measure ν on [ G [ (cid:101) q ] ] such that for all n ∈ N , ν k | K n ∗ (cid:42) ν | K n . And, by the choice of thesequence ( K n ) n ∈ N , for all f ∈ C c ([ G [ (cid:101) q ] ]) , there exists n ∈ N such that Supp( f ) ⊂ K n , and then ( ν k ( f )) k ∈ N = ( ν k | K n ( f )) k ∈ N converges to ν | K n ( f ) = ν ( f ) . Hence, we have ν k ∗ (cid:42) ν and ϕ ∗ is properon the space of Radon measures.Finally, by definition of ϕ , the measure ϕ ∗ ν belongs to M Γ (cid:101) Σ ([ G (cid:101) m ]) if and only if ν belongs to M Γ (cid:101) Σ ([ G [ (cid:101) q ] ]) . Moreover, the space M Γ (cid:101) Σ ([ G (cid:101) m ]) is closed (see [Bon2, Prop. 3 et 17]), and since ϕ ∗ iscontinuous, its preimage is closed. Hence, the restriction of ϕ ∗ to these spaces is continuous, properand surjective. (cid:3) The map ϕ ∗ is not injective. Considering the lack of injectivity of ϕ , the preimage by ϕ ∗ of aRadon measure on [ G (cid:101) m ] whose support contains no geodesic whose projection in (Σ , m ) is a closedgeodesic, consists in a unique Radon measure on [ G [ (cid:101) q ] ] whose support contains no geodesic whoseprojection in (Σ , [ q ]) is a periodic local geodesic. However, the preimage by ϕ ∗ of the Γ (cid:101) Σ -orbit ofa Dirac measure, whose support is the Γ (cid:101) Σ -orbit of a geodesic (cid:101) λ , of mass δ , whose projection in (Σ , m ) is a closed geodesic, is the set of Radon measures on [ G [ (cid:101) q ] ] , whose support is the Γ (cid:101) Σ -orbitof a closed subset F of the set of geodesics of ( (cid:101) Σ , [ (cid:101) q ]) having the same ordered pair of points atinfinity that (cid:101) λ , such that the mass of F is δ . If there exist at least two such geodesics, then this set s the set of parallel geodesics contained in a flat strip, of width L > , hence it is homeomorphicto [0 , L ] . Hence, the set F is homeomorphic to a closed subset of [0 , L ] . Consequently, the preimageof 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 (cid:101) Σ ([ G (cid:101) m ]) and to M Γ (cid:101) Σ ([ G [ (cid:101) q ] ]) , hence ϕ ∗ defines a continuous map ψ : M L p (Σ) → M L h (Σ) which is proper andsurjective. The group R + ∗ acts on these two spaces by multiplication of the measures. We denoteby PM L p (Σ) and PM L h (Σ) the quotient spaces for these actions. Since ψ is equivariant bythese actions, it defines a continuous map ψ : PM L p (Σ) → PM L h (Σ) which is surjective andproper. We deduce from this the following lemmas. Lemma 12
The space
PM L p (Σ) is compact. Proof.
The space
PM L h (Σ) is compact (see [Bon2, Cor. 5 and Prop. 17]) and ψ is proper. (cid:3) If Σ is compact, a measured cylinder lamination is a measured flat lamination having a uniquecomponent which is a cylinder component (see [Mor, §6]). Lemma 13 If Σ is compact, the set of measured cylinder laminations having finitely many leavesis 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 aDirac 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 denoteby α : [0 , T ] → C ( T > ) a geodesic arc such that the interior of the maximal flat cylinder C thatcontains the support of Λ , endowed with the induced distance, is isometric to α (]0 , T [) × S . 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 ofRadon measures on [0 , T ] , there exists a sequence ( µ n,α ) n ∈ N of Radon measures with finite supporton 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 ∗ (cid:42) µ α . Moreover, Λ n = [Supp( µ n,α )] is a flat lamination and µ n,α defines a transversemeasure on Λ n such that the sequence (Λ n , µ n ) n ∈ N converges to (Λ , µ ) . (cid:3) 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 measuredhyperbolic lamination having no closed leaf consists in a unique measured flat lamination havingno 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 > , the preimageof (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 freelyhomotopic to λ contains at least two elements, it foliates a maximal flat cylinder. Then, this setis homeomorphic to [0 , L ] , with L is the height of the flat cylinder, so F is homeomorphic to aclosed subset of [0 , L ] . Hence, the preimage of (Λ m , µ m ) by ψ is homeomorphic to the set of Borelmeasures on [0 , L ] of mass δ . Since ϕ ∗ is equivariant for the addition of measures, we see that if ameasured hyperbolic lamination has some closed leaves λ , . . . , λ p , of respective masses δ , . . . , δ p ( p is always at most g − b ), then the preimage of (Λ m , µ m ) by ψ is homeomorphic to theCartesian product of the sets of Borel measures on [0 , L i ] , (cid:54) i (cid:54) p , where L i is the height of themaximal flat cylinder, union of the local geodesics of (Σ , [ q ]) freely homotopic to λ i , whose totalmass is δ i .Since Σ is compact, the projectified space PM L h (Σ) is homeomorphic to the sphere S g − b (see [Bon1, Th. 17]). If the support of the measured hyperbolic laminations in the equivalence classof 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 , . . . λ p belong to the support of the measured hyperbolic laminations in theequivalence class of x , and if L , . . . , L p are the heights of the maximal flat cylinders, unions of the mages of the periodic local geodesics of (Σ , [ q ]) that are freely homotopic to λ , . . . , 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 . In particular, since the set of measured hyperbolic laminationswhose support contains a closed leaf is projectively dense in the space of measured hyperboliclaminations, the subset of PM L h (Σ) of points whose preimages by ψ are not a single point isdense in PM L h (Σ) . If g : R → (cid:101) Σ is a geodesic of ( (cid:101) Σ , [ (cid:101) q ]) or of ( (cid:101) Σ , (cid:101) m ) , the orientation of (cid:101) Σ allows to define the twosides (arbitrarily) + and − of g ( R ) (except if g ( R ) is a boundary component of ∂ (cid:101) Σ , in which case g ( R ) has one side). We denote by C + ( g ) and C − ( g ) the two complementary sides which are theunions of g ( R ) with the connected components of (cid:101) Σ − g ( R ) corresponding to one and the other sidesof g ( R ) ( (cid:101) Σ − g ( R ) may have more than two connected components if g ( R ) is not disjoint from ∂ (cid:101) Σ ).If (cid:101) Λ is a geodesic lamination of ( (cid:101) Σ , [ (cid:101) q ]) or ( (cid:101) Σ , (cid:101) m ) , and if g is a leaf of (cid:101) Λ , then the image of anotherleaf of (cid:101) 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 and g be two leaves of (cid:101) Λ . We denote by C i the complementary side of g i ( R ) that contains g i +1 ( R ) (with i ∈ Z / Z ). We define C ( g , g ) = C ∩ C . Let c be a geodesic segment joining theirimages (if the images are not disjoint, the segment c may be a point). A leaf of (cid:101) Λ intersects c nontrivially if it is contained in C ( g , g ) and if it intersects both complementary components of theimage of c in C ( g , g ) , and we denote by B ( g , g ) = B (cid:101) Λ ( g , g ) the set of leaves of (cid:101) Λ intersecting c non trivially. Lemma 14
The compact set B ( g , g ) does not depend on the choice of c . Proof.
The set of leaves whose images are contained in C ∩ C is compact and the requirement tointersect c non trivially is closed in this set, hence B ( g , g ) is compact. Let c (cid:48) be another geodesicsegment joining the images of g and g . Since both c and c (cid:48) separate C ∩ C in two connectedcomponents and since the intersection of the images of two geodesic segments of ( (cid:101) Σ , [ (cid:101) q ]) (or betweenthe image of a geodesic and a point) is connected, every leaf that intersects c non trivially alsointersects c (cid:48) non trivially, and conversely. (cid:3) The set of free homotopy classes of simple closed curves on Σ endowed with transverse measureswhich are Dirac measures of positive masses, embeds into M L h (Σ) , and the intersection number onthis set can be extended, in a unique way, to a continuous map i : M L h (Σ) × M L h (Σ) → R + (see[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 laminationand let ν µ [ q ] ∈ M Γ (cid:101) Σ ([ G [ (cid:101) q ] ]) be the measure defined by (Λ [ q ] , µ [ q ] ) on [ G [ (cid:101) q ] ] (see Corollary 10). Wedefine the intersection number between (Λ [ q ] , µ [ q ] ) and α by i [ q ] ( µ [ q ] , α ) = i ( ψ (Λ [ q ] , µ [ q ] ) , α ) . If k ∈ N , we have i [ q ] ( µ [ q ] , α k ) = k i [ q ] ( µ [ q ] , α ) . Hence, we assume that α is primitive (i.e. if thereexists a free homotopy class α such that α = α k , then k = ± ). We denote by α [ q ] a periodiclocal flat geodesic in the class of α , and by (cid:101) α [ q ] a lift of α [ q ] in (cid:101) Σ . Let γ ∈ Γ (cid:101) Σ − { e } be one of thetwo primitive hyperbolic elements of Γ (cid:101) Σ whose translation axis is (cid:101) α [ q ] ( R ) , and let ( (cid:101) Λ [ q ] , (cid:101) µ [ q ] ) be thepreimage of (Λ [ q ] , µ [ q ] ) in (cid:101) Σ . Lemma 15
Let (cid:101) (cid:96) be a leaf of (cid:101) Λ [ (cid:101) q ] which is interlaced with (cid:101) α [ q ] . The number i [ q ] ( µ [ q ] , α ) is equal to ν µ [ q ] ( B (cid:101) Λ ( (cid:101) (cid:96), γ (cid:101) (cid:96) ) − γ (cid:101) (cid:96) ) . If there exists no such leaf, then i [ q ] ( µ [ q ] , α ) = 0 . roof. The number i ( ψ (Λ [ q ] , µ [ q ] ) , α ) is equal to ϕ ∗ ν µ [ q ] ( F m ) , where F m is the set of leaves of themeasured hyperbolic lamination ( (cid:101) Λ (cid:101) m , (cid:101) µ m ) defined by ϕ ∗ ν µ [ q ] which intersect a segment I = [ a, γa [ transversally, with a ∈ (cid:101) α m ( R ) , where (cid:101) α m ( R ) is the translation axis of γ in ( (cid:101) Σ , (cid:101) m ) , which is afundamental domain of (cid:101) α m ( R ) for the action by translations of γ Z (see [Bon2, Prop. 3]). Since thechoice of a is arbitrary, if F m is not empty, we may assume that a is an intersection point between aleaf (cid:101) λ of (cid:101) Λ (cid:101) m and (cid:101) α m ( R ) . Hence, F m = B (cid:101) Λ (cid:102) m ( (cid:101) λ, γ (cid:101) λ ) − γ (cid:101) λ . Moreover ϕ ∗ ν µ [ q ] ( F m ) = ν µ [ q ] ( ϕ − ( F m )) ,and by definition of ϕ , ϕ − ( F m ) = B (cid:101) Λ [ (cid:101) q ] ( (cid:101) (cid:96), γ (cid:101) (cid:96) ) − γ (cid:101) (cid:96) , with (cid:101) (cid:96) ∈ (cid:101) Λ [ (cid:101) q ] belonging to ϕ − ( λ ) (since ν µ [ q ] is γ -invariant, if several leaves belong to ϕ − ( λ ) , we can choose (cid:101) (cid:96) arbitrarily in this set). Finally, if F m is empty, no leaf of (cid:101) Λ [ (cid:101) q ] is interlaced with (cid:101) α [ q ] and i ( ψ (Λ [ q ] , µ [ q ] ) , α ) = 0 . (cid:3) Remark.
We could define the intersection number between a free homotopy class of closed curveswith (Λ [ q ] , µ [ q ] ) by the infimum of the masses given by the measured flat lamination to the closedcurves that are piecewise transverse to the lamination, similarly to the intersection number witha measured foliation, but this infimum would not be necessarly attained since the periodic localgeodesics are generally not piecewise transverse to the lamination.Furthermore, in the case of compact surfaces endowed with a half-translation structure, whoseboundary is empty, contrarily to measured hyperbolic lamination (see [Ota, Th. 2]), the intersectionnumbers with the free homotopy classes of closed curves of Σ do not separate the measured flat la-minations, but only their images in M L h (Σ) . In particular, the topology defined after Definition 6is not equivalent to the one induced by the product topology on R H , with H the set of free homo-topy classes of closed curves, on the image of M L p (Σ) by the map (Λ [ q ] , µ [ q ] ) (cid:55)→ ( i ( µ [ q ] , α )) α ∈ H . Let (Σ , [ q ]) be a compact, connected, orientable surface with (possibly empty) boundary, en-dowed with a half-translation structure, such that χ (Σ) < . Let p : ( (cid:101) Σ , [ (cid:101) q ]) → (Σ , [ q ]) be a locallyisometric universal cover and let (Λ , µ ) be a measured flat lamination on (Σ , [ q ]) . We denote by ( (cid:101) Λ , (cid:101) µ ) its preimage in ( (cid:101) Σ , [ (cid:101) q ]) and by ν (cid:101) µ the Borel measure that it defines on G [ (cid:101) q ] (see Lemma 9).We first assume that ν (cid:101) µ has no atom.If { (cid:101) (cid:96) , (cid:101) (cid:96) } is a pair of leaves of (cid:101) Λ , we define (cid:101) d (cid:101) Λ ( (cid:101) (cid:96) , (cid:101) (cid:96) ) = ν (cid:101) µ ( B ( (cid:101) (cid:96) , (cid:101) (cid:96) )) (see Section 5.1 for thedefinition of B ( (cid:101) (cid:96) , (cid:101) (cid:96) )) ). Then (cid:101) d (cid:101) Λ ( (cid:101) (cid:96) , (cid:101) (cid:96) ) (cid:62) and (cid:101) d (cid:101) Λ ( (cid:101) (cid:96) , (cid:101) (cid:96) ) = (cid:101) d (cid:101) Λ ( (cid:101) (cid:96) , (cid:101) (cid:96) ) . Moreover, if (cid:101) (cid:96) , (cid:101) (cid:96) and (cid:101) (cid:96) are three leaves of (cid:101) Λ and c , c and c are some geodesic segments joining respectively the imagesof (cid:101) (cid:96) and (cid:101) (cid:96) , (cid:101) (cid:96) and (cid:101) (cid:96) , and (cid:101) (cid:96) and (cid:101) (cid:96) (denoted by c , , on the figure, cases , and ), then eithernone of the three leaves separates the other ones (case , see Section 5.1 for the definition of a leafseparating to other leaves) or one separates the other ones (case , and ). In case , every leafof B ( (cid:101) (cid:96) , (cid:101) (cid:96) ) intersects c or c non trivially, hence B ( (cid:101) (cid:96) , (cid:101) (cid:96) ) ⊆ B ( (cid:101) (cid:96) , (cid:101) (cid:96) ) ∪ B ( (cid:101) (cid:96) , (cid:101) (cid:96) ) . In case , wehave B ( (cid:101) (cid:96) , (cid:101) (cid:96) ) = B ( (cid:101) (cid:96) , (cid:101) (cid:96) ) ∪ B ( (cid:101) (cid:96) , (cid:101) (cid:96) ) . In case , we have B ( (cid:101) (cid:96) , (cid:101) (cid:96) ) ⊆ B ( (cid:101) (cid:96) , (cid:101) (cid:96) ) and in case , wehave B ( (cid:101) (cid:96) , (cid:101) (cid:96) ) ⊆ B ( (cid:101) (cid:96) , (cid:101) (cid:96) ) . In any case, we have (cid:101) d (cid:101) Λ ( (cid:101) (cid:96) , (cid:101) (cid:96) ) (cid:54) (cid:101) d (cid:101) Λ ( (cid:101) (cid:96) , (cid:101) (cid:96) ) + (cid:101) d (cid:101) Λ ( (cid:101) (cid:96) , (cid:101) (cid:96) ) . Hence (cid:101) d (cid:101) Λ is apseudo-distance on (cid:101) Λ . c c c case case case case (cid:101) (cid:96) (cid:101) (cid:96) (cid:101) (cid:96) (cid:101) (cid:96) (cid:101) (cid:96) (cid:101) (cid:96) (cid:101) (cid:96) c , , (cid:101) (cid:96) (cid:101) (cid:96) c , , (cid:101) (cid:96) (cid:101) (cid:96) (cid:101) (cid:96) c , , e denote by ( T, d T ) the quotient metric space ( (cid:101) Λ , (cid:101) d (cid:101) Λ ) / ∼ , where (cid:101) (cid:96) ∼ (cid:101) (cid:96) (cid:48) if and only if (cid:101) d (cid:101) Λ ( (cid:101) (cid:96), (cid:101) (cid:96) (cid:48) ) =0 , and if F is a set of leaves of (cid:101) Λ , we denote by F T its image by the quotient map. Remark 16
Let (cid:101) (cid:96) and (cid:101) (cid:96) be two distinct leaves of (cid:101) Λ . Since ν (cid:101) µ has no atom and its support is (cid:101) Λ , we have ν (cid:101) µ ( B ( (cid:101) (cid:96) , (cid:101) (cid:96) )) = 0 if and only if B ( (cid:101) (cid:96) , (cid:101) (cid:96) ) = { (cid:101) (cid:96) , (cid:101) (cid:96) } , and the topology defined by thedistance d T is equivalent to the quotient topology of the topology induced by the geodesic topologyon (cid:101) Λ , by the equivalence relation (cid:101) (cid:96) R (cid:101) (cid:96) if and only if B ( (cid:101) (cid:96) , (cid:101) (cid:96) ) = { (cid:101) (cid:96) , (cid:101) (cid:96) } .Finally, since the image of a leaf has two complementary sides (except if its image is a boundarycomponent), and the image of each of the other leaves is contained in one of them, the relationdefined on B ( (cid:101) (cid:96) , (cid:101) (cid:96) ) by (cid:101) (cid:96) (cid:22) (cid:101) (cid:96) (cid:48) if (cid:101) (cid:96) belongs to B ( (cid:101) (cid:96) , (cid:101) (cid:96) (cid:48) ) is a total order which is compatible with R ,hence defines a total order on B ( (cid:101) (cid:96) , (cid:101) (cid:96) ) T . Lemma 17
The metric space ( T, d T ) is an R -tree. Proof.
Let (cid:101) (cid:96) and (cid:101) (cid:96) be two leaves of (cid:101) Λ . The map f : B ( (cid:101) (cid:96) , (cid:101) (cid:96) ) → R + defined by f ( (cid:101) (cid:96) ) = (cid:101) d (cid:101) Λ ( (cid:101) (cid:96) , (cid:101) (cid:96) ) isnondecreasing (with respect to (cid:22) ) and continuous since ν (cid:101) µ has no atom. Moreover, it is compatiblewith the equivalence relation R and defines an increasing continuous map f : B ( (cid:101) (cid:96) , (cid:101) (cid:96) ) T → R + .Since B ( (cid:101) (cid:96) , (cid:101) (cid:96) ) T is compact, it is a homeomorphism onto its image. Assume that its image is notan interval. Since it is a compact subset of R , if U is a bounded complementary component of f ( B ( (cid:101) (cid:96) , (cid:101) (cid:96) )) in R + , then its closure is an interval [ a, b ] with a < b . Let (cid:101) (cid:96) a and (cid:101) (cid:96) b be some leavesof f − ( a ) and f − ( b ) . If there exists a leaf (cid:101) (cid:96) ∈ B ( (cid:101) (cid:96) a , (cid:101) (cid:96) b ) − { (cid:101) (cid:96) a , (cid:101) (cid:96) b } , since ν (cid:101) µ has no atom and itssupport is (cid:101) Λ , we have a < f ( (cid:101) (cid:96) T ) < b . Hence B ( (cid:101) (cid:96) a , (cid:101) (cid:96) b ) = { (cid:101) (cid:96) a , (cid:101) (cid:96) b } , which is impossible since thenwe would have (cid:101) d ( (cid:101) (cid:96) a , (cid:101) (cid:96) b ) = 0 and thus a = b . Hence, the image of f is the interval [0 , d T ( (cid:101) (cid:96) T , (cid:101) (cid:96) (cid:48) T )] ,and f − : [0 , d T ( (cid:101) (cid:96) T , (cid:101) (cid:96) (cid:48) T )] → T is a geodesic segment between (cid:101) (cid:96) T and (cid:101) (cid:96) (cid:48) T . Moreover, up toreparametrization, it is the unique arc joining (cid:101) (cid:96) T to (cid:101) (cid:96) (cid:48) T . Indeed, let g : [0 , → T be another arcjoining (cid:101) (cid:96) T and (cid:101) (cid:96) (cid:48) T . If a leaf (cid:101) (cid:96) belongs to B ( (cid:101) (cid:96), (cid:101) (cid:96) (cid:48) ) then it separates (cid:101) (cid:96) and (cid:101) (cid:96) (cid:48) (in the sense of Section5.1) and since g is continuous for the quotient topology of the geodesic topology by the equivalencerelation R , the point (cid:101) (cid:96) T belongs to the image of g . Hence B ( (cid:101) (cid:96), (cid:101) (cid:96) (cid:48) ) T is contained in the image of g .Assume that there exists an element x = g ( t ) , with t ∈ ]0 , , in the image of g that does notbelong to B ( (cid:101) (cid:96), (cid:101) (cid:96) (cid:48) ) T and let (cid:101) (cid:96) x be a leaf representing x . Assume for a contradiction that (cid:101) (cid:96) x ( R ) iscontained in C ( (cid:101) (cid:96), (cid:101) (cid:96) (cid:48) ) (case below, see Section 5.1 for the definition of C ( (cid:101) (cid:96), (cid:101) (cid:96) (cid:48) ) ). Then B ( (cid:101) (cid:96), (cid:101) (cid:96) x ) isthe union of the compact sets B ( (cid:101) (cid:96), (cid:101) (cid:96) x ) ∩ B ( (cid:101) (cid:96), (cid:101) (cid:96) (cid:48) ) and B ( (cid:101) (cid:96), (cid:101) (cid:96) x ) ∩ B ( (cid:101) (cid:96) x , (cid:101) (cid:96) (cid:48) ) , whose intersection is { (cid:101) (cid:96) x } .Let (cid:101) (cid:96) y be the closest element to (cid:101) (cid:96) in the compact set ( B ( (cid:101) (cid:96), (cid:101) (cid:96) x ) − B ( (cid:101) (cid:96), (cid:101) (cid:96) (cid:48) ) ∩ B ( (cid:101) (cid:96), (cid:101) (cid:96) x )) ∪{ (cid:101) (cid:96) x } (for (cid:22) ). Byassumption on (cid:101) (cid:96) x , we see that (cid:101) (cid:96) y is also the closest element to (cid:101) (cid:96) (cid:48) in B ( (cid:101) (cid:96) x , (cid:101) (cid:96) (cid:48) ) − B ( (cid:101) (cid:96), (cid:101) (cid:96) (cid:48) ) ∩ B ( (cid:101) (cid:96) (cid:48) , (cid:101) (cid:96) x ) . If (cid:101) (cid:96) y (cid:54) = (cid:101) (cid:96) x , the leaf (cid:101) (cid:96) y separates (cid:101) (cid:96) from (cid:101) (cid:96) x and (cid:101) (cid:96) x from (cid:101) (cid:96) (cid:48) . Since g is continuous, the element (cid:101) (cid:96) Ty belongsto g ([0 , t ]) and to g ([ t, . If B ( (cid:101) (cid:96) y , (cid:101) (cid:96) x ) (cid:54) = { (cid:101) (cid:96) y , (cid:101) (cid:96) x } , then (cid:101) (cid:96) Tx (cid:54) = (cid:101) (cid:96) Ty , and (cid:101) (cid:96) Ty would belong to g ([0 , t [) and to g (] t, , thus g would not be injective. If B ( (cid:101) (cid:96) y , (cid:101) (cid:96) x ) = { (cid:101) (cid:96) y , (cid:101) (cid:96) x } , we denote by (cid:101) (cid:96) z the closestelement to (cid:101) (cid:96) y in the compact set B ( (cid:101) (cid:96), (cid:101) (cid:96) y ) ∩ B ( (cid:101) (cid:96), (cid:101) (cid:96) (cid:48) ) . By definition of (cid:101) (cid:96) y , we have B ( (cid:101) (cid:96) z , (cid:101) (cid:96) y ) = { (cid:101) (cid:96) z , (cid:101) (cid:96) y } ,thus (cid:101) (cid:96) Ty = (cid:101) (cid:96) Tz , and since (cid:101) (cid:96) Tx = (cid:101) (cid:96) Ty , we have (cid:101) (cid:96) Tx ∈ B ( (cid:101) (cid:96), (cid:101) (cid:96) (cid:48) ) T , which is a contradiction.Hence (cid:101) (cid:96) x ( R ) is contained in the complementary side of (cid:101) (cid:96) ( R ) that does not contain (cid:101) (cid:96) (cid:48) ( R ) , orthe opposite (cases and below). However, since g is continuous and since (cid:101) (cid:96) separates (cid:101) (cid:96) x and (cid:101) (cid:96) (cid:48) (or (cid:101) (cid:96) (cid:48) separates (cid:101) (cid:96) x and (cid:101) (cid:96) ), there would then exist t ∈ ]0 , such that (cid:101) (cid:96) (or (cid:101) (cid:96) (cid:48) ) represents g ( t ) , et g would not be injective. case case (cid:101) (cid:96) (cid:48) (cid:101) (cid:96) z (cid:101) (cid:96) x (cid:101) (cid:96) (cid:101) (cid:96) y (cid:101) (cid:96) x (cid:101) (cid:96) (cid:48) (cid:101) (cid:96) (cid:48) (cid:101) (cid:96) x (cid:101) (cid:96) (cid:101) (cid:96) Hence, if g is an arc between (cid:101) (cid:96) T and (cid:101) (cid:96) (cid:48) T , it has the same image as f − . Thus, up to repa-rametrization, the unique arc between (cid:101) (cid:96) T and (cid:101) (cid:96) (cid:48) T is f − which is isometric to [0 , d T ( (cid:101) (cid:96) T , (cid:101) (cid:96) (cid:48) T )] byconstruction. Since it is true for all pair of leaves, the metric space ( T, d T ) is an R -tree. (cid:3) Assume that the measure ν (cid:101) µ defined by ( (cid:101) Λ , (cid:101) µ ) has an atom (cid:101) (cid:96) . We replace (cid:101) (cid:96) ( R ) by a flat stripof width ν (cid:101) µ ( (cid:101) (cid:96) ) by gluing isometrically each of the complementary sides of (cid:101) (cid:96) ( R ) on the boundarycomponents of FS( (cid:101) (cid:96) ) = R × [0 , ν (cid:101) µ ( (cid:101) (cid:96) )] , endowed with the Euclidean distance. By doing the samething for every atom of ν (cid:101) µ , we get a surface (cid:101) Σ (cid:48) endowed with a complete CAT(0) metric (cid:101) d (cid:48) and theisometric action of the covering group Γ (cid:101) Σ on (cid:101) Σ minus the images of the atoms of ν (cid:101) µ extends in aunique way to an isometric action on ( (cid:101) Σ (cid:48) , (cid:101) d (cid:48) ) . Note that ( (cid:101) Σ , (cid:101) d (cid:48) ) is a complete CAT(0) metric spacewhose boundary at infinity is endowed with a (total) cyclic order, compatible with the coveringgroup action, hence it enters in the generalized framework of the geodesic laminations introducedin [Mor]. However, the distance (cid:101) d (cid:48) does not necessarily come from a half-translation structure, sincethe angles of the conical singular points belonging to the boundary of an added flat strip may notbe multiples of π .Let (cid:101) (cid:96) be an atom of ν (cid:101) µ and let F (cid:101) (cid:96) be the maximal set of geodesics whose images are containedin the corresponding flat strip FS( (cid:101) (cid:96) ) of ( (cid:101) Σ (cid:48) , (cid:101) d (cid:48) ) , that are parallel to its boundary. Let α be ageodesic segment of FS( (cid:101) (cid:96) ) that orthogonally joins its boundary components. Then, the map r : F (cid:101) (cid:96) → Image( α ) defined by r ( g ) = g ( R ) ∩ Image( α ) is a homeomorphism. Hence, we can endow F (cid:101) (cid:96) with the measure ν (cid:101) (cid:96) = ( r − ) ∗ dx α , where dx α is the Lebesgue measure on Image( α ) , of totalmass ν (cid:101) µ ( (cid:101) (cid:96) ) . The canonical isometric embedding of every connected component of (cid:101) Σ − (cid:83) ν (cid:101) µ ( (cid:101) (cid:96) ) > (cid:101) (cid:96) ( R ) into (cid:101) Σ (cid:48) induces an embedding of (cid:101) Λ − (cid:83) ν (cid:101) µ ( (cid:101) (cid:96) ) > (cid:101) (cid:96) into the space [ G (cid:101) d (cid:48) ] of geodesics of ( (cid:101) Σ (cid:48) , (cid:101) d (cid:48) ) definedup to changing origin. We denote by ν (cid:48) c the push forward of ν (cid:101) µ | (cid:101) Λ − (cid:83) ν (cid:101) µ ( (cid:101) (cid:96) ) > (cid:101) (cid:96) by this embedding.Then, we define ν (cid:48) (cid:101) µ = ν (cid:48) c + (cid:80) ν (cid:101) µ ( (cid:101) (cid:96) ) > ν (cid:101) (cid:96) . The measure ν (cid:48) (cid:101) µ is a Γ (cid:101) Σ -invariant Radon measure on theset of geodesics (defined up to changing origin) of ( (cid:101) Σ (cid:48) , (cid:101) d (cid:48) ) . Its support is the union of the image of (cid:101) Λ − (cid:83) ν (cid:101) µ ( (cid:101) (cid:96) ) > (cid:101) (cid:96) by the canonical embedding, and of the sets F (cid:101) (cid:96) , with ν ( (cid:101) (cid:96) ) > , defined above, whichis a geodesic lamination (cid:101) Λ (cid:48) of ( (cid:101) Σ (cid:48) , (cid:101) d (cid:48) ) .Since, by construction, the Radon measure ν (cid:48) (cid:101) µ has no atom, we can define the tree associatedto the measure ν (cid:48) (cid:101) µ , exactly as above, even if its support is not a flat lamination as defined inthis paper, since it only matters than ( (cid:101) Σ (cid:48) , (cid:101) d (cid:48) ) is complete, CAT(0) , and the support (cid:101) Λ (cid:48) of ν (cid:48) (cid:101) µ is ageodesic lamination in the generalized sense of [Mor]. We will call tree associated to (Λ , µ ) the treeassociated to ν (cid:48) (cid:101) µ defined in this way. In this section, we use the definitions and notations of Section 6 and we consider the canonicalaction of the covering group Γ (cid:101) Σ on the tree ( T, d T ) associated to the preimage ( (cid:101) Λ , (cid:101) µ ) of the measuredflat lamination (Λ , µ ) in (cid:101) Σ . We may assume that ν (cid:101) µ has no atom, up to proceeding as in the last aragraph of Section 6. Since (cid:101) Λ is fixed, we will use the notation B ( (cid:101) (cid:96), (cid:101) (cid:96) (cid:48) ) instead of B (cid:101) Λ ( (cid:101) (cid:96), (cid:101) (cid:96) (cid:48) ) forevery pair of leaves (cid:101) (cid:96), (cid:101) (cid:96) (cid:48) of (cid:101) Λ .The covering group Γ (cid:101) Σ acts on (cid:101) Σ by isometries. Hence, it defines an action on the set [ G [ (cid:101) q ] ] of geodesics of ( (cid:101) Σ , [ (cid:101) q ]) that are defined up to changing origin. Since (cid:101) Λ is Γ (cid:101) Σ -invariant, this actiondefines an action on (cid:101) Λ . Since for every γ ∈ Γ (cid:101) Σ we have γ ∗ ν (cid:101) µ = ν (cid:101) µ and γB ( (cid:101) (cid:96), (cid:101) (cid:96) (cid:48) ) = B ( γ (cid:101) (cid:96), γ (cid:101) (cid:96) (cid:48) ) , forevery pair of leaves (cid:101) (cid:96), (cid:101) (cid:96) (cid:48) of (cid:101) Λ , it defines an isometric action of Γ (cid:101) Σ on the tree ( T, d T ) associated to ( (cid:101) Λ , (cid:101) µ ) defined in Lemma 17. Lemma 18
For every primitive element γ ∈ Γ (cid:101) Σ − { e } , if (cid:101) α γ ( R ) is a tranlation axis of γ in ( (cid:101) Σ , [ (cid:101) q ]) and α γ is the projection of (cid:101) α γ in Σ , then the translation distance (cid:96) T ( γ ) of γ in ( T, d T ) is equal to i [ q ] ( µ, α γ ) . Moreover, if (cid:96) T ( γ ) > , the translation axis of γ is the image in T of the set of leavesof (cid:101) Λ which are interlaced with (cid:101) α γ . Proof. Case (1) . Assume that i [ q ] ( µ, α γ ) > . Then (cid:101) α γ is interlaced with at least one leaf (cid:101) (cid:96) of (cid:101) Λ . Hence, acccording to Lemma 15 and since ν (cid:101) µ ( { γ (cid:101) (cid:96) } ) = 0 , we have i [ q ] ( µ, α γ ) = ν (cid:101) µ ( B ( (cid:101) (cid:96), γ (cid:101) (cid:96) )) = (cid:101) d ( (cid:101) (cid:96), γ (cid:101) (cid:96) ) .Moreover, the set F of leaves of (cid:101) Λ whose images are contained in C ( (cid:101) (cid:96), γ (cid:101) (cid:96) ) , minus γ (cid:101) (cid:96) , is afundamental domain of (cid:101) Λ for the action of γ Z (see Section 5.1 for the definition of C ( (cid:101) (cid:96), γ (cid:101) (cid:96) ) ). If (cid:101) (cid:96) (cid:48) is a leaf of (cid:101) Λ , there exists a unique n ∈ Z such that γ n (cid:101) (cid:96) (cid:48) belongs to F . (cid:101) (cid:96) (cid:48) A (cid:101) (cid:96) (cid:48) a γaγ (cid:101) (cid:96) (cid:101) α γ (cid:101) (cid:96) (cid:48)− (cid:101) (cid:96) Let (cid:101) (cid:96) (cid:48) = γ n (cid:101) (cid:96) (cid:48) , (cid:101) (cid:96) (cid:48) = γ (cid:101) (cid:96) (cid:48) and (cid:101) (cid:96) (cid:48)− = γ − (cid:101) (cid:96) (cid:48) . Then (cid:101) (cid:96) (cid:48)− and (cid:101) (cid:96) (cid:48) do not belong to F . Since theleaves of (cid:101) Λ are pairwise non interlaced, the leaves of B ( (cid:101) (cid:96), γ (cid:101) (cid:96) ) belong to B ( (cid:101) (cid:96) (cid:48)− , (cid:101) (cid:96) (cid:48) ) ∪ B ( (cid:101) (cid:96) (cid:48) , (cid:101) (cid:96) (cid:48) ) , and γ ( B ( (cid:101) (cid:96) (cid:48)− , (cid:101) (cid:96) (cid:48) ) ∩ B ( (cid:101) (cid:96), γ (cid:101) (cid:96) ) − (cid:101) (cid:96) ) ⊆ B ( (cid:101) (cid:96) (cid:48) , (cid:101) (cid:96) (cid:48) ) − B ( (cid:101) (cid:96) (cid:48) , (cid:101) (cid:96) (cid:48) ) ∩ B ( (cid:101) (cid:96), γ (cid:101) (cid:96) ) , since F is a fundamental domain of (cid:101) Λ for the action of γ Z . Since ν (cid:101) µ is Γ (cid:101) Σ -invariant, we have ν (cid:101) µ ( B ( (cid:101) (cid:96) (cid:48) , γ (cid:101) (cid:96) (cid:48) )) = ν (cid:101) µ ( B ( (cid:101) (cid:96) (cid:48) , (cid:101) (cid:96) (cid:48) )) (cid:62) ν (cid:101) µ ( B ( (cid:101) (cid:96), γ (cid:101) (cid:96) ) ∩ B ( (cid:101) (cid:96) (cid:48) , (cid:101) (cid:96) (cid:48) )) + ν (cid:101) µ ( γ ( B ( (cid:101) (cid:96), γ (cid:101) (cid:96) ) ∩ B ( (cid:101) (cid:96) (cid:48)− , (cid:101) (cid:96) (cid:48) )))= ν (cid:101) µ ( B ( (cid:101) (cid:96), γ (cid:101) (cid:96) ) ∩ B ( (cid:101) (cid:96) (cid:48) , (cid:101) (cid:96) (cid:48) )) + ν (cid:101) µ ( B ( (cid:101) (cid:96), γ (cid:101) (cid:96) ) ∩ B ( (cid:101) (cid:96) (cid:48)− , (cid:101) (cid:96) (cid:48) )) (cid:62) ν (cid:101) µ ( B ( (cid:101) (cid:96), γ (cid:101) (cid:96) )) since B ( (cid:101) (cid:96), γ (cid:101) (cid:96) ) ⊆ B ( (cid:101) (cid:96) (cid:48)− , (cid:101) (cid:96) (cid:48) ) ∪ B ( (cid:101) (cid:96) (cid:48) , (cid:101) (cid:96) (cid:48) )= (cid:101) d ( (cid:101) (cid:96), γ (cid:101) (cid:96) ) Hence (cid:96) T ( γ ) = (cid:101) d ( (cid:101) (cid:96), γ (cid:101) (cid:96) ) = i [ q ] ( µ, α γ ) . In particular, we have (cid:96) T ( γ ) > and the isometric actionof γ on ( T, d T ) is hyperbolic, and hence admits a tranlation axis. Moreover, if (cid:101) (cid:96) (cid:48) is not interlacedwith (cid:101) α γ , then neither are (cid:101) (cid:96) (cid:48)− , (cid:101) (cid:96) (cid:48) and (cid:101) (cid:96) . However, we have seen that B ( (cid:101) (cid:96), γ (cid:101) (cid:96) ) is contained in B ( (cid:101) (cid:96) (cid:48)− , (cid:101) (cid:96) (cid:48) ) ∪ B ( (cid:101) (cid:96) (cid:48) , (cid:101) (cid:96) (cid:48) ) and that B ( (cid:101) (cid:96) (cid:48) , (cid:101) (cid:96) (cid:48) ) contains the union of the sets B ( (cid:101) (cid:96), γ (cid:101) (cid:96) ) ∩ B ( (cid:101) (cid:96) (cid:48) , (cid:101) (cid:96) (cid:48) ) and ( B ( (cid:101) (cid:96), γ (cid:101) (cid:96) ) ∩ B ( (cid:101) (cid:96) (cid:48)− , (cid:101) (cid:96) (cid:48) )) , and their intersection is { γ (cid:101) (cid:96) } . Assume that (cid:101) d ( (cid:101) (cid:96) (cid:48) , (cid:101) (cid:96) (cid:48) ) is equal to (cid:101) d ( (cid:101) (cid:96), γ (cid:101) (cid:96) ) ,i.e. that ν (cid:101) µ ( B ( (cid:101) (cid:96) (cid:48) , (cid:101) (cid:96) (cid:48) )) = ν (cid:101) µ ( B ( (cid:101) (cid:96), γ (cid:101) (cid:96) ) ∩ B ( (cid:101) (cid:96) (cid:48) , (cid:101) (cid:96) (cid:48) )) + ν (cid:101) µ ( γ ( B ( (cid:101) (cid:96), γ (cid:101) (cid:96) ) ∩ B ( (cid:101) (cid:96) (cid:48)− , (cid:101) (cid:96) (cid:48) ))) . Since the supportof ν (cid:101) µ is (cid:101) Λ , there are two possibilities.In the first possibility, the intersection ( F ∪ γ (cid:101) (cid:96) ) ∩ ( B ( (cid:101) (cid:96) (cid:48) , (cid:101) (cid:96) (cid:48) ) ∪ γ − B ( (cid:101) (cid:96) (cid:48) , (cid:101) (cid:96) (cid:48) )) = ( F ∪ γ (cid:101) (cid:96) ) ∩ ( B ( (cid:101) (cid:96) (cid:48)− , (cid:101) (cid:96) (cid:48) ) ∪ B ( (cid:101) (cid:96) (cid:48) , (cid:101) (cid:96) (cid:48) )) is equal to B ( (cid:101) (cid:96), γ (cid:101) (cid:96) ) . Thus, no leaf of F is contained in the complementaryside A of (cid:101) (cid:96) (cid:48) ( R ) that contains (cid:101) α γ ( R ) , if it is non interlaced with (cid:101) α γ (there is no leaf as representedby a dotted line points on the above picture). Consequently, there exists (cid:101) (cid:96) ” ∈ B ( (cid:101) (cid:96), γ (cid:101) (cid:96) ) such thatthe set B ( (cid:101) (cid:96) (cid:48) , (cid:101) (cid:96) ”) is reduced to { (cid:101) (cid:96) (cid:48) , (cid:101) (cid:96) ” } and (cid:101) d ( (cid:101) (cid:96) ” , (cid:101) (cid:96) (cid:48) ) = 0 . Thus (cid:101) (cid:96) (cid:48) T = (cid:101) (cid:96) ”‘ T ∈ B ( (cid:101) (cid:96), γ (cid:101) (cid:96) ) T .In the second possibility, there is a minimum, denoted by m in B ( (cid:101) (cid:96) (cid:48) , (cid:101) (cid:96) ) , for (cid:22) defined in Remark16, different from (cid:101) (cid:96) (cid:48) . Then (cid:101) d ( m, (cid:101) (cid:96) (cid:48) ) = 0 and since m belongs to B ( (cid:101) (cid:96) (cid:48) , (cid:101) (cid:96) ) ⊂ F ∩ ( B ( (cid:101) (cid:96) (cid:48)− , (cid:101) (cid:96) ) , it isinterlaced with (cid:101) α γ by assumption. Hence, the image of (cid:101) (cid:96) (cid:48) in T belongs to the translation axis of γ in T and, by γ -invariance, so does the image of (cid:101) (cid:96) ’. Hence, if i [ q ] ( µ, α γ ) > , the translation distance (cid:96) 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 leavesof (cid:101) Λ which are interlaced with a translation axis of γ in ( (cid:101) Σ , [ (cid:101) q ]) . Case (2) . Assume that i [ q ] ( µ, α γ ) = 0 , or equivalently that (cid:101) α γ is interlaced with no leaf of (cid:101) Λ . If (cid:101) α γ has the same ordered pair of points at infinity as a leaf (cid:101) (cid:96) of (cid:101) Λ , then γ (cid:101) (cid:96) = (cid:101) (cid:96) and (cid:101) (cid:96) T is afixed point of γ in T . Otherwise, we define ( S, N ) = ( (cid:101) α γ ( −∞ ) , (cid:101) α γ (+ ∞ )) . According to Lemma 7,no leaf of Λ is positively periodic unless it is periodic, hence according to [Mor, Lem. 4.13 et 4.14],neither N nor S is a point at infinity of any leaf of (cid:101) Λ . We recall that the (total) cyclic order o on ∂ ∞ (cid:101) Σ , defined by the orientation of (cid:101) Σ , defines a total order (cid:54) on ∂ ∞ (cid:101) Σ defined by S < η for every η ∈ ∂ ∞ (cid:101) Σ − { S } and η (cid:54) η if and only if o ( η , η , S ) ∈ { , } for every η , η ∈ ∂ ∞ (cid:101) Σ − { S } (see[Mor, Rem. 2.9] for the definition of o and [Wol, Déf. 2.23] for the definition of (cid:54) ).Let (cid:101) (cid:96) be a leaf of (cid:101) Λ and ( a, b ) = ( (cid:101) (cid:96) ( −∞ ) , (cid:101) (cid:96) (+ ∞ )) . Since no leaf of (cid:101) Λ is interlaced with (cid:101) α γ , theunion (cid:101) Λ ∪ { (cid:101) α γ , (cid:101) α − γ } is a flat lamination, and we can define B ( (cid:101) (cid:96), (cid:101) α γ ) . We replace (cid:101) (cid:96) by the maximumof (the compact set) B ( (cid:101) (cid:96), (cid:101) α γ ) − (cid:101) α γ , for (cid:22) . Then B ( (cid:101) (cid:96), (cid:101) α γ ) = { (cid:101) (cid:96), (cid:101) α γ } . However, the action of γ Z on ∂ ∞ (cid:101) 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 ∂ ∞ (cid:101) Σ , we can assume that the action of γ Z on the γ Z -orbits of a and b is increasing.Hence a < γa , b < γb and since (cid:101) (cid:96) and γ (cid:101) (cid:96) are not interlaced, we alsohave b (cid:54) γa (cid:54) γb . Assume that there exists a leaf (cid:101) (cid:96) (cid:48) ∈ B ( (cid:101) (cid:96), γ (cid:101) (cid:96) ) −{ (cid:101) (cid:96), γ (cid:101) (cid:96) } , whose ordered pair of points at infinity is ( a (cid:48) , b (cid:48) ) . Since (cid:101) (cid:96) (cid:48) is not interlaced with (cid:101) α γ and by assumption on (cid:101) (cid:96) , up to replacing (cid:101) (cid:96) (cid:48) by its opposite, we have b (cid:54) a (cid:48) (cid:54) γa and γb (cid:54) b (cid:48) < N , and ( a (cid:48) , b (cid:48) ) (cid:54) = ( γa, γb ) . But then S < γ − a (cid:48) (cid:54) a and b (cid:54) γ − b (cid:48) < N ,with ( γ − a (cid:48) , γ − b (cid:48) ) (cid:54) = ( a, b ) , thus γ − (cid:101) (cid:96) (cid:48) ∈ B ( (cid:101) (cid:96), (cid:101) α γ ) −{ (cid:101) (cid:96), (cid:101) α γ } , whichis a contradiction to the assumption on (cid:101) (cid:96) . Hence B ( (cid:101) (cid:96), γ (cid:101) (cid:96) ) = { (cid:101) (cid:96), γ (cid:101) (cid:96) } and (cid:101) d ( (cid:101) (cid:96), γ (cid:101) (cid:96) ) = 0 . Hence (cid:101) (cid:96) T is a fixed point of γ in T . (cid:3) γb (cid:101) Σ γ (cid:101) (cid:96)∂ ∞ (cid:101) Σ (cid:101) (cid:96) (cid:48) Na b γa (cid:101) α γ S γ − (cid:101) (cid:96) (cid:48) (cid:101) (cid:96) b (cid:48) a (cid:48) In this section 8, we use the same notation as in Section 6. We denote by m a hyperbolicmetric with totally geodesic boundary on Σ and by (cid:101) m the lifted hyperbolic metric on (cid:101) Σ . Webegin by recalling the definition of the dual tree to a measured hyperbolic lamination (see forexample [MS2, §1]), with a new presentation that allows to construct a Γ (cid:101) Σ -equivariant isometrybetween the tree associated to a measured flat lamination and the dual tree to the correspondingmeasured hyperbolic lamination. Let (Λ m , µ m ) be a measured hyperbolic lamination of (Σ , m ) nd let ( (cid:101) Λ (cid:101) m , (cid:101) µ m ) be its preimage in (cid:101) Σ . Then ( (cid:101) Λ (cid:101) m , (cid:101) µ m ) is Γ (cid:101) Σ -invariant and defines a measure ν (cid:101) µ m ∈ M Γ (cid:101) Σ ([ G (cid:101) m ]) (see [Bon2, Prop. 17 p. 154]). If (cid:101) λ is an atom of ν (cid:101) µ m , we replace (cid:101) λ by a flat strip FS( (cid:101) λ ) of width ν (cid:101) µ m ( (cid:101) λ ) foliated by geodesic lines parallel to its boundary. Proceeding similarly forevery atom of ν (cid:101) µ m , we get a geodesic lamination (cid:101) Λ (cid:48) on a surface endowed with a distance ( (cid:101) Σ (cid:48) , d (cid:48) ) ,which is CAT(0) , and the isometric action of Γ (cid:101) Σ on ( (cid:101) Σ , (cid:101) m ) extends uniquely to an isometric actionon ( (cid:101) Σ (cid:48) , d (cid:48) ) . Then, it is the same framework as at the end of Section 6. We can similarly defines themeasure ν (cid:101) λ of total mass ν (cid:101) µ m ( (cid:101) λ ) on the parallel leaves foliating the flat strip FS( (cid:101) λ ) associated toan atom (cid:101) λ of ν (cid:101) µ m . The measure ν (cid:48) which is equal to ν (cid:101) µ m outside of the atoms and equal to ν (cid:101) λ onthe set of leaves foliating the flat stip associated to (cid:101) λ , for every atom (cid:101) λ of ν (cid:101) µ m , is a Radon measureon [ G d (cid:48) ] which is Γ (cid:101) Σ -invariant, atomless, and whose support is equal to (cid:101) Λ (cid:48) .We also define a pseudo-distance (cid:101) d (cid:101) Λ (cid:48) on (cid:101) Λ (cid:48) by (cid:101) d (cid:101) Λ (cid:48) ( (cid:101) λ , (cid:101) λ ) = ν (cid:48) ( B (cid:101) Λ (cid:48) ( (cid:101) λ , (cid:101) λ )) for every pair ofleaves (cid:101) λ , (cid:101) λ of (cid:101) Λ (cid:48) (see Section 5.1 for the definition of B (cid:101) Λ (cid:48) ( (cid:101) λ , (cid:101) λ ) ), and the quotient of ( (cid:101) Λ (cid:48) , d (cid:101) Λ (cid:48) ) bythe equivalence relation (cid:101) λ ∼ (cid:101) λ if and only if d (cid:101) Λ (cid:48) ( (cid:101) λ , (cid:101) λ ) = 0 (or equivalently B ( (cid:101) λ , (cid:101) λ ) = { (cid:101) λ , (cid:101) λ } )is an R -tree ( T, d T ) called the tree dual to ( (cid:101) Λ (cid:101) m , (cid:101) µ m ) . For every γ ∈ Γ (cid:101) Σ , we have γ ∗ ν (cid:48) = ν (cid:48) and γB ( (cid:101) λ , (cid:101) λ ) = B ( γ (cid:101) λ , γ (cid:101) λ ) for every pair of leaves (cid:101) λ and (cid:101) λ of (cid:101) Λ (cid:48) . Hence, the action of Γ (cid:101) Σ on (cid:101) Λ (cid:101) m defines an isometric action on ( T, d T ) . It is easy to check (compare for instance with [ ? ]) thatthere exists a Γ (cid:101) Σ -equivariant isometry from the dual tree constructed in this way, and the oneconstructed for example in [MS2, §1], by identification of the leaves of (cid:101) Λ (cid:48) with the complementaryconnected component of the support of (cid:101) Λ (cid:48) that they bound.Let ( (cid:101) Λ [ (cid:101) q ] , (cid:101) µ [ (cid:101) q ] ) be a measured flat lamination on ( (cid:101) Σ , [ (cid:101) q ]) , let ν (cid:101) µ [ q ] be its associated measureon [ G [ (cid:101) q ] ] and let ν (cid:101) µ m be its image by ϕ ∗ (see Lemma 11). We denote by ( (cid:101) Λ (cid:101) m , (cid:101) µ m ) the measuredhyperbolic lamination defined by ν (cid:101) µ m , and by (cid:101) Λ (cid:48) and ν (cid:48) the geodesic lamination on ( (cid:101) Σ (cid:48) , d (cid:48) ) andthe Radon measure on [ G d (cid:48) ] defined by ( (cid:101) Λ (cid:101) m , (cid:101) µ m ) as above. We assume (up to proceeding as in thelast paragraph of Section 6) that ν (cid:101) µ [ q ] has no atom.If (cid:101) λ is an atom of ν (cid:101) µ m , and if F (cid:101) λ is the set of leaves of (cid:101) Λ [ (cid:101) q ] to which corresponds (cid:101) λ (see [Mor,§4.2]), there exist maximal flat strips in ( (cid:101) Σ , [ (cid:101) q ]) and in ( (cid:101) Σ (cid:48) , d (cid:48) ) that contain respectively F (cid:101) λ andthe set F (cid:48) (cid:101) λ of leaves of (cid:101) Λ (cid:48) corresponding to (cid:101) λ in the above construction. We denote by (cid:101) (cid:96) and (cid:101) (cid:96) (resp. (cid:101) λ and (cid:101) λ ) the extremal leaves of F (cid:101) λ (resp. F (cid:48) (cid:101) λ ), i.e. the leaves that satisfy F (cid:101) λ = B (cid:101) Λ (cid:102) m ( (cid:101) (cid:96) , (cid:101) (cid:96) ) and F (cid:48) (cid:101) λ = B (cid:101) Λ (cid:48) ( (cid:101) λ , (cid:101) λ ) . Then there exists a unique map φ (cid:101) λ : F (cid:101) λ → F (cid:48) (cid:101) λ such that for every (cid:101) (cid:96) ∈ F (cid:101) λ ,we have ν (cid:48) ( B (cid:101) Λ (cid:48) ( φ (cid:101) λ ( (cid:101) (cid:96) ) , φ (cid:101) λ ( (cid:101) (cid:96) )) = ν (cid:101) µ [ q ] ( B (cid:101) Λ [ (cid:101) q ] ( (cid:101) (cid:96) , (cid:101) (cid:96) )) . We denote by φ the map from (cid:101) Λ [ (cid:101) q ] to (cid:101) Λ (cid:48) equalto ϕ | (cid:101) Λ [ (cid:101) q ] outside the preimages of the atoms of ν (cid:101) µ m and equal to φ (cid:101) λ on the sets F (cid:101) λ where (cid:101) λ is an atom of ν (cid:101) µ m . Then, by construction, for every pair of leaves (cid:101) (cid:96) and (cid:101) (cid:96) of (cid:101) Λ [ (cid:101) q ] , we have B (cid:101) Λ (cid:48) ( φ ( (cid:101) (cid:96) ) , φ ( (cid:101) (cid:96) )) = φ ( B (cid:101) Λ [ (cid:101) q ] ( (cid:101) (cid:96) , (cid:101) (cid:96) )) . Hence the map φ defines a map φ T : ( T [ q ] , d T [ q ] ) → ( T m , d T m ) ,where ( T [ q ] , d T [ q ] ) and ( T m , d T m ) are respectively the tree associated to ( (cid:101) Λ [ (cid:101) q ] , (cid:101) µ [ q ] ) and the dual treeto ( (cid:101) Λ (cid:101) m , (cid:101) µ m ) . Lemma 19
The map φ T : ( T [ q ] , d T [ q ] ) → ( T m , d T m ) is a Γ (cid:101) Σ -equivariant isometry. Proof. If (cid:101) (cid:96) and (cid:101) (cid:96) are two leaves of (cid:101) Λ [ (cid:101) q ] , we have φ ( B (cid:101) Λ [ (cid:101) q ] ( (cid:101) (cid:96) , (cid:101) (cid:96) )) = B (cid:101) Λ (cid:48) ( φ ( (cid:101) (cid:96) ) , φ ( (cid:101) (cid:96) )) and since ν (cid:101) µ m = ϕ ∗ ν (cid:101) µ [ q ] , the map φ T is isometric. Moreover, φ is surjective hence, by taking the quotient,so is φ T and φ T is an isometry. Finally, as ϕ is, the map φ is Γ (cid:101) Σ -equivariant and by taking thequotient, so is φ T . (cid:3) éférences [BH] M. R. Bridson and A. Haefliger. Metric spaces of non-positive curvature . Grund. math.Wiss. , Springer Verlag, 1999.[Bon1] F. Bonahon.
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