Counting closed geodesics in globally hyperbolic maximal compact AdS 3-manifolds
aa r X i v : . [ m a t h . M G ] J a n Counting closed geodesics in globally hyperbolic maximalcompact AdS 3-manifolds.
Olivier Glorieux12th September 2018
Abstract
We propose a definition for the length of closed geodesics in a globally hyperbolic max-imal compact (GHMC) Anti-De Sitter manifold. We then prove that the number of closedgeodesics of length less than R grows exponentially fast with R and the exponential growthrate is related to the critical exponent associated to the two hyperbolic surfaces coming fromMess parametrization. We get an equivalent of three results for quasi-Fuchsian manifolds inthe GHMC setting : R. Bowen’s rigidity theorem of critical exponent, A. Sanders’ isolationtheorem and C. McMullen’s examples lightening the behaviour of this exponent when thesurfaces range over Teichmüller space. A classical problem in Riemannian geometry is to count the number of closed geodesics on amanifold, or estimate its growth. For compact negatively curved manifolds, this number growsexponentially fast and we know a very precise estimate since G. Margulis’ thesis, [Mar69], showingthe relation with volume entropy. This exponential growth rate is called critical exponent, this isalso the abscissa of convergence of the Poincaré series of the fundamental group of the manifoldacting on the universal cover.For a wide class of manifolds we understand quite well this invariant. For hyperbolic compactmanifolds M , it is constant equal to dim( M ) − . More generally for hyperbolic convex-cocompactmanifolds it is equal to the Hausdorff dimension of the limit set on the sphere at infinity, see[BJ97] for an even more general result. A vast category of convex-cocompact 3-manifolds aregiven by quasi-Fuchsian manifolds. These are hyperbolic 3-manifolds whose limit set of theirfundamental group on the sphere at infinity is a topological circle. They are topologically theproduct of a surface of genus greater than 2 and R . The geometry of a quasi-Fuchsian is encodedby two points in the Teichmüller space of S , Teich( S ) , through the so called Bers simultaneousuniformization [Ber72].The behaviour of the critical exponent for quasi-Fuchsian manifolds has been deeply studied.A theorem of R. Bowen [Bow79] says that the critical exponent is greater or equal to , andequality occurs if and only if the two points are the same, which geometrically says that thelimit set is a round circle of the sphere at infinity. Quantitative theorems have also been shown.A. Sanders in [San14b] proved an isolation theorem: he showed that the critical exponent of asequence of quasi-Fuchsian manifolds encoded, through Bers simultaneous uniformisation, by twosequences of point goes to 1 if and only if the two points tend to the same limit in the Teichmüllerspace. Finally C. McMullen in [McM99] studied the behaviour of the critical exponent forsequences of quasi-Fuchsian manifolds parametrized by a fixed surface and a sequence going tothe boundary of the Teichmüller space. 1or Lorentzian manifolds, the same counting problem makes no sense in general, since thelength of a curve is not necessarily well defined. However, there is a subclass of Lorentzianmanifolds called Globally Hyperbolic Anti-de Sitter manifolds , which are the analogous to quasi-Fuchsian manifolds, for which we propose in this article a natural definition for critical exponent.The aim of this article is to define, study and prove the counterpart of the three theorems givenin the last paragraph, for this critical exponent.We will give a quick review of the Lorentzian manifolds we are interested in. It will turnout that their geometry is also encoded by two points in the Teichmüller space and a largepart of this article will study the action of two Fuchsian representations on the product of twohyperbolic spaces: H × H . A preprint with D. Monclair and the author, [GM16], continues theinvestigation of these invariants in a more Lorentzian perspective.We will finally recall the usual definition of critical exponent and explain how it is related toother classical invariants. A lot of work has been done on
Globally Hyperbolic (GH)
Anti-de Sitter ( AdS ) manifolds duringthe last decades, based on the pioneering work of G. Mess [Mes07] describing the geometry ofsuch manifolds. A concise and complete presentation of the geometric background for GH
AdS manifolds can be found in [Dia14], and I will follow this text. More detailed ones are [BBZ07]and [BB09].Recall that on Lorentzian manifold there are three types of tangent vectors classified by thesign of the quadratic form. Let Q denotes the quadratic form coming from the Lorentzian scalarproduct and v a tangent vector. We say that v is spacelike if Q ( v ) > , lightlike if Q ( v ) = 0 and timelike if Q ( v ) < . This vocabulary extends to C curves : we say that a C curve isspacelike (respectively lightlike, timelike), if all its tangent vectors are spacelike (respectivelylightlike, timelike).A Cauchy surface in a Lorentzian 3-manifold is a spacelike surface which intersects everyinextendable timelike and lightlike curves exactly once. If a Lorentzian manifold contains aCauchy surface it is said globally hyperbolic . Moreover it is said maximal if there is no isometricembedding in a strictly larger space time, sending a Cauchy surface on a Cauchy surface.It follows from global hyperbolicity that every Cauchy surfaces are homeomorphic and we willsuppose in this paper that Cauchy surfaces are compact surfaces of genus g ≥ . The manifoldssatisfying those last two conditions are called globally hyperbolic maximal compact (GHMC).The Anti-de Sitter space is a maximal symmetric Lorentzian space of constant curvature − , this is the equivalent of the hyperbolic space in the Lorentzian setting. Let us present thefollowing linear model for AdS . Consider M ( R ) the space of 2 by 2 real matrices endowed withthe scalar product η induced by the quadratic form − det . The signature of − det is (2,2). Thelevel − of this quadratic form is SL ( R ) , and the restriction of η is of signature (2 , , hence it is aLorentzian manifold called Anti-de Sitter space . It is a Lorentzian space time of constant negativecurvature − . We will consider 3-manifolds locally modeled on AdS (Isom (AdS ) , AdS ) − structure, where Isom (AdS ) is the connected componentof ( Id, Id ) in the group of isometries of AdS . We can see that Isom (AdS ) is isomorphic to SL ( R ) × SL ( R ) / ( − Id, − Id ) , [BBZ07], where an element ( γ L , γ R ) ∈ SL ( R ) × SL ( R ) / ( − Id, − Id ) acts by left and right multiplication. This action of SL ( R ) × SL ( R ) on M ( R ) preserves thequadratic form and therefore − det induces a bi-invariant Lorentzian metric on SL ( R ) .It will sometimes be more convenient for us to use projective model : η induces on PSL ( R ) aLorentzian structure for which the isometry group is PSL ( R ) × PSL ( R ) . Hence the holonomyof a AdS M , is naturally given by two representations of π ( M ) in PSL ( R ) . We2ill call these two representations left and right : ( ρ L , ρ R ) .Let M be a GHMC, AdS manifold. Recall that we supposed the Cauchy surface to be a com-pact surface S of genus greater than 2. Then from global hyperbolicity we have π ( M ) = π ( S ) .G. Mess, showed in [Mes07] that for GHMC, the two representations ( ρ L , ρ R ) are faithful anddiscrete, in other words, they are points in the Teichmüller space of S , Teich( S ) . Conversely, heexplained how to construct a GHMC manifold with these two points as left and right holonomies.His construction gives a parametrization of GHMC manifolds by the product of two Teichmüllerspaces of S .This can be considered as the Bers simultaneous unifomization of quasi-Fuchsian manifolds,and GHMC manifolds can be seen as the Lorentzian counterpart of quasi-Fuchsian manifolds inmany aspects. Both are homeomorphic to a compact surface time R . Their holonomy is givenby product of Teichmüller representations. The holonomy of a GHMC manifolds gives a limit seton the boundary of AdS , this limit set is a curve which is a round circle if and only if the left andright representations are the same in
Teich( S ) . As for quasi-Fuchsian manifolds, GHMC oneshave a convex core, for which the boundaries components are two hyperbolic pleated surfaces.There are open questions on the geometry of those boundaries : we know in both cases thatwe can prescribe the metric on them (from the work of Epstein and Marden, [Mar], for quasi-Fuchsian case, and from the work of Diallo, [Dia14], for GHMC case) and it is still an openquestion to know whether or not the metrics on the boundaries determine the entire manifold(these questions have been raised by W. Thurston for quasi-Fuchsian and G. Mess for GHMC). There are a lot of dynamical invariants associated to quasi-Fuchsian manifolds, for example theHausdorff dimension of the limit set, the volume entropy of the convex core, the critical exponent,the growth rate of the number of closed geodesics... It is a standard fact these invariants areequal in the context of quasi-Fuchsian manifolds, see for example the introduction of [McM99].A natural question is to know if those invariants can be defined in the GHMC setting and if theysatisfy nice properties ?Let us recall the known results in the quasi-Fuchsian setting we want to obtain in the GHMCsetting. We call δ QF ( S , S ) the critical exponent associated to the quasi-Fuchsian manifoldparametrized by S and S through Bers simultaneous uniformization. We would at least expectan invariant which distinguishes the Fuchsian case, ie. when ρ L is conjugated to ρ R . In thequasi-Fuchsian settings, it is given by the following rigidity theorem due to R. Bowen Theorem 1.1. [Bow79] δ QF ( S , S ) ≥ . Moreover the equality occurs if and only if S = S . It is moreover known that δ QF < , for any quasi-Fuchsian manifolds.In fact, there is a quantitative result of this last theorem given by A. Sanders. Let us recallbefore stating this theorem what we mean by the thick part of the set of quasi-Fuchsian manifolds.We say that a sequence of quasi-Fuchsian manifolds stays in the thick part of the set of Quasi-Fuchsian manifolds if the injectivity radius is bounded below by ǫ , for some ǫ > . Recently, A.Sanders showed that if a quasi-Fuchsian manifolds M stays in the thick part of the Teichmüllerspace of S , the value for the critical exponent is isolated around Fuchsian locus : Theorem 1.2. [San14b, Theorem 5.5] Fix ǫ > , and suppose M is a quasi-Fuchsian manifold n the ǫ thick part, parametrized by S and S . For every ǫ there exists η ( ǫ, ǫ ) such that if δ QF ( S , S ) ≤ η, then d T eich ( S , S ) ≤ ǫ, where d T eich is the Teichmüller distance on
Teich( S ) . We can rephrase it with sequences :
Theorem. [San14b, Theorem 5.5] Let ( M n ) be a sequence of quasi-Fuchsian manifolds in the ǫ thick part, parametrized by ( S n ) and ( S ′ n ) , then: lim n →∞ δ QF ( S n , S ′ n ) = 1 , iff lim n →∞ d T eich ( S n , S ′ n ) = 0 , Finally we also expect a good behaviour at infinity and C. McMullen [McM99] gave examplesof quasi-Fuchsian sequences for which we know the behaviour of critical exponent.
Theorem 1.3. [McM99] Let S be a fixed point on Teich( S ) . • Let A be a pseudo-Anosov diffeomorphism S and call S n := A n S . Then lim n →∞ δ QF ( S , S n ) = 2 • Let S n be the surface obtained after pinching a disjoint set of simple closed curves on S .Then lim n →∞ δ QF ( S , S n ) = α < • Let c be a simple closed curve, and τ the Dehn twist along c . Call S n := τ n S . Then lim n →∞ δ QF ( S , S n ) = α. • Let S t be a surfaces path obtained by Fenchel twist along a simple closed geodesic c . Thenthere exists a ℓ S ( c ) -periodic function δ such that: lim t →∞ | δ QF ( S , S t ) − δ ( t ) | = 0 . The purpose of this paper is to give analogous statements in the GHMC setting.Let us say a few words on the others invariants. There are several definitions for criticalexponent which all agree for quasi-Fuchsian manifolds: δ QF = lim sup R →∞ R log Card { γ ∈ Γ | d ( γo, o ) ≤ R } = lim sup R →∞ R log Card { c ∈ C | ℓ QF ( c ) ≤ R } = Hdim(Λ) . Generally, the critical exponent for a group Γ acting on metric space ( X, d ) is defined bythe exponential growth rate of { γ ∈ Γ | d ( γo, o ) ≤ R } where o is any point in X . As Lorentzianmanifolds are not metric spaces, we have to set a definition for the "distance" between two points.Hausdorff dimension is a numerical invariant showing how wild is the limit set and is definedthrough a metric on the boundary of the hyperbolic space. Here again for Lorentzian manifoldswe need to find a good class of metrics on the boundary.4ur work with D. Monclair [GM16] explains how these two invariants can be generalised toGHMC manifolds (called AdS quasi-Fuchsian manifolds in the latter) and shows that we haveequality between them.Finally, let us mention that the work of D. Sullivan for hyperbolic manifolds shows the criticalexponent is related to the spectrum of the Laplacian. It has been studied for compact AdS manifolds by F. Kassel and T. Kobayashi [KK12], however nothing is known for GHMC manifolds.In this article we will consider the exponential growth rate of the number of closed geodesics,which is the definition that can be translate in the Lorentzian setting in the most straightforwardway.
Let M be a GHMC AdS manifold and S a Cauchy surface in M . Let C be the set of freehomotopy classes of closed curves on S , this corresponds to the conjugacy classes of π ( S ) . Itis well known that if S is endowed with a negatively curved metric there is an unique geodesicrepresentative for any c ∈ C . We will show in the next Section this is also true on M endowedwith its AdS metric. Hence, it gives for any c ∈ C a number ℓ Lor ( c ) which is the length of thegeodesic representative of c in M . In a GH manifold there is neither time nor lightlike closedgeodesic, hence this length is positive. This is also the translation length of the conjugacy classof an element γ ∈ Γ associated to c . Let ρ L and ρ R be the two representations defining M .Recall that these representations are faithful and discrete in PSL ( R ) henceforth they definetwo hyperbolic marked surfaces S L := H /ρ L (Γ) and S R := H /ρ R (Γ) . The following gives therelation between the length of a closed geodesic in M and on the pair ( S L , S R ) . Proposition (Proposition 2.3, and 2.4) . For every c ∈ C , there is a unique geodesic represent-ative of c in M . Denoting by ℓ Lor ( c ) its lorentzian length we have moreover. ℓ Lor ( c ) = ℓ S L ( c ) + ℓ S R ( c )2 . The proof is two folded. First we need to show the uniqueness result which follows from theknowledge of the geodesics in
AdS . Then we need to compute its length, which results froman easy algebraic computation. It follows from this proposition that the counting problem inGHMC
AdS manifolds is well defined. We define critical exponent by δ Lor ( M ) := lim sup T →∞ T log Card { c ∈ C | ℓ Lor ( c ) ≤ T } . The aim of this paper is the study of δ Lor ( M ) , when ( S L , S R ) ranges over Teich( S ) × Teich( S ) .Proposition 2.3 allows us to translate the purely Lorentzian problem into the counting problemon product of hyperbolic planes. Let us define for a pair of diffeomorphic marked hyperbolicsurfaces ( S L , S R ) the critical exponent by δ ( S L , S R ) := lim sup T →∞ T log Card { c ∈ C | ℓ S L ( c ) + ℓ S R ( c ) ≤ T } , where ℓ ∗ ( c ) designed the length of the unique closed geodesic in the free homotopy class of c on S L or S R . From Proposition 2.3, it it clear that δ Lor ( M ) = 2 δ ( S L , S R ) . We will then make the study of δ ( S L , S R ) . 5he invariant δ ( S L , S R ) has already been studied by C. Bishop and T. Steger in [BS91] wherethey showed the following rigidity result [BS91] δ ( S , S ) ≤ / . (1)Moreover the equality occurs if and only if S = S .We can rephrase it in Lorentzian setting by, Theorem (Theorem 2.5) . Let M be a GHMC manifold parametrized by ( S L , S R ) , then δ Lor ( M ) ≤ . Moreover the equality occurs if and only if S L = S R . This is the counterpart of Bowen’s rigidity theorem for GHMC. We give a completly Lorent-zian proof of this result in [GM16].The counterpart for C. McMullen’s Theorem is given by the following behaviour for δ Lor ,they are explained with details in Section 4.4.
Examples (Section 4.4) . Let S be a fixed point in Teich( S ) .1. Let A be a pseudo-Anosov diffeomorphism on S and call S n := A n S . Then lim n →∞ δ ( S , S n ) = 0 .
2. Let S n be the surface obtained after shrinking one simple closed geodesic on S . Then lim inf n →∞ δ ( S , S n ) = α >
3. Let τ be a Dehn twist around a simple closed curve and call S n := τ n S . Then lim n →∞ δ ( S , S n ) = α < / .
4. Let S t be a surfaces path obtained by Fenchel twist along a simple closed geodesic c . Thenthere exists a ℓ S ( c ) -periodic function δ such that: lim t →∞ | δ ( S , S t ) − δ ( t ) | = 0 . Finally, the counterpart Sanders’ Theorem is our main theorem. It gives the isolation of thecritical exponent δ Lor , Theorem (Theorem 5.1) . Let ( M n ) be a sequence of GHMC manifolds, parametrized by ( S , S n ) ,where S is a fixed hyperbolic surface, then lim n →∞ δ Lor ( M n ) = 1 iff lim n →∞ d T h ( S , S n ) = 0 , where d T h is the (symmetrized) Thurston distance on the Teichmüller space.
Let
Teich ǫ ( S ) be the thick part of Teichmüller space, that is the surfaces for which no closedgeodesic has length less than ǫ > . A sequence X n ∈ Teich( S ) is said to stay in the thick partif there exists a ǫ > such that X n ∈ Teich ǫ ( S ) for all n ∈ N .We copy this definition and call the thick part of the set of GHMC AdS manifolds, a set for whichthe length of the smallest geodesic is bounded below by a fixed constant. From Proposition 2.3,the sequence parametrized by ( S , S n ) in the previous Theorem stays in the thick part, since thelength of the systole of the AdS manifold is bounded below by the systole of S divided by 2.We deduce the following corollary thanks to Mumford’s compactness Theorem and the in-variance of critical exponent by diagonal action of the mapping class group6 heorem (Corollary 5.3) . Let ( M n ) be a sequence of GHMC manifolds, parametrized by ( S n , S ′ n ) .If one of the sequences stays in the thick part of Teich( S ) then lim n →∞ δ Lor ( M n ) = 1 iff lim n →∞ d T h ( S n , S ′ n ) = 0 . Our hypothesis on the thick part is a bit more restrictive than just saying ( M n ) stays in thethick part of GHMC AdS manifolds ie. there is no closed Lorentzian geodesic of arbitrary smalllength. Indeed from 2.3 the length of a geodesic on M n goes to zero if and only if it goes to zeroon both hyperbolic surfaces. Actually we found a counter example if ( M n ) does not stay in anythick part, cf Section 4.4.6: Theorem (Theorem 4.31) . There exists a sequence ( M n ) of GHMC manifolds, (not staying inthe thick part as AdS manifolds) parametrized by ( S n , S ′ n ) such that lim n →∞ δ Lor ( M n ) = 1 and lim n →∞ d T h ( S n , S ′ n ) = + ∞ , We summarize the results previously cited in the following table :Quasi-Fuchsian setting GHMC AdS setting. δ QF ∈ [1 , δ Lor ∈ (0 , δ QF = 1 iff S = S δ Lor = 1 iff S = S .[Bow79] Theorem 2.5. lim n →∞ δ QF ( S , A n S ) = 2 lim n →∞ δ Lor ( S , A n S ) = 0 . where A is pseudo Anosov. where A is pseudo Anosov.[McM99] Proposition 4.26. lim n →∞ δ QF ( S , τ n S ) = α lim n →∞ δ Lor ( S , τ n S ) = α > . where τ is a Dehn twist along where τ is a Dehn twist alonga simple closed curve. a simple closed curve.[McM99] Section 4.4.3 lim t →∞ δ QF ( S , S t ) = α < n →∞ δ Lor ( S , S t ) = α > . where S t is obtained by pinching where S t is obtained by pinchingdisjoint simple closed curves. disjoint simple closed curves.[McM99] Section 4.4.4 lim t →∞ | δ QF ( S , S t ) − δ ( t ) | = 0 lim t →∞ | δ Lor ( S , S t ) − δ ( t ) | = 0 where S t is obtained by Fenchel twist where S t is obtained by Fenchel twistalong a simple closed curve c along a simple closed curve c and δ ( t ) is ℓ ( c ) -periodic. and δ ( t ) is ℓ ( c ) -periodic.[McM99] Section 4.4.5
Let M n be a sequence of quasi-Fuchsian Let M n be a sequence of GHMC AdSmanifolds parametrized by ( S n , S ′ n ) manifolds parametrized by ( S n , S ′ n ) staying in the thick part. with one of the surface in thick part of Teich( S ) Then : lim n →∞ δ QF ( M n ) = 1 Then : lim n →∞ δ Lor ( M n ) = 1 iff lim n →∞ d T eich ( S n , S ′ n ) = 0 iff lim n →∞ d T h ( S n , S ′ n ) = 0 where d T eich is the Teichmüller distance. where d T h is the Thurston distance.[San14b]
Corollary 5.3
Let us say a few words about the proof of Theorem 5.1. There are mainly three ingredientsin the proof. • A Anti-de Sitter geometry problem that we reduced to a problem on a product of hyperbolicplanes. 7
Understand critical exponent of pairs of Teichmüller representations acting on H × H through the Manhattan curve. • See how Teichmüller representations degenerate thanks to earthquakes.
Sketch of proof of Theorem 5.1
In Section 2, we see that the action on
AdS can be rein-terpreted in terms of the diagonal action on the product of two hyperbolic planes and we willthen study the critical exponent δ ( S L , S R ) instead of δ Lor ( M ) .Then the proof of Theorem 5.1 goes as follow. First, remark that one way is trivial andfollows from the continuity of critical exponent. For the converse, let ( S n ) such that δ ( S , S n ) → / . Two cases may happen: ( S n ) stays in a compact set or ( S n ) goes out of every compactset of Teich( S ) . For the first case, using rigidity theorem and continuity we easily show that ( S n ) must converge to S . We will show that the second case cannot happen. By Thurston’sgeology Theorem, we can connect S to any point via an earthquake and we will show that alongearthquake paths the critical exponent cannot tend to / . This will conclude the proof.The idea to show that critical exponent cannot tend to / along earthquake is to adapt theDehn twist example. A Dehn twist τ is a particular case of earthquake and we can show usingconvexity of the geodesic length along earthquake paths that the critical exponent δ ( S , τ n S ) is decreasing, hence admits a limit which must be strictly less than / by rigidity. The trickin this example is to symmetrize the length function : we show in fact that ℓ ( τ n c ) + ℓ ( τ − n c ) isincreasing for all c ∈ C . From the invariance of δ ( · , · ) by the diagonal action of the mapping classgroup, it implies that δ ( S , τ n S ) = δ ( τ − n S , τ n S ) is decreasing.We cannot use the same argument for general earthquakes. Indeed, there is no reason tohave equality between δ ( S , E t L ( S )) and δ ( E − t L ( S ) , E t L ( S )) because an earthquake changes thelength spectrum. To take care of this difficulty we use a large deviation Theorem of geodesicflow, showing that "most" curves on S grows along earthquake. The next difficulty is toshow a correlation between the behaviour of "most" curves on S and "most" curves on the pair ( S , S n ) . This will be done using what we call critical exponent with slope , which is definedin a similar fashion as the critical exponent but where we impose the ratio ℓ n ( c ) ℓ ( c ) to be almostconstant. The proof ends using estimates we found in Section 3 on critical exponents with slopeand maximal slope.The plan of the paper is the following. The next section is devoted to the Lorentzian geometryof GHMC AdS manifolds. The aim is to show how we can pass from a Lorentzian problem to aquestion on product of hyperbolic planes. We study the critical exponent for diagonal action onproduct of hyperbolic planes in Section 3. We introduce the Manhattan curve as defined by M.Burger [Bur93] and make a careful study of the different invariants that can be read out of thiscurve: in particular, many inequalities between these invariants are deduced from the convexityof the Manhattan curve. At the end of Section 3, we will prove as a by pass a generalisation of aTheorem of R. Sharp and R. Schwarz about pair of geodesics on two surfaces. The Section 4 isa description of geometric results about currents, laminations and earthquakes. We will followthe presentation of F. Bonahon [Bon88] in order to get a clear picture of the results we need,namely, Corollary 4.20. This section contains also examples where we can compute either thelimit of the critical exponent of some sequences, or at least get some bounds. The last section isdevoted to the proof of the isolation Theorem 5.1 and Corollary 5.3. A property P is satisfied by most curves on S if there is η > such that Card { c ∈C | ℓ ( c ) ≤ R }∩ P Card { c ∈C | ℓ ( c ) ≤ R } = o ( e − ηR ) We replace { c ∈ C | ℓ ( c ) ≤ R } by { c ∈ C | ℓ ( c ) + ℓ n ( c ) ≤ R } in the previous definition cknowledgements This work is a part of my Ph. D. thesis and I am very grateful to GillesCourtois for his support and advices. I would like to thank Maxime Wolff for his help aboutTeichmüller space, laminations, and Thurston compactification. I am also grateful to Jean-MarcSchlenker who suggested me the Anti-de Sitter interpretation of my work. Finally I would liketo thank the referees for many useful comments.
As we said in the introduction, the geometry of GHMC,
AdS manifolds is encoded throughMess parametrization by two Fuchsian representations, ie. faithful and discrete representationsin
PSL ( R ) . The aim of the section is to recall some basic facts about AdS geometry and toexplain how closed geodesics on Lorentzian manifolds are related to the closed geodesics on thehyperbolic surfaces used in Mess parametrization.Our main result in this section is the Theorem 2.5, which is the counterpart of Bowen’sTheorem for globally hyperbolic manifolds.
Proof of Theorem 2.5.
First we compute the length of a closed geodesic in the
AdS manifoldparametrized by two Fuchsian representations, in terms of the lengths of closed geodesics inthe corresponding hyperbolic surfaces: Proposition 2.3. Then we prove that there is a uniqueclosed geodesic in every non trivial isotopy class of closed curve: Proposition 2.4. Applying theTheorem of Bishop-Steger [BS91] we get Theorem 2.5.
The model for
AdS we will use is the projective model [BBZ07], namely
AdS ≃ PSL ( R ) endowedwith the restriction of the quadratic form − det (cid:18) a bc d (cid:19) = − ad + bc . The isometry group of AdS is PSL ( R ) × PSL ( R ) ; an element γ = ( A, B ) ∈ PSL ( R ) × PSL ( R ) acts on X ∈ AdS by, γ · X = AXB − . The topological boundary of AdS is the set of projective non-invertiblematrices. It is parametrized by RP L × RP R , through the Segre embedding : to u L = (cid:20) x L y L (cid:21) and u R = (cid:20) x R y R (cid:21) we associated the projective non-invertible matrix u L u tR = (cid:20) x L x R x L y R y L x R y L y R (cid:21) The action on
AdS extends on a continuous action on the boundary given for γ = ( A, B ) ∈ PSL ( R ) × PSL ( R ) by γ · u L u tR = Au L u tR B − = Au L ( B ∗ u R ) t , where B ∗ = ( B − ) t . AdS
In a Lorentzian manifolds we can defined geodesics in the same manner as in a Riemannianmanifold by parallel transport with respect to the Levi-Civita connection. In our model of
AdS space there is a simple description given in [BBZ07]. Namely, geodesics in
AdS are theintersection of projective lines and
AdS . The type (timelike, null or spacelike) is invariant byleft and right multiplication by elements of
PSL ( R ) . It is determined by the signature of − det restricted to the plane defining the projective line. It is then easy to see that the one passingthrough identity are given by 1-parameter subgroups of PSL ( R ) . • They are time-like if the corresponding 1-parameter subgroup is elliptic, ie conjugated to (cid:20) cos( θ ) sin( θ ) − sin( θ ) cos( θ ) (cid:21) . These geodesics are entirely contained in
AdS .9 They are light-like if the corresponding 1-parameter subgroup is parabolic, ie conjugatedto (cid:20) x (cid:21) . These geodesics are tangent to ∂AdS . • They are space-like if the corresponding 1-parameter subgroup is hyperbolic, ie conjug-ated to (cid:20) e t e − t (cid:21) . These geodesics have two endpoints in ∂AdS x (+ ∞ ) = (cid:20) (cid:21) = (cid:20) (cid:21) × (cid:20) (cid:21) t and x ( −∞ ) = (cid:20) (cid:21) = (cid:20) (cid:21) × (cid:20) (cid:21) t . Let γ ∈ PSL ( R ) × PSL ( R ) be given by (cid:18)(cid:20) e λ e − λ (cid:21) , (cid:20) e − µ e µ (cid:21)(cid:19) , λ, µ > , A computation shows that the element g fixes the geodesic (cid:20) e t e − t (cid:21) t ∈ R , and acts bytranslation on this geodesic. g · (cid:20) (cid:21) = (cid:20) e λ e − λ (cid:21) (cid:20) (cid:21) (cid:20) e − µ e µ (cid:21) − = (cid:20) e λ + µ e − ( λ + µ ) (cid:21) We can then define the attracting fixed point to be g + = x (+ ∞ ) = (cid:20) (cid:21) and the repelling fixed point to be g − = x ( −∞ ) = (cid:20) (cid:21) . Remark that the geodesic t → (cid:20) e t e − t (cid:21) is parametrized by unit speed. Hence the trans-lation length for the induced metric on this geodesic is equal to λ + µ .Recall that any hyperbolic element of PSL ( R ) is conjugated to (cid:20) e λ e − λ (cid:21) , hence a pairof hyperbolic elements ( g L , g R ) acting on AdS has a well defined space like axis.
Definition 2.1.
We call axis of a pair of hyperbolic elements g = ( g L , g R ) acting on AdS theunique spacelike geodesic joining g − to g + Let us remark that this is not the unique geodesic fixed by g . Indeed for the element (cid:18)(cid:20) e λ e − λ (cid:21) , (cid:20) e − µ e µ (cid:21)(cid:19) , the geodesic (cid:20) e t − e − t (cid:21) t ∈ R , is also preserved. This geodesicjoins the attracting fixed point of (cid:20) e λ e − λ (cid:21) , to the repelling fixed point of (cid:20) e − µ e µ (cid:21) andtherefore does not have a satisfying geometric meaning. We will indeed show in the next sectionthat for a GHMC manifolds this geodesic is not in the range of the developing map.As in the hyperbolic setting, we define the translation length of g :10 efinition 2.2. We call translation length of an isometry g consisting of a pair of hyperbolicelements g = ( g L , g R ) , and we denote ℓ Lor ( g ) the distance (for the induced metric) between g and g · o for any point o on the axis of g . The Lorentzian translation length is invariant by conjugation as well as the hyperbolic trans-lation. Moreover this last one for (cid:20) e λ e − λ (cid:21) is equal to λ . We have proved the following Proposition 2.3.
Let γ = ( γ , γ ) be an isometry of AdS defined by a pair of hyperbolic elementsthen ℓ Lor ( γ ) = ℓ H ( γ )+ ℓ H ( γ )2 . In Subsection 2.3, we are going to prove the 1-to-1 correspondence between the set of freeisotopy classes of closed curves on Cauchy surfaces and closed geodesics in globally hyperbolicmanifolds.
Our aim is to show that in any free homotopy class of closed curve, there is exactly one geodesic.A GHMC, M , is topogically the product R × S , hence the set C of free homotopy classes of closedcurves on M is the same as the set of free homotopy classes of closed curves on S . The proofrelies on the knowledge of geodesics in the range of the developing map, D : ˜ M → AdS .The only closed geodesics of
AdS are timelike, and it is proven in [BBZ07, Corollary 5.20 andRemark 5.16 ] that D ( ˜ M ) does not contain any timelike geodesic. Hence it does not contain anyclosed geodesic.From the work of G. Mess in [Mes07], we know that the holonomy ρ is given by the productof two Fuchsian representations. Since S is supposed to be compact, it follows that for every γ ∈ Γ , ρ ( γ ) = ( γ , γ ) ∈ PSL ( R ) × PSL ( R ) is a pair of hyperbolic isometries. We have seenthat such an isometry fixes two geodesics in AdS , we have to show that only one of them is in D ( ˜ M ) , which is the axis of ρ ( γ ) .For this, recall that we defined the axis as the geodesic joining the attracting and the repellingpoint of ρ ( γ ) , those points are in Λ the limit set of Γ on AdS . As before, without loss of generality,we can conjugate ρ such that ρ ( γ ) = (cid:18)(cid:20) e λ e − λ (cid:21) , (cid:20) e − µ e µ (cid:21)(cid:19) . The axis (cid:20) e t e − t (cid:21) ( t ∈ R ) is in the convex core of the limit set, since once again (cid:20) (cid:21) and (cid:20) (cid:21) are points of thelimit sets. In particular, this implies that Id = (cid:20) (cid:21) is in the convex core. From Lemma6.17 of [BBZ07], for every point in the convex core, its dual plane is disjoint from the D ( ˜ M ) .The dual plane of a point p is the intersection of the orthogonal for the quadratic form with SL ( R ) , which is for Id the plane T r ( X ) = 0 . I follows that the other geodesic fixed by ρ ( γ ) : (cid:20) e t − e − t (cid:21) ( t ∈ R ) , is not contained in D ( ˜ M ) .Finally, we conclude the proof as Lemma B.4.5 of [BP12], that we reproduce for the conveni-ence of the reader. Proposition 2.4.
In every non trivial free homotopy class of closed curves in M there is aunique geodesic.Proof. Let α be a non trivial closed loop. Let γ be a representative in π ( M ) of the class of α .We have seen that ρ ( γ ) fixes exactly one geodesic ˜ g in D ( ˜ M ) . The projection of ˜ g is a geodesicloop which is in the same class as α . 11onversely, let g be a geodesic loop in M representing the same class as α . Let ˜ g denotea lift, and γ the element in Γ such that ρ ( γ )( ˜ g (0)) = ˜ g (1) . Since ˜ g is a geodesic it followsthat it is invariant by ρ ( γ ) . Moreover, there exists h such that ρ ( γ ) = ρ ( h − γh ) , whence ˜ g = ρ ( h − )(˜ g ) and from uniqueness of the geodesic fixed by ρ ( γ ) , we have ˜ g = ˜ g. The proof and Proposition 2.3 shows that for any c ∈ C there is a unique geodesic whoselength is equal to ℓ Lor = ℓ L ( c )+ ℓ R ( c )2 .We then defined critical exponent for GHMC AdS manifolds : δ ( M ) := lim sup R →∞ R log Card { c ∈ C | ℓ Lor ( c ) ≤ R } . Moreover, if M is parametrized by ( ρ L , ρ R ) we have δ ( M ) = lim sup R →∞ R log Card { c ∈ C | ℓ L ( c ) + ℓ R ( c )2 ≤ R } (2) = 2 lim sup R →∞ R log Card { c ∈ C | ℓ L ( c ) + ℓ R ( c ) ≤ R } (3)Now using Bishop-Steger Theorem [BS91] we prove the result equivalent to Bowen’s resultfor quasi-Fuchsian manifold. Theorem 2.5.
Let M be a GHMC manifold parametrized by ( S L , S R ) . We have δ Lor ( M ) ≤ Moreover the equality occurs if and only if S L = S R . The aim of the rest of this paper is to study δ when ρ L and ρ R move in Teich( S ) . Weare not going to use the Lorentzian interpretation anymore. From now on, we will study thedistribution of closed geodesics on the two hyperbolic surfaces associated to ρ L and ρ R . Thiscorresponds geometrically to the diagonal action on H × H endowed with the Manhattan metric, d M , which is the sum of the two hyperbolic metrics. The first tool we are going to introduce isthe Manhattan curve defined by M. Burger in [Bur93] for which we will replace, for the sake ofcoherence, the "left" and "right" notations by "1" and "2", since it is a curve in R , for exampleManhattan metric will be denoted by d M = d + d .Manhattan curve is a nice tool which allow to visualize different invariants associated to twoFuchsian representations. We can easily read on this curve the critical exponent for the diagonalaction, δ ( S , S ) = lim sup R →∞ R log Card { c ∈ C | ℓ ( c ) + ℓ ( c ) ≤ R } , which is also equal to the exponential growth of the cardinal of an orbit in H × H : δ ( S , S ) = δ ( ρ , ρ ) = lim sup R →∞ R log Card { γ ∈ Γ | d ( ρ ( γ ) o, o ) + d ( ρ ( γ ) o, o ) ≤ R } . This last invariant is actually the one which is called critical exponent usually. Moreover on theManhattan curve, we can read out what we will call critical exponent with slope , the maximalslope , and geodesic stretch . These three invariants are related by different means to the ratio ofthe geodesic lengths on the two surfaces ℓ ( c ) ℓ ( c ) . From the convexity of Manhattan curve it is easyto deduce inequalities between all of them. With D. Monclair, we propose a Lorentzian proof of the previous result independent of the one of BishopSteger in [GM16]. this follows from a small modifications of the arguments in [Kni83] Manhattan curve
Let S be a compact surface of genus g ≥ , Γ := π ( S ) its the fundamental group, ρ , ρ ,two Fuchsian representations of Γ and C the set of free homotopy classes of closed curves. Wewill denote by Γ i := ρ i (Γ) the corresponding subgroups of PSL(2 , R ) and S i := H / Γ i thehyperbolic surfaces homeomorphic to S . We will often forget ρ i and denote ρ i ( γ ) by γ i . Welook at the diagonal action on H × H , ie. γ. ( x, y ) = ( ρ ( γ ) x, ρ ( γ ) y ) , that we endow withdifferent Manhattan distances, defined by the weighted sum of the two hyperbolic distances oneach factor, d x,yM := xd + yd . Since x, y can be negative it is not necessarily genuine distances,however we will often restrict ourselves to x, y ∈ (0 , . Let o = ( p, q ) ∈ H × H be a point fixedonce for all. Definition 3.1.
The
Poincaré series is defined by P M [ ρ , ρ , x, y ]( s ) := X γ ∈ Γ e − sd x,yM ( γo,o ) . There is a convenient way to normalize the pair ( x, y ) which is the definition of the Manhattancurve : Definition 3.2.
The Manhattan curve is defined by C M := { ( x, y ) ∈ R | The abscissa of convergence of P M [ ρ , ρ , x, y ]( s ) is 1 } . Recall the definition of the critical exponent associated to the pair ( ρ , ρ ) . Definition 3.3.
We define the critical exponent of ρ and ρ by δ ( ρ , ρ ) := inf { s > | P M [ ρ , ρ , , s ) < ∞} . By triangular inequality δ ( ρ , ρ ) depends neither on o ∈ H × H , nor on the conjugacy classof ρ and ρ , hence it defines a function on Teich( S ) × Teich( S ) , still denoted by δ .We could have defined it in the following equivalent way : Proposition 3.4.
The following are equal :1. δ ( ρ , ρ ) as defined in Definition 3.3.2. lim sup R →∞ R log Card { γ ∈ Γ | d ( ρ ( γ ) o, o ) + d ( ρ ( γ ) o, o ) ≤ R } .3. The abscissa of convergence of P c ∈C e − s ( ℓ ( c )+ ℓ ( c )) . lim sup R →∞ R log Card { c ∈ C | ℓ ( c ) + ℓ ( c ) ≤ R } . From the geometric nature of points 3 and 4, we choose to endow the Teichmüller space of S with the Thurston distance. Definition 3.5. [Thu98] Let ( S , S ) be two hyperbolic surfaces. The Thurston distance isdefined by d ( S , S ) = log sup c ∈C max (cid:18) ℓ ( c ) ℓ ( c ) ; ℓ ( c ) ℓ ( c ) (cid:19)
13e define dil + := sup c ∈C ℓ ( c ) ℓ ( c ) and dil − := inf ℓ ( c ) ℓ ( c ) , therefore d ( S , S ) = log max (cid:0) dil + , − (cid:1) . Two remarks have to be pointed out. First, we choose the symmetric distance instead of theusual asymmetric because our isolation result is stronger in this way. Second, it is usually definedtaking the supremum over simple closed curves, but from Thurston’s work [Thu98, Proposition3.5], the two definitions coincide.We begin this section by some inequalities between critical exponent and intersection numberthen we study the correlation number with slope. The intersection between two hyperbolicsurfaces will be defined in Definition 4.10. This is a generalisation of the classical geometricintersection between two closed curves. In this section, the precise definition does not matter,we will just prove some inequalities coming from the Theorem of M. Burger [Bur93] about theManhattan curve. For all this section we fix S and S two hyperbolic surfaces and we call I the geodesic stretch function on Teich( S ) × Teich( S ) defined by, I : ( S , S ) → i ( S ,S ) i ( S ,S ) , where i ( S i , S j ) is the intersection number between S i and S j . The function I has a natural geometricinterpretation, it corresponds to the ratio ℓ ( c ) ℓ ( c ) for "typical curves" on S . More precisely, if ( c n ) is a sequence of closed curves on S which becomes equidistributed with respect to the Liouvillemeasure on T S , we have I ( S , S ) = lim ℓ ( c n ) ℓ ( c n ) . Finally let λ ( x ) be the slope of a normal vectorto C M at the abscissa x . Recall the following, Theorem 3.6. [Bur93, Theorem 1] The Manhattan curve is the straight line containing (1 , and (0 , if and only if S and S are equal in Teich( S ) . Moreover • λ (1) = I ( S , S ) . • lim x → + ∞ λ ( x ) = dil + , lim x →−∞ λ ( x ) = dil − . The other result we will need is the
Theorem 3.7. [Sha98] The Manhattan curve is real analytic.
For concision we write δ for δ ( S , S ) and I for I ( S , S ) .We first begin by recalling some basic facts about δ and C M . Proposition 3.8.
1. The points (1 , and (0 , are on C M .2. The intersection point between C M and the line y = x has coordinates ( δ, δ ) .3. The Manhattan curve is convex and strictly convex if S = S .4. If S = S then λ : R → (dil − , dil + ) is one-to-one.Proof.
1. Follows from the compactness of S . Indeed the critical exponent of P c ∈C e − sl ( c ) is for a compact surface.2. The intersection point has coordinates of the form ( x, x ) since it is on the line y = x . Since ( x, x ) is on C M , P c ∈C e − sx ( ℓ ( c )+ ℓ ( c )) has critical exponent equal to , this exactly meansthat x = δ
3. Follows from the convexity of the exponential map. More precisely let ( x , y ) , ( x , y ) betwo points of C M then by Hölder’s inequality P [ S , S , tx + (1 − t ) x , ty + (1 − t ) y ]( s ) ≤ ( P [ S , S , x , y ]( s )) t ( P [ S , S , x , y ]( s )) − t . By definition, both series of the right hand side have critical exponent equal to , hence P [ S , S , tx + (1 − t ) x , ty + (1 − t ) y ]( s ) has critical less than , which exactly meansthat ( tx + (1 − t ) x , ty + (1 − t ) y ) is above C M . The strict convexity follows from theanaliticity of C M . 14. By strict convexity, λ is strictly increasing and it is continuous hence one-to-one.The convexity of C M and Theorem 3.6 imply the rigidity Theorem of C. Bishop and S. Stegerwe mentioned in the introduction, δ ≤ with equality if and only if S = S .The last statement in Proposition 3.8, shows that for every λ ∈ (dil − , dil + ) there is a pointon the curve for which the slope of the normal is exactly λ . We set the following notation Definition 3.9.
The point on the Manhattan curve for which the slope of the normal is λ ∈ (dil − , dil + ) , will be noted ( x ( λ ) , y ( λ )) ∈ C M . Definition 3.10.
We call maximal slope associated to S and S the slope of a normal vectorat ( δ, δ ) . This is λ ( δ ) . By convexity C M is above the line y = − λ ( δ ) ( x − δ ) + δ . Since (0 , ∈ C M and (1 , ∈ C M , weobtain the two following inequalities : δ − δ ≤ λ ( δ ) ≤ − δδ . We will use the following corollary of these inequalities.
Corollary 3.11.
Let S n and S ′ n be two sequences of hyperbolic surfaces. If lim n →∞ δ ( S n , S ′ n ) = then lim n →∞ λ ( δ ( S n , S ′ n )) = 1 . By convexity again, the line y = − I ( x − is below C M . Hence taking the intersection with y = x we get δ ≥ − I ( δ − , which is equivalent to for every S , S ∈ Teich( S ) : δ ( S , S ) ≥
11 + I ( S , S ) . This gives the following,
Corollary 3.12.
Let S n and S ′ n be two sequences of hyperbolic surfaces. If lim n →∞ I ( S n , S ′ n ) =1 , then lim n →∞ δ ( S n , S ′ n ) = . Moreover, δ ≥ I gives, I ≥ δ − , hence we have the following corollary which is in thepaper of M. Burger, Corollary 3.13. [Bur93, Corollary 1] We have I ( S , S ) ≥ , with equality if and only if S = S . A central idea in our work is to study the proportionality factors between the two lengths of thegeodesic corresponding to a closed curve on S and S .The principal result in this Section is a formula for the critical exponent with slope λ in termsof the Manhattan curve.For λ ∈ (dil − , dil + ) and ǫ > , let C ( λ, ǫ ) := (cid:26) c ∈ C | (cid:12)(cid:12)(cid:12)(cid:12) ℓ ( c ) ℓ ( c ) − λ (cid:12)(cid:12)(cid:12)(cid:12) ≤ ǫ (cid:27) . (4)To this set is naturally associated a critical exponent, namely δ ( S , S , λ, ǫ ) := inf s > | X c ∈C ( λ,ǫ ) e − s ( ℓ ( c )+ ℓ ( c )) < + ∞ . efinition 3.14. The critical exponent with slope λ , is defined by δ ( S , S , λ ) := lim ǫ → δ ( S , S , λ, ǫ ) . As the critical exponent we could have defined it by δ ( S , S , λ ) = lim ǫ → lim sup T → + ∞ T log Card { c ∈ C ( λ, ǫ ) | ℓ ( c ) + ℓ ( c ) ≤ T } . Theorem 3.15.
Let ( x, y ) ∈ C M , and λ ∈ (dil − , dil + ) . We have the following inequality δ ( λ ) ≤ x + λy λ , equality occurs if and only if ( x, y ) = ( x ( λ ) , y ( λ )) , the point on the Manhattan curve for whichthe slope of the normal vector is equal to λ , see Definition 3.9. First we are going to prove the inequality, which is a simple algebraic manipulation. Thiswill be done in the following Lemma 3.16. The equality case is a bit harder and we delay itsproof for the next subsection. The Theorem 3.15 will finally be proven after Corollary 3.26.This kind of results implicitly appears, but for Riemannian metric, in the work of G. Link[Lin04]. We propose a proof since our context is a little bit different and since recognizing that[Lin04, Lemma 3.7] and Lemma 3.16 from our work are similar is not obvious.
Lemma 3.16.
Compare to [Lin04, Lemma 3.7]. For every ( x, y ) ∈ C M and every λ ∈ (dil − , dil + ) we have the inequality δ ( λ ) ≤ x + λy λ .Proof. Let ( x, y ) ∈ C M , λ ∈ (dil − , dil + ) , and c ∈ C ( λ, ǫ ) . First we suppose that x > and y > .Then xℓ ( c ) + yℓ ( c ) = (cid:18) x
11 + ℓ ( c ) /ℓ ( c ) + y ℓ ( c ) /ℓ ( c )1 + ℓ ( c ) /ℓ ( c ) (cid:19) ( ℓ ( c ) + ℓ ( c )) (5) ≤ x + ( λ + ǫ ) y λ − ǫ ( ℓ ( c ) + ℓ ( c )) (6)This implies for the Poincaré series that : X c ∈C e − s ( xℓ ( c )+ yℓ ( c )) ≥ X c ∈C ( λ,ǫ ) e − s ( xℓ ( c )+ yℓ ( c )) (7) ≥ X c ∈C ( λ,ǫ ) e − s ( x +( λ + ǫ ) y λ − ǫ ) ( ℓ ( c )+ ℓ ( c )) (8)Since ( x, y ) ∈ C M the critical exponent of the left hand side is equal to 1, therefore ≥ δ ( λ, ǫ ) (cid:18) λ − ǫx + ( λ + ǫ ) y (cid:19) (9) δ ( λ, ǫ ) ≤ x + ( λ + ǫ ) y λ − ǫ (10)passing to the limit gives, δ ( λ ) ≤ x + λy λ . (11)This end the proof for x > and y > . 16f x < and y > , the inequality (6) would become xℓ ( c ) + yℓ ( c ) ≤ (cid:18) x λ + ǫ + ( λ + ǫ ) y λ − ǫ (cid:19) ( ℓ ( c ) + ℓ ( c )) and then the inequality (9) would become ≥ δ ( λ, ǫ ) (cid:18) x λ + ǫ + ( λ + ǫ ) y λ − ǫ (cid:19) . Passing to the limit as in 11 ends the proof in the case x < and y > .The case x > and y < can be treated similarly.The equality case in Theorem 3.15 will be our goal for the last part of this section and provenin Corollary 3.26. We are actually going to show the equality case using an extension of a formula due to R. Sharp,[Sha98] about the correlation number. This result of R. Sharp uses thermodynamic formalismfor the geodesic flow. We will make a brief survey of results on thermodynamic formalism andgeodesic flow which ends with Theorems 3.21 and 3.23. Finally, we will prove Theorem 3.25extending the Theorem of R. Sharp. The proof is very similar to the original one and we includeit for the sake of completeness.Here again S and S will be fixed hyperbolic surfaces. Definition 3.17.
For λ ∈ (dil − , dil + ) , the correlation number with slope λ is m ( S , S , λ ) := lim T →∞ T log Card { c ∈ C , ℓ ( c ) ∈ [ T, T + 1) and ℓ ( c ) ∈ [ λT, λT + 1) } . The bound λT + 1 in the interval [ λT, λT + 1) has no consequences on m , we could havetaken λT + k for any k > .Since S and S are fixed, we will note m ( λ ) instead of m ( S , S , λ ) for the correlation numberof slope λ , cf Definition 3.17 and δ ( λ ) instead of δ ( S , S , λ ) for the directional critical exponent,cf Definition 3.14.We first begin to prove an inequality : Lemma 3.18. m ( λ ) ≤ δ ( λ )(1 + λ ) .Proof. Let CC ( T, λ ) := { c ∈ C , ℓ ( c ) ∈ [ T, T + 1) and ℓ ( c ) ∈ [ λT, λT + 1) } the set of closed and"correlated" curves. Let ǫ > and c ∈ CC ( T, λ ) then, for T > max(1 ,λ ) ǫ , (cid:12)(cid:12)(cid:12) ℓ ( c ) ℓ ( c ) − λ (cid:12)(cid:12)(cid:12) ≤ ǫ , that is tosay c ∈ C ( λ, ǫ ) (cf. eq (4)). X k ≥ max(1 ,λ ) /ǫ X c ∈CC ( k,λ ) e − s ( ℓ ( c )+ ℓ ( c )) ≤ X c ∈C ( λ,ǫ ) e − s ( ℓ ( c )+ ℓ ( c )) . The right hand side has critical exponent equal to δ ( λ, ǫ ) and for c ∈ CC ( T, λ ) we have ℓ ( c ) ≤ λT + 1 ≤ λℓ ( c ) + 1 , hence the left hand side satisfies X k ≥ max(1 ,λ ) /ǫ X c ∈CC ( k,λ )) e − s ( ℓ ( c )+ ℓ ( c )) ≥ X k ≥ max(1 ,λ ) /ǫ X c ∈CC ( k,λ ) e − sℓ ( c )(1+ λ ) − s . Card CC ( k, λ ) is larger than e ( m ( λ ) − η ) k for all η > , there is k ∈ N suchthat for every k ≥ k Card CC ( k, λ ) ≥ e ( m ( λ ) − η ) k . Set k := max( k , max(1 , λ ) /ǫ ) , X k ≥ max(1 ,λ ) /ǫ X c ∈CC ( k,λ ) e − s ( ℓ ( c )+ ℓ ( c )) ≥ X k ≥ max(1 ,λ ) /ǫ X c ∈CC ( k,λ ) e − sℓ ( c )(1+ λ ) − s ≥ X k ≥ k X c ∈CC ( k,λ )) e − s ( k +1)(1+ λ ) − s ≥ e − s (1+ λ ) − s X k ≥ k e ( m ( λ ) − η ) k e − sk (1+ λ ) In fine , the critical exponent of P k ≥ max(1 ,λ ) /ǫ P c ∈CC ( k,λ ) e − s ( ℓ ( c )+ ℓ ( c )) is larger than m ( λ ) − η λ ,therefore we have for all ǫ > and η > , that m ( λ ) − η ≤ δ ( λ, ǫ )(1 + λ ) . We conclude since η and ǫ are arbitrary.Our proof of equality case use thermodynamic formalism that we survey in the next para-graph. Reparametrization of geodesic flow.Definition 3.19.
Let ϕ t be the geodesic flow on T S and τ be a periodic orbit for ϕ t . Let also ψ : T S → R be any Hölder continuous function. We note ω ( ψ, τ ) := R τ ψ the integral of ψ with respect to the arc length along the geodesic associated to τ . For example if ψ = 1 , we have ω ( ψ, τ ) = ℓ ( c ) , where c is the closed geodesic on S whosesupport is τ . We are going to construct a function ψ such that ω ( ψ, τ ) = ℓ ( c ) . This repara-metrization is classic for example R. Schwartz and R. Sharp suggest a method to construct sucha function. Let’s describe this construction. The classical references for what we are going tointroduce are [Led94, Sam14a, Sam15].A Hölder cocycle is a map u : Γ × ∂ Γ → R satisfying : u ( γ ′ γ, ξ ) = u ( γ, ξ ) + u ( γ ′ , γξ ) for every pair γ, γ ′ ∈ Γ and ξ ∈ ∂ Γ , which is a Hölder continuous map in the variable ξ . Sincethe surface S is compact the boundary of Γ can be identified with S that is the boundary of H .The period of a cocycle u , ℓ u ( γ ) is by definition u ( γ, γ + ) where γ + is the attracting fixed pointof γ . From [Led94, Corollary 1 p.106] for every Hölder cocycle u there exists a function ψ suchthat ω ( ψ, τ ) = ℓ u ( γ ) , where γ is the elements corresponding to the closed geodesic of support τ . Now, the following Busemann cocycle defined by β ( γ, ξ ) := lim x → ξ d ( o, x ) − d ( ρ ( γ ) o, x ) ,where o is any point of H , satisfies ℓ β ( γ ) = ℓ ( γ ) . Definition 3.20.
We call ψ ρ the (any) Hölder continuous function defined thanks to [Led94,Corollary 1 p.106] and the Busemann cocycle. We then have X c ∈C e − xℓ ( c ) − yℓ ( c ) = X τ e R τ − x − yψ ρ , where the sum of the right hand side is taken over all closed orbit of ϕ t .18iven a Hölder continuous function f : T S → R , we define the pressure of f to be P ( f ) = sup ν (cid:26) h ( ν ) + Z T S f dν (cid:27) , where the supremum is taken over all ( ϕ t )-invariant probability measures, and h ( ν ) is the entropyof the geodesic flow with respect to ν . This supremum is achieved by a unique such measure µ f , called the equilibrium state for f . In our setting of a surface of constant curvature, theequilibrium state for f ≡ is the Liouville measure, which is the local product of the Lebesguemeasure for the metric associated to S and the arc length along the fibre.We say that a function v : T S → R is continuously differentiable with respect to ϕ t if thelimit v ′ ( y ) := lim t → v ( ϕ t ( y )) − v ( y ) t exists everywhere, and is continuous. Two Hölder functions f and g are said to be cohomologous if f − g = v ′ for some continuously differentiable function v .W. Parry and M. Pollicott showed in their book [PP90, Proposition 4.8], that if f is a Höldercontinuous function which is not cohomologous to a constant, then the function t → P ( tf ) isanalytic, strictly convex and furthermore, [PP90, Proposition 4.12] P ′ ( tf ) := ddt P ( tf ) = Z T S ψdµ tf holds for each value of t ∈ R . A version of this theorem more adapted to our notations can befound in [SS93, p. 429].The following is a classical theorem in thermodynamic theory, and has been proved by P.Walters in [Wal75, Theorem 4.1] ; here again, the following version of this theorem in [Led94, p.106 eq. 10] is more adapted : Theorem 3.21.
Let f : T S → R be a Hölder continuous function, then lim T → + ∞ T log X τ, ω (1 ,τ ) ≤ T e R τ f = P ( f ) . By the previous discussion the Manhattan curve is the set of ( x, y ) ∈ R such that P τ e − s R τ x + yψ ρ has critical exponent equal to 1. Applying [Led94, Lemma 1, p106] and theremark after it, to the function f := x + yψ ρ , we see that critical exponent is equal to if andonly if P ( − f ) = 0 that is P ( − x − yψ ρ ) = 0 . Hence C M = { ( x, y ) , P ( − x − yψ ρ ) = 0 } = { ( x, y ) , P ( − yψ ρ ) = x } . The independance Lemma of [SS93] shows that ψ ρ is not cohomologous to a constant assoon as ρ and ρ are not conjugated. Hence we will thereafter suppose that ρ and ρ are notconjugated. Moreover, the function ψ ρ satisfies the property that along any geodesic segment s on S , R s ψ ρ = ℓ ( s ) , hence the function ψ ρ is strictly positive. Taking the derivative of t → P ( − tψ ρ ) gives ∂P ( − tψ ρ ) ∂t = − R T S ψ ρ dµ − tf = 0 and finally the implicit function Theoremassures the existence of an analytic function q ( t ) defined by P ( − q ( t ) ψ ρ ) = t . By definition C M is the graph of q .We can then compute the derivative of q , as it is done in [SS93] : ddt P ( − q ( t ) ψ ρ ) = (cid:18) − Z ψ ρ dµ − q ( t ) ψ ρ (cid:19) dqdt = 1 dqdt = − R ψ ρ dµ − q ( t ) ψ ρ . Definition 3.22.
We set J ( f ) := { P ′ ( qf ) , q ∈ R } . By strict convexity , if λ ∈ J ( f ) , there is aunique real number noted q λ such that P ′ ( q λ f ) = λ . The next theorem is due to S.P Lalley [Lal87],
Theorem 3.23. [Lal87, Theorem 1][SS93, Proposition p.429] Let f : T S → R be a Höldercontinuous function, which is not cohomologous to a constant. Let λ ∈ J ( f ) , there exists K > such that, Card { τ | ω (1 , τ ) ∈ [ T, T + 1) and ω ( f, τ ) ∈ [ λT, λT + 1) } ∼ T →∞ K e h ( µ qλf ) T T / . Applying this theorem to the function f = ψ ρ , defined in Definition 3.20, which is notcohomologous to a constant, gives for every λ ∈ J ( ψ ρ ) , the existence of K > such that, Card CC ( T, λ ) ∼ x →∞ K e h ( µ qλψρ ) T T / , hence if λ ∈ J ( ψ ρ ) , we have m ( λ ) = h ( µ q λ ψ ρ ) .This next proposition does not figure in the original paper of R. Sharp but the proof isessentially the same as in the remarks at the end of [SS93] Proposition 3.24.
For the function ψ ρ defined in Definition 3.20, we have the following in-clusion : (dil − , dil + ) ⊂ J ( ψ ρ ) . Proof.
Let λ ∈ (dil − , dil + ) . By definition there exists closed geodesics c and c ′ such that ℓ ( c ) < λℓ ( c ) and ℓ ( c ′ ) > λℓ ( c ′ ) . They correspond to closed orbits of ϕ t , τ and τ ′ , hence ω ( ψ ρ , τ ) /λ (1 , τ ) < λ et ω ( ψ, τ ′ ) /ω (1 , τ ′ ) > λ .Let I ( ψ ρ ) denote the set of values R ψ ρ dν where ν ranges over invariant probability meas-ures. Clearly I ( ψ ρ ) is a closed interval. If we take ν to be the probability measure whose supporteither c or c ′ we see that ω ( ψ ρ , τ ) /ω (1 , τ ) ∈ I ( ψ ρ ) , and ω ( ψ ρ , τ ′ ) /ω (1 , τ ′ ) ∈ I ( ψ ρ ) . Hence ] λ − ǫ, λ + 2 ǫ [ ∈ I ( ψ ρ ) for some ǫ > .Since λ ± ǫ ∈ I ( ψ ρ ) , we have, ∀ t ∈ R , λt + | t | ǫ ∈ tI ( ψ ρ ) . And by definition of pressure if y ∈ I ( ψ ρ ) then P ( tψ ρ ) ≥ ty . Hence ∀ t ∈ R , P ( tψ ρ ) ≥ sup tI ( ψ ρ ) . Combining the last two inequalities gives ∀ t ∈ R , P ( tψ ρ ) − λt ≥ | t | ǫ. Consider the function Q ( t ) := P ( tψ ρ ) − λt ; we have Q (0) = P (0) = 1 and we just provedthat for all | t | > /ǫ , Q ( t ) > . Therefore Q has a minimum q λ ∈ [ − /ǫ, /ǫ ] , where Q ′ ( q λ ) = 0 which is to say P ′ ( q λ ψ ) = λ and λ ∈ J ( ψ ρ ) . 20ow we are going to prove the formula that extends the result of R. Sharp [Sha98]. Theorem 3.25.
Let λ ∈ (dil − , dil + ) . There exists
K > and m ( λ ) such that Card CC ( T, λ ) ∼ K e m ( λ ) T T / . Moreover m ( λ ) = x ( λ ) + λy ( λ ) .Proof. Let λ ∈ (dil + , dil − ) , by Proposition 3.24, we know that λ ∈ J ( ψ ρ ) . The remark after theTheorem 3.23 says that m ( λ ) = h ( µ q λ ψ ρ ) . We have to show that h ( µ q λ ψ ρ ) = x ( λ ) + λy ( λ ) . Bydefinition of h ( µ q λ ψ ρ ) we have that m ( λ ) = h ( µ q λ ψ ρ ) = P ( q λ ψ ρ ) − Z q λ ψ ρ dµ q λ ψ ρ . Set x such that q λ = − q ( x ) , this gives, m ( λ ) = P ( − q ( x ) ψ ρ ) + q ( x ) Z ψ ρ dµ − q ( x ) ψ ρ = x + q ( x ) Z ψ ρ dµ − q ( x ) ψ ρ . But q λ is such that R ψ ρ dµ q λ ψ ρ = P ′ ( q λ ψ ρ ) = λ , hence m ( λ ) = x + q ( x ) λ. Moreover, dqdt ( x ) = − R ψ ρ dµ − q ( x ) ψ ρ = − λ . This is exactly where the slope of a normal vector at q is equal to λ , that is x = x ( λ ) , and m ( λ ) = x ( λ ) + λq ( x ( λ )) = x ( λ ) + λy ( λ ) . We finally conclude the proof of the equality case in Theorem 3.15.
Corollary 3.26.
Let λ ∈ (dil − , dil + ) , we have m ( λ )1 + λ = δ ( λ ) = x ( λ ) + λy ( λ )1 + λ . Proof of Corollary 3.26 and Theorem 3.15.
From Lemma 3.16 and Lemma 3.18 we have for all ( x, y ) ∈ C M and all λ ∈ (dil − , dil + ) that m ( λ ) ≤ δ ( λ )(1 + λ ) ≤ x + λy and with Theorem 3.25,equality occurs for ( x ( λ ) , y ( λ )) .We now show that the inequality δ ( λ ) ≤ x + λy λ is strict for ( x, y ) = ( x ( λ ) , y ( λ ) . This is asimple application of the strict convexity of the Manhattan curve. Let x → q ( x ) be the functionwhich graph is the Manhattan curve. Fix λ and defined g ( x ) := x + λq ( x )1 + λ . The derivative of g is g ′ ( x ) = λq ′ ( x )1+ λ . By Definition λ = − /q ′ ( x λ ) , hence g ′ ( x ) = q ′ ( x ( λ )) − q ′ ( x ) q ′ ( x ( λ )) − .By strict convexity of q , g ′ ( x ) = 0 if and only if x = x ( λ ) , which finally implies that the inequalityis strict for the others values of ( x, y ) ∈ C M . 21ere again, the following corollary implicitly belongs to the work of G. Link [Lin04]. However,G. Link considers H × H endowed with the Riemannian metric, hence for the Manhattan metricthis corollary was not known. Corollary 3.27. [Lin04, compare to Theorem 3.12, Theorem 5.1] For all λ ∈ (dil − , dil + ) wehave δ ( S , S , λ ) ≤ δ ( S , S ) , and equality occurs for a unique λ : the maximal slope (cf. Definition 3.10).Proof. The inequality is obvious by definition. For the strict inequality we are going to usea similar method as the previous corollary. We use the formula from Corollary 3.26, δ ( λ ) = x ( λ )+ λy ( λ )1+ λ . Recall that λ ( x ) is the slope of a normal vector at ( x, q ( x )) ∈ C M . Hence δ ( λ ( x )) = x + λ ( x ) q ( x )1+ λ ( x ) = xq ′ ( x ) − q ( x ) q ′ ( x ) − , since λ ( x ) = − /q ′ ( x ) . Denote by h the function h ( x ) = xq ′ ( x ) − q ( x ) q ′ ( x ) − . The derivative of h is h ′ ( x ) = q ′′ ( x )( − x + q ( x ))( q ′ ( x ) − , which has the same the sign as − x + q ( x ) . ByProposition 3.8, δ is the intersection of C M and the line y = x , this implies that the uniquesolution of − x + q ( x ) = 0 is x = q ( x ) = δ , and h ′ ( x ) < for all x > δ and h ′ ( x ) > for all x < δ. Hence for all x = δ we have δ ( λ ( x )) = h ( x ) < h ( δ ) = δ . Corollary 3.28.
Let S n and S ′ n be two sequences in the Teichmüller space of S . Then thefollowing is equivalent : • lim n →∞ δ ( S n , S ′ n ) = 0 • lim n →∞ δ ( S n , S ′ n ,
1) = 0 • lim n →∞ m ( S n , S ′ n ,
1) = 0
Proof.
From Corollary 3.26, δ ( S n , S ′ n ,
1) = m ( S n ,S ′ n , hence the second and third statements areclearly equivalent. We then show the equivalence of the first and the second ones.By definition ≤ δ ( S n , S ′ n , ≤ δ ( S n , S ′ n ) , hence δ ( S n , S ′ n ) tends to zero, implies that δ ( S n , S ′ n , tends also to zero.Conversely, by Corollary 3.26 δ ( S n , S ′ n ,
1) = a n (1)+ a n (1)2 , where ( a n (1) , a n (1)) is the point onthe Manhattan curve C nM for the surfaces S n and S ′ n , where the slope of one normal vector is .From the convexity of C nM we have that a n (1) > and a n (1) > . Hence if lim n →∞ δ ( S n , S ′ n ,
1) =0 it follows that lim n →∞ a n (1) = 0 and lim n →∞ a n (1) = 0 . By continuity of C nM , this impliesthat there are points in C nM which are as close as we want from the origin, and since the criticalexponent is equal to the abscissa of the intersection of C nM and the line y = x , this finally impliesthat the critical exponent can be made as small as we want. Corollary 3.29.
Let S n and S ′ n be two sequences in the Teichmüller space of S . Then thefollowing is equivalent : • lim n →∞ δ ( S n , S ′ n ) = 1 / • lim n →∞ δ ( S n , S ′ n ,
1) = 1 / • lim n →∞ m ( S n , S ′ n ,
1) = 1 roof. Here again the second and the third statement are equivalent from the formula obtainedin Corollary 3.26.We now show the equivalence of the first two statements. By definition δ ( S n , S ′ n , ≤ δ ( S n , S ′ n ) ≤ / hence the second statement implies the first.Conversely, if lim δ ( S n , S ′ n ,
1) = 1 / by Corollary 3.11, lim λ ( δ ( S n , S ′ n , since thethe Manhattan curve is C , this implies that lim( x n (1) , y n (1)) = lim( δ ( S n , S ′ n ) , δ ( S n , S ′ n )) =(1 / , / . Hence lim n →∞ δ ( S n , S ′ n ,
1) = lim n →∞ x n (1)+ y n (1)2 = lim n →∞ δ ( S n ,S ′ n , δ ( S n ,S ′ n , =1 / . The notion of geodesic currents has been introduced by F. Bonahon in [Bon88] in order to get,in Bonhanon’s words " a better understanding of homotopy classes of unoriented closed curveson S ". But it is also a beautiful tool to understand Thurston compactification of Teichmüllerspace by measured geodesic laminations, [Bon92], or the ends of three manifolds as in [Bon86].Geodesic currents are a generalisation of closed geodesics where the set of geodesic measuredlaminations as well as Teichmüller space Teich( S ) have natural embeddings. All the material ofthis Section is well known and can be found in [Bon92, Bon88, Bon86, Ota90, Šar05]. We surveythis material in order to introduce notations and will follow the lines of [Bon88].Let S be a surface of genus g ≥ , that we endow with a hyperbolic metric, and C the setof closed unoriented geodesics on S . In this Section, geodesics will be primitive elements, ierepresented by an indivisible element of Γ . There is a bijective correspondence between the setof homotopy classes of unoriented closed curves and the set of unoriented closed geodesics withmultiplicity. We didn’t make this distinction in the previous Section since according to Knieper’stheorem, the exponential growth of the number of primitive geodesics and non-primitive geodesicsare the same, hence the resulting Manhattan curve and critical exponent are the same.Consider e S the universal covering of S , and G e S the set of geodesics of e S . Any closed geodesiclifts to a Γ -invariant set of G e S . We can take the multiplicity into account if we identify the liftswith a Dirac measure on this discrete subset, where the Dirac measures are multiplied by themultiplicity of the geodesic. This is equivalent to put weights on geodesics. Hence a weightedsum of geodesics of S is naturally a Γ invariant measure of G e S .There is a parametrization of unoriented geodesics by the boundary at infinity of e S , say ∂ e S ,that is G e S ≃ ∂ e S × ∂ e S − ∆ / Z , where ∆ is the diagonal and Z acts by exchanging the two factors.Giving a hyperbolic metric on S gives an identification between e S and H , whose boundary iscanonically S . Hence G e S is naturally identified with S × S − ∆ / Z . Definition 4.1.
A geodesic current is a Γ invariant borelian measure on G e S . The set GC ofgeodesic currents is endowed with the weak* topology defined by the family of semi-distances d f where f ranges over all continuous functions f : G e S → R with compact support and where d f ( α, β ) = | α ( f ) − β ( f ) | . By the previous discussion the set of homotopy classes of closed curves on S is embedded in GC . Proposition 4.2. [Bon88, Proposition 2] The space GC is complete and the real multiples ofhomotopy classes of closed curves are dense in it. Definition 4.3. If [ α ] and [ β ] are two homotopy classes of closed curves on S , their geometricintersection is the infinimum of the cardinality of α ′ ∩ β ′ for all α ′ ∈ [ α ] and β ′ ∈ [ β ] . α ′ and β ′ the geodesic representatives, [FLP79, Ch 3, Propos-ition 10] .The crucial fact is that the geometric intersection can be extended to geodesic currents. Theorem 4.4. [Bon86, Paragraph 4.2][Bon88, Proposition 3] There is a continuous symmetricbilinear function i : GC × GC → R + such that, for any two homotopy classes of closed curves α, β ∈ GC , i ( α, β ) is the above geometric intersection. If we fix a current α whose support is sufficiently large (if it fills S ), then the set of currentswhose intersections with α is bounded, is compact. More precisely we have the following, Theorem 4.5. [Bon88, Proposition 4] Let α be a geodesic current with the property that everygeodesics of e S transversely meets some geodesic which is in the support of α . Then the set { β ∈ GC , i ( α, β ) ≤ } is compact in GC . Taking a current which fills S gives the following, Corollary 4.6. [Bon88, Corollary 5] The space
PGC of projective geodesic currents is compact.
We are not going to use exactly this fact but a related one : the set of geodesic currents oflength less or equal to one is compact. This leads us to define a length for any geodesic current.Let F be the foliation of T S by the orbit of geodesic flow. For each ϕ t invariant finite measure µ there exists an associated transverse measure to F , e µ , normalized by the requirement that itis locally a product : µ = e µ × dt , where dt is the Lebesgue measure along orbits of ϕ t in F .By definition e µ is a geodesic current. Hence there is a bijective correspondence between the setof invariant measures by the geodesic flow and the set of currents. In particular the Liouvillemeasure gives rise to a geodesic current, L . This current depends on the hyperbolic metric andwe will write the metric as a subscript if there is an ambiguity on the metric we consider on S .For example L m will be the Liouville geodesic current on ( S, m ) . Theorem 4.7. [Bon88, Proposition 14] Let ( S, m ) be a hyperbolic surface. For every closedcurve c ∈ GC , we have i ( L m , c ) = ℓ m ( c ) . This remarkable property allows us to define the length of any geodesic current once we haveset a hyperbolic metric on S , by the extension of i to any geodesic current. Let m be a hyperbolicmetric, for every β ∈ GC , by definition ℓ m ( β ) := i ( L m , β ) Since the Liouville current has the property to intersect transversely every geodesics of e S applying Theorem 4.5 we get, Theorem 4.8.
For any hyperbolic metric m , the set of geodesic currents of m -length equal toone is compact. That is the set { β ∈ GC , i ( β, L m ) = 1 } is compact. The Liouville currents are particularly interesting since they allow to embed Teichmüllerspace into the set of geodesic currents.
Theorem 4.9. [Bon88, Theorem 12] The map m → L m is a proper embedding of Teichmüllerspace into the space of geodesic currents. Definition 4.10.
Let S = ( S, m ) and S = ( S, m ) be two surfaces endowed with hyperbolicmetric.Then the intersection between S and S is defined by i ( S , S ) := i ( L m , L m ) . For all m ∈ Teich( S ) we have that i ( L m , L m ) = π | χ ( S ) | [Bon88, Proposition 15] hence if L m = kL m ′ then k = 1 and m = m ′ . 24 orollary 4.11. [Bon88, Corollary 16] The composition Teich( S ) → GC → PGC is an embed-ding. Finally here is the corollary we are going to use and which is clearly equivalent to the previousone, since there is a homeomorphism between the set of projective currents and the set of currentsof length 1.
Corollary 4.12.
Let ( S , m ) be a fixed hyperbolic surface, then the map m → L m i ( m ,L m ) is anembedding of Teichmüller space into the space of geodesic currents of S length 1. Let us introduce geodesic laminations in the context of geodesic currents. The simplest definitionin this setting is,
Definition 4.13.
A measured geodesic lamination L is a geodesic current whose self-intersection i ( L , L ) is equal to . The set of measured geodesic laminations will be denoted by ML . The original definition of measured laminations due to W. Thurston [TM79] has been shownto be equivalent to the above by F. Bonahon. Recall that according to Thurston, a measuredgeodesic lamination is a closed subset Λ which is the union of simple disjoint geodesics endowedwith a transverse measure, ie. a measure on every arc transverse to Λ invariant by holonomy.The Thurston’s topology on the set of measured laminations is defined by the semi-distances d γ ( α, β ) = | i ( α, γ ) − i ( β, γ ) | where γ ranges over all simple closed curves on S . Theorem 4.14. [Bon88, Proposition 17] The set { α ∈ GC , i ( α, α ) = 0 } has a natural homeo-morphic identification with Thurston’s space of measured laminations. Let
PML be the set of projective measured laminations, that is the quotient of the spaceof geodesic measured laminations by R ∗ + , where the equivalence is given by ( L , µ ) ∼ ( L , xµ ) for x ∈ R ∗ + . The topology of Thurston on Teich( S ) ∪ PML is defined as follows: Teich( S ) is openin Teich( S ) ∪ PML and a sequence m j ∈ Teich( S ) converges to L if and only if the ratio ℓ mj ( α ) ℓ mj ( β ) converges to i ( L ,α ) i ( L ,β ) for every pair of simple closed curves α, β with i ( L , β ) = 0 . Theorem 4.15. [Bon88, Theorem 18] The topology of
Teich( S ) ∪ PML seen as a subspace of PGC is the same as the Thurston’s topology.
By Corollary 4.12 if we fix a metric m , the set of geodesic laminations of m -length 1 iscompact.The simplest example of measured geodesic lamination is, of course, a weighted simplegeodesic, or a disjoint union of such measures. Moreover, the next theorem, which is a ver-sion for laminations of Proposition 3.2, shows that these examples are dense. Theorem 4.16. [Bon86, Proposition 4.9] [TM79] ML is the closure of the linear combinationsof closed disjoint geodesics. We introduced geodesic laminations in order to present a central tool in Teichmüller theory, theearthquake map. They are an extension of the notion of Fenchel-Nielsen twists. Let c be a closedsimple geodesic on ( S, m ) , take a tubular neighborhood U ≃ S × [ − ε, ε ] of c , such that S × { } is isometrically sent to c , and every z × [ − ε, ε ] corresponds to a geodesic arc orthogonal to c .25onsider a smooth function ξ : [0 , ε ] → R equal to on a neighborhood of ε and equal to on aneighborhood of . Then, for t ∈ R consider the map φ t : S → S which is the identity on S − U and on S × [ − ε, ⊂ U and which is defined by φ t ( e iθ , u ) = (cid:0) e iθ − itξ ( u ) , u (cid:1) on S × [0 , ε ] ⊂ U . Themap φ t is a diffeomorphism on S − c , discontinuous along c . The metric φ ∗ t ( m ) on S − c coincideswith the original metric m if we are sufficiently close to c , and it extends to a hyperbolic metricover all S . We denote the new metric by E tc ( m ) , which is, in Thurston terminology, the metricobtained from m by a left earthquake of amplitude t along the curve c . When c is a simple closedcurve, this transformation is more classically called Fenchel-Nielsen twist, the extension to anygeodesic lamination has been studied by W. Thurston, W. Kerckhoff, and F. Bonahon amongothers. Theorem 4.17. [Ker92][Bon92, Proposition 1] There is a continuous function E : ML × R × Teich( S ) → Teich( S ) , associating to m ∈ Teich( S ) an element E t L ( m ) ∈ Teich( S ) such that E tλ L ( m ) = E λt L ( m ) , for all λ > and all L ∈ ML , and coincide with a Fenchel-Nielsen twistwhen L is a closed geodesic. Moreover S. Kerckhoff showed that the length function is convex along earthquake.
Theorem 4.18. [Ker83, p253, Theorem 1] Let m be a hyperbolic metric, L be a measuredgeodesic lamination, and c be a closed curve. Then the function t → ℓ ( E t L ( m ) c ) = i ( E t L ( m ) , c ) isconvex. The last tool we are going to use is the "geology" Theorem of Thurston saying that any twohyperbolic metrics are linked by an earthquake.
Theorem 4.19. [Ker83, Appendix] Let ( S , m ) and ( S , m ) be two hyperbolic metrics on S ,then there exists a unique measured lamination L such that E L ( m ) = m . We are going to use the equivalent following corollary, which is just a renormalization of themeasured lamination seen as current.
Corollary 4.20.
Let ( S , m ) and ( S , m ) be two hyperbolic metrics on S , then there exists aunique measured lamination L of m -length and a T ∈ R + such that : E T L ( m ) = m . In the paper [Sha98], R. Sharp is asking about the behaviour of the correlation number m ( S , S , as S and S range over Teich( S ) . By Corollaries 3.28 and 3.29, the asymptotic behavior of m ( S , S , , δ ( S , S ) and δ ( S , S , are the same. Hence our examples answer his question ina quantitative way.Moreover, this section establishes the counterpart of McMullen’s examples for globally hy-perbolic AdS manifolds.
Proof of Examples
The main ingredients for the different proofs are Proposition 4.22, andthe convexity of the length function along earthquake paths, Theorem 4.18. Example 1 is provedin Proposition 4.26, we use the fact that a pseudo-Anosov diffeomorphism dilates the length ofthe curves along one lamination, and its inverse dilates along another lamination. Those twolaminations fill the surface, therefore we conclude with Proposition 4.22.26xample 2 is proved in 4.4.4. It is the more straightforward. Since there exists some curves forwhich the length is bounded in both surfaces, this immediately gives the bound on the criticalexponent.Example 3 is particularly interesting for our work, since the proof gives the idea for Theorem 5.1.It uses the convexity of length function along earthquakes and invariance of critical exponentthrough the diagonal action of the mapping class group, see Section 4.4.3.Example 4 follows is proved in 4.4.5 from the fact that the critical exponent is uniformly con-tinuous, Proposition 4.30. The proof of this last fact is not so direct, it uses the machinery ofgeodesic currents and earthquakes from the previous sections.Finally we prove also the counter example of Corollary 5.3, when the surfaces do no stay in thethick part of the Teichmüller space, Theorem 4.31. This is done in 4.4.6. The main argumentused is the fact that the Thurston and the Weil-Petersson distances are not comparable, moreprecisely that the Weil-Petersson distance is not complete.
In this first example, we iterate a pseudo-Anosov diffeomorphism to show that the critical expo-nent can tend to . Definition 4.21. [CT07, Section 13] A diffeomorphism A : S → S is said pseudo-Anosov ifthere exists two measured geodesic laminations ( L + , µ + ) , ( L − , µ − ) , one transverses to the others,and a constant k > such that :1. L + ∪ L − fills S .2. A ( L ± ) = L ± . A ∗ µ + = kµ + . (inverse image of µ + by A )4. A ∗ µ − = k µ − . (inverse image of µ − by A )The laminations L + , L − are called attracting and repelling, the constant k is called the dilatationof A . Let ( S, m ) be a hyperbolic surface. Let A be a pseudo-Anosov diffeomorphism and L ± its associated laminations, k its dilatation. We consider the following sequences of hyperbolicsurfaces : S n := ( S, A ∗ n ( m )) et S ′ n := ( S, A ∗− n ( m )) Proposition 4.22.
Let α be a geodesic current which fills S and β any geodesic current. Thereexists K > , such that for every c ∈ C we have : i ( α, c ) ≥ K i ( β, c ) . Proof.
Indeed, the function c → i ( β, c ) is continuous on the set of geodesic currents. As { c ∈GC | i ( α, c ) = 1 } is compact, by Theorem 4.5, there exists K > such that for every c ∈ { c ∈GC | i ( α, c ) = 1 } we have i ( β, c ) ≤ K. We conclude by homogeneity of this last formula.27n particular, using the preceding proposition with α = L , the Liouville current associatedto the metric m and β = L ± we have Corollary 4.23.
There exists
K > such that for every c ∈ C we have ℓ S ( c ) ≥ K i ( L ± , c ) . We also need the following property of the mapping class group . This shows that they acton intersection forms as "isometries". Proposition 4.24.
Let D be a diffeomorphism of S . For every pairs of geodesic current ( α, β ) ∈GC we have i ( Dα, Dβ ) = i ( α, β ) . Proof.
These is true for every currents which are union of closed curves, since D is a diffeomorph-ism and does not change the geometric interesction. We conclude by density and continuity ofthe intersection.We finally gives the bounds on the length of a geodesic on S n and S ′ n . Proposition 4.25.
There exists
K > such that for every c ∈ C and every n ∈ N we have ℓ n ( c ) ≥ k n K i ( L + , c ) .ℓ ′ n ( c ) ≥ k n K i ( L − , c ) . Proof.
Indeed, we just saw that ℓ n ( c ) = i ( A − n L , c ) = i ( L , A n c ) . Hence, from Corollary 4.23,there exists K > such that ℓ n ( c ) ≥ K i ( L + , A n c ) . Thanks to the Proposition 4.24, we have i ( L + , A n c ) = i ( A − n L + , c ) . The third property of pseudo-Anosov diffeomorphisms implies that i ( A − n L + , c ) = k n i ( L + , c ) . These three relations shows the first inequality of the Proposition 4.25. The second inequalitycan be shown by the same method.
Proposition 4.26.
Let A be a pseudo-Anosov diffeomorphism, then lim n →∞ δ Lor ( S , A n S ) = 0 . (Mapping class group is defined by MCG = Diff ( S ) /Diff ( S ) , where Diff ( S ) is the group of diffeomorph-isms isotopic to the identity) roof. We just showed there exists
K > such that for every c ∈ C ℓ n ( c ) + ℓ ′ n ( c ) ≥ k n K i ( L + ∪ L − , c ) As L + ∪ L − fills S , Proposition 4.22 shows there exists K ′ > such that for every c ∈ C wehave i ( L + ∪ L − , c ) ≥ K ′ i ( L , c ) = ℓ ( c ) K ′ . Hence, we have ℓ n ( c ) + ℓ ′ n ( c ) ≥ k n KK ′ ℓ ( c ) . Finally, there exists K ′′ > such that X c ∈C e − s ( ℓ n ( c )+ ℓ ′ n ( c )) ≤ X c ∈C e − sk n K ′′ ℓ ( c ) Hence, critical exponent associated to ( S n , S ′ n ) , satisfies δ ( S n , S ′ n ) ≤ k n K ′′ and tends to .Finally, remark that we can fix S . Indeed, let D be an element of the mapping class group.The critical exponent δ ( S, S ′ ) is equal to δ ( D ( S ) , D ( S ′ )) , since the length spectrum is invariantby M CG . Let S be a hyperbolic surface and consider the critical exponent associated to ( S , A n S ) . This exponent is then equal to δ ( A − n S , A n S ) , which tends to from what wejust saw. Let S be a hyperbolic surface. Take two pants decomposition of S , P and P ′ , such that P ∪ P ′ fills up S (ie the complement of P ∪ P ′ consists of topological disks). Definition 4.27.
A decomposition of a surface S by pair of pants P give Fenchel-Nielsen co-ordinates, "length" and "twist" coordinates. A surface is said to be shrinked along a simplegeodesic c ∈ P if the length coordinates of c tends to and the others do not change. It is said shrinked by a factor x if the length coordinates of c ∈ P is replace by xℓ ( c ) . Shrinking is defined via the hyperbolic structure of S . In Section 4.4.6, we will call pinching the equivalent procedure defined via the complex structure as defined in [Wol75]. The importantfact is the existence of a very small geodesic after shrinking/pinching.We consider the surfaces S n and S ′ n , defined by shrinking the geodesics of P respectively P ′ bya factor e − n . By the [Hal81, collar Lemma], there exists M such that for every c ∈ C , ℓ n ( c ) ≥ C | log( e − n ) | i ( c, P ) = Ci ( c, P ) n , since the length of the geodesics in P are less than e − n . We alsohave ℓ ′ n ( c ) ≥ M | log( e − n ) | i ( c, P ′ ) = nM i ( c, P ′ ) . These two inequalities give, ℓ n ( c ) + ℓ ′ n ( c ) ≥ nM i ( c, P ∪ P ′ ) , By Proposition 4.22 and since
P ∪ P ′ fills up S , there exists K > such that for all c ∈ C wehave i ( c, P ∪ P ′ ) ≥ K ℓ ( c ) , hence ℓ n ( c ) + ℓ ′ n ( c ) ≥ nM K ℓ ( c ) . Again δ ( S n , S ′ n ) ≤ nMK goes to zero. 29 .4.3 Dehn twists The next example contains the basic idea for the proof of Theorem 5.1 : we show that along asequence obtained by iteration of a Dehn twist, the critical exponent is decreasing. Let S :=( S, m ) be a hyperbolic surface and α a simple closed curve on S . Let t α be the Dehn twist along α and define S n (respectively S ′ n ) by S n := t nα S ( respectively S ′ n := t − nα S ). By definition wehave S n = E nℓ ( α ) α ( S ) . We denote for t ∈ R the surface S t := E tℓ ( α ) α ( S ) . Let c be a closedcurve on S and f be the function f : t → ℓ t ( c ) , that is the length of the geodesic representativeof c on S t . By Theorem 4.18 the function f is convex. Finally let g ( t ) = f ( t ) + f ( − t ) and remarkthat g ( n ) = f ( n ) + f ( − n ) (12) = ℓ n ( c ) + ℓ ′ n ( c ) (13) g is convex and g ′ (0) = 0 hence, g is an increasing function on R + . Hence g ( n ) ≥ g ( n − orequivalently, ℓ n ( c ) + ℓ ′ n ( c ) ≥ ℓ n − ( c ) + ℓ ′ n − ( c ) This implies that n → δ ( S n , S ′ n ) is decreasing. Moreover, by Theorem 2.5, δ ( S , S ′ ) < / and we can then conclude lim δ ( S n , S ′ n ) < / . The limit exists since δ ( S n , S ′ n ) is decreasing.As in the first example, this shows that lim δ ( S , τ nα S ) < / . This result is the one we wantto generalise thanks to earthquakes.
Let P be a pants decomposition of a hyperbolic surface S . Let α i , i ∈ [1 , g − be the geodesicsboundaries of P . We call S t the surface shrinked along α by a factor e − t . This means that on S t the length of α is ℓ t ( α ) = e − t ℓ ( α ) , and the length of α i , i ∈ [2 , g − is ℓ t ( α i ) = ℓ ( α i ) .Let ρ t be the sequence of representations of Γ in PSL ( R ) such that H /ρ t (Γ) = S t . Let g i ∈ Γ , i ∈ [1 , g − be elements of the fundamental group projecting to α i , i ∈ [1 , g − . We canchoose ρ t in order that ρ t ( α i ) is fixed for all i ≥ . By definition the critical exponent associatedto ( S , S t ) is larger than the critical exponent associated to ( S \ { α } , S t \ { α } ) , in which wedo not count the curves meeting α .Moreover, the critical exponent associated to ( S \ { α } , S t \ { α } ) is constant and positive. Thisshows that lim inf t →∞ δ ( S , S t ) > . This example as to be compared to the equivalent of [McM99, p.3 Example 3].
We show there is a periodic limiting function for critical exponent along Fenchel-Nielsen twist. Let α be a simple closed curve and E tα the Fenchel-Nielsen twist along α , recall that τ := E ℓ ( α ) α is theDehn twist along α . Let S be a hyperbolic surface and defined S t := E tα ( S ) . Fix t ∈ [0 , ℓ ( α )) ,then as the previous example shows we see that δ ( S , S ℓ ( α ) n + t ) = δ ( τ − n S , τ n S t ) is decreasing,hence we can consider the following function : δ ( t ) := lim n →∞ δ ( S , S ℓ ( α ) n + t ) . Obviously δ is ℓ ( α ) -periodic. 30 heorem 4.28. Let S t be a surfaces path obtained by Fenchel-Nielsen twist along a simple closedgeodesic α . Then there exists a ℓ S ( α ) -periodic function δ such that: lim t →∞ | δ ( S , S t ) − δ ( t ) | = 0 . We need the following result of real analysis
Lemma 4.29.
Let f : R → R be a continuous real function such that F ( t ) := lim n →∞ f ( t + n ) exists for all t ∈ [0 , . If f is uniformly continuous then F is continuous and lim t →∞ | F ( t ) − f ( t ) | = 0 . Proof.
Let ǫ > . Let δ > such that for all t ∈ R , all t ′ ∈ ( t − δ, t + δ ) , we have | f ( t ′ ) − f ( t ) | ≤ ǫ. We then have for all n ∈ N , for all t, t ′ ∈ R such that | t − t ′ | ≤ δ | f ( t + n ) − f ( t ′ + n ) | ≤ ǫ. Taking the limit in n shows that | F ( t ) − F ( t ′ ) | ≤ ǫ, and the continuity of F follows.Now for all t ∈ [0 , , there exists N > such that for all n > N we have: | f ( t + n ) − F ( t ) | ≤ ǫ. Since F is -periodic, we have: | f ( t + n ) − F ( t + n ) | ≤ ǫ. From uniform continuity, for all t ′ ∈ ( t − δ, t + δ ) we have | f ( t ′ + n ) − F ( t ′ + n ) | ≤ ǫ. Covering [0 , by a finite number of segments of the form ( t − δ, t + δ ) , shows there is N ∈ N such that for all n > N and all t ∈ [0 , we have: | f ( t + n ) − F ( t + n ) | ≤ ǫ. In other words, there is
N > such that for all t ≥ N we have: | f ( t ) − F ( t ) | ≤ ǫ. This shows the second part of the proposition.The following proposition will finish the proof Theorem 4.28
Proposition 4.30.
The function t δ ( S , S t ) is uniformly continuous.Proof. We define the following function g ( t ) := sup c ∈C i ( c, α ) ℓ S t ( c ) . This function is well defined by Proposition 4.22. By homogeneity of i ( c,α ) ℓ St ( c ) , we could have takenfor definition the supremum on geodesic currents of S -length equal 1. This set is compact, hencethe function g is continuous. Moreover it is ℓ ( α ) -periodic : g ( t + ℓ ( α )) = sup c ∈C i ( c, α ) ℓ S t + ℓ α ) ( c ) . τ = E ℓ ( α ) α () we have g ( t + ℓ ( α )) = sup c ∈C i ( c, α ) ℓ S t ( τ ( c )) . As i is invariant by the diagonal action of the mapping class group 4.24, g ( t + ℓ ( α )) = sup c ∈C i ( τ c, τ α ) ℓ S t ( τ ( c )) . And α is invariant by τ , hence : g ( t + ℓ ( α )) = sup c ∈C i ( τ c, α ) ℓ S t ( τ ( c )) . We conclude by a change of variable τ c = c ′ g ( t + ℓ ( α )) = sup c ′ ∈C i ( c ′ , α ) ℓ S t ( c ′ )= g ( t ) . Hence g is bounded on R , we call M its maximum.From convexity of earthquake paths, we have for all t ∈ R , ε > and c ∈ C ℓ S t ( c ) − εi ( c, α ) ≤ ℓ S t + ε ( c ) ≤ ℓ S t ( c ) + εi ( c, α ) . We then have ℓ S t ( c ) − εM ℓ S t ( c ) ≤ ℓ S t + ε ( c ) ≤ ℓ S t ( c ) + εM ℓ S t ( c ) . Therefore: ( ℓ S t ( c ) + ℓ S ( c ))(1 − εM ) ≤ ℓ S t + ε ( c ) + ℓ S ( c ) ≤ ( ℓ S t ( c ) + ℓ S ( c ))(1 + εM ) , − M ε ≤ ℓ S t + ε ( c ) + ℓ S ( c ) ℓ S t ( c ) + ℓ S ( c ) ≤ M ε.
Passing to critical exponent we finally obtain that for all η > there exists ε > such that forall t ∈ R . | δ ( S , S t + ε ) − δ ( S , S t ) | ≤ η. This is the result equivalent to [McM99, Theorem 9.6] for GHMC manifolds. Whereas C.McMullen said that the function t → δ ( t ) should not be constant in the quasi-Fuchsian case, itseems legit to think in our context that t → δ ( t ) is constant over R . The reason for this is wehave a natural candidate for the limit of δ ( S , S t ) . Indeed, the length of all the curves crossing α growth to infinity (but not uniformly) hence we can conjecture that t → δ is constant and equalto the abscissa of convergence of X c ∈C , i ( c,α )=0 e − s ℓ S ( c ) . .4.6 Pinching at different speed Finally we give an example of a family of surfaces ( S t , S t ′ ) for which the Thurston distancebetween the two representations is bounded below (or even tends to infinity), but the criticalexponent tends to / . Let S be a hyperbolic surface, and c be a simple closed curve. Let S t the hyperbolic surface obtained by pinching c as it is explained in Wolpert article [Wol75]. Wedon’t want to explain this construction since it uses Jenkins-Strebel differential, which are farfrom the subject of the present paper. The two things we have to know for our example is, firstthat the Weil-Petterson lenght of the path t → S t is finite, and second that the length of thegeodesic c converges to . Theorem 4.31.
There is a sequence t n such that the family d T h ( S n , S t n ) → + ∞ , and such that lim n →∞ δ ( S n , S t n ) = 1 / .Proof. The length of c tends to 0, therefore, for all n there is t n > n such that ℓ S n ( c ) ≥ ℓ S tn ( c ) e n .In this way, the Thurston symmetric distance between ( S n , S t n ) is bigger than log( ℓ Sn ( c ) ℓ Stn ( c ) ≥ n .The result on the critical exponent is due to two facts. The first one is, as we already said,that the Weil-Petersson distance of the path p : R + → Teich( S ) , p ( t ) = S t is finite, this is provedby Wolpert in [Wol75]. The second fact is the link between the intersection and Weil-Peterssonmetric. Bonahon showed in [Bon88, Theorem 19] that i ( m, m ′ ) = i ( m, m ) + o ( d W P ( m, m ′ )) .The first fact assures that d W P ( S n , S t n ) → as n → ∞ . With the second fact, wecan conclude that i ( m n , m t n ) → i ( m n , m n ) . Now by Corollary 3.12 we can conclude that lim n →∞ δ ( S n , S t n ) = 1 / . Finally we will use a Theorem of large deviation for orbits of geodesic flow due to Y. Kifer [Kif94](in a much more general context). Let P be the set of ϕ t -invariant probability on T S . Froma closed geodesic c we can defined a borelian, invariant measure ˆ c as follow. Let E ⊂ T S anyborelian, χ the characteristic function of E and v any vector of ˙ c : ˆ c ( E ) = Z l ( c )0 χ E ( ϕ t ( v )) dt. Every closed geodesic c can be considered as an element of P , through ˆ cl ( c ) . By the discussionin Section 4.1, the set of invariant probability measures is in bijection with the set of geodesiccurrents of length equal to 1. Let b L be the Liouville measure on T S (which corresponds to theLiouville current L ). We give a name for the set of geodesics of S of length less than T : C ( T ) := { c ∈ C | ℓ ( c ) ≤ T } . (14) Theorem 4.32.
For any open neighborhood U of b L in P , there exists η > such that C ( T ) Card (cid:26) γ ∈ C ( T ) , ˆ cℓ ( c ) / ∈ U (cid:27) = O ( e − ηT ) . Moreover η := inf ν / ∈U { − h ( ν ) } where h ( ν ) is the entropy of ϕ t with respect to ν . The fact that η > is a consequence that ˆ L is the only measure of maximal entropy and h ( ˆ L ) = 1 .This theorem has been used by Y. Herrera in his thesis [HJ13] to estimate the self-intersectionnumber of a random geodesic. His method inspired the proof of our main result.33 Isolation theorem
We are now ready to enter into the proof of the main isolation theorems.
Theorem 5.1.
Let S be a fixed hyperbolic surface and S n be a sequence of hyperbolic surfaces.Then lim n →∞ δ ( S , S n ) = 1 / if and only if lim n →∞ d T h ( S , S n ) = 0 . Or in the Lorentzian language :
Theorem.
Let M n be a sequence of GHMC AdS manifolds parametrize by ( ρ , ρ n ) then lim n →∞ δ Lor ( M n ) = 1 if and only if lim n →∞ d T h ( S , S n ) = 0 .Proof. One way is just the continuity of critical exponent at S . Let’s prove it briefly. If d T h ( S n , S ) → , for all ǫ > , there is a n such that for all c ∈ C , and all n ≥ n , − ǫ < ℓ n ( c ) ℓ ( c ) < ǫ. Hence P c ∈C e − s ( ℓ ( c )+ ℓ n ( c )) > P c ∈C e − s ( ℓ ( c )+(1+ ǫ ) ℓ ( c )) This implies that for any ǫ > , and for n ≥ n we have δ ( S , S n ) ≥
12 + ǫ .
Therefore, lim δ ( S , S n ) ≥ . Recalling that / ≥ δ ( S , S n ) for any surfaces gives the result.Let us show the converse. Suppose by contradiction that d T h ( S , S n ) doesn’t tend to 0. If S n stays in a compact subset of Teich( S ) , it admits a converging subsequence, that we still denoteby S n . Denote by S ∞ its limit which by hypothesis is different of S , then δ ( S , S ∞ ) < / byrigidity Theorem 3.6. By continuity of critical exponent for the Thurston metric, lim δ ( S , S n ) = δ ( S , S ∞ ) < / which is absurd, hence we can assume that S n leaves every compacts set of Teich( S ) .By earthquake’s Theorem 4.20, there is a path from S to S n in Teich( S ) following an earth-quake line. The first step then consists in proving that along every earthquake paths, the lengthof "most" curves on S are increasing. This will imply that the Poincaré’s series over these curveshas a decreasing critical exponent. Consider the following function, f : ML ( S ) × GC → R ( L , ν ) i ( E L ( m ) ,ν ) i ( L ,ν ) which is continuous, since earthquakes and intersection are continuous, Theorems 4.17 and 4.4.And where m is the metric on S , ML ( S ) is the set of laminations of m -length 1 and L isthe Liouville current associated to m . The set ML ( S ) is compact hence g ( ν ) := min L f ( L , ν ) is well defined and continuous.The compactness of ML ( S ) implies also the existence of L ∈ ML ( S ) such that g ( L ) =min L f ( L , L ) = i ( E L ( m ) ,L ) i ( L ,L ) . Remark that E L ( m ) = m since L is not the trivial lamination,hence by Corollary 3.13 it follows that g ( L ) > . Let ǫ > such that g ( L ) = 1 + ǫ . Recall thatfor a geodesic current µ we denote by b µ its corresponding ϕ t -invariant probability obtained bythe product of µ and the length along fibers. Let U be the neighbourhood of c L in P defined by U := { ˆ µ ∈ P , | g ( µ ) − g ( L ) | < ǫ/ } , and consider the set C U := (cid:26) c ∈ C , ˆ cℓ ( c ) ∈ U (cid:27) .
34y definition, if c is in C U , it satisfies, g ( c ) − g ( L ) > − ǫ/ , that is min i ( E L ( m ) ,c ) i ( m ,c ) > ǫ/ .Equivalently, we have for any L ∈ ML ( S ) ℓ ( E L ( m )( c )) ℓ ( c ) > ǫ/ . By convexity of length along earthquake, Theorem 4.18, we have for all
L ∈ ML ( S ) , all t > and all c ∈ C U , ℓ ( E t L ( m )( c )) ≥ (1 + tǫ/ ℓ ( c ) . This last inequality is what we mathematically meant, by saying that the length of "most"curves are increasing. Indeed Kifer’s result, Theorem 4.32, says that
Card C c U ∩ C ( k ) has smallerexponential growth than Card C ( k ) .Let’s look to the Poincaré series associated to ( S , S n ) , P ,n ( s ) := X c ∈C e − s ( ℓ ( c )+ ℓ n ( c )) . By Corollary 4.20, there exists L n ∈ ML ( S ) and t n such that E t n L n ( m ) = m n , hence P ,n ( s ) = X c ∈C e − s ( ℓ ( c )+ ℓ ( E tn L n ( m )( c ))) . Now we divide this sum into two parts, the curves which are in C U and the others.We claim that the critical exponent of the Poincaré series associated to the curves whichare in C U , tends to . Indeed, S n goes out of every compacts of Teich( S ) , hence by continuityof ( t, L ) → E t L ( m ) , the sequences t n must tends to infinity, in particular is greater than .Therefore: X c ∈C U e − s ( ℓ ( c )+ l ( E tn L n ( m )( c ))) < X c ∈C U e − s ( ℓ ( c )(1+ ǫt n / It implies that the series P c ∈C U e − s ( ℓ ( c )+ ℓ ( E tn L n ( m )( c ))) has critical exponent strictly less than ǫt n / . The fact that t n → ∞ finishes the proof of the claim.In the second step of the proof we get an upper bound on the critical exponent of the Poincarésum over C c U . This relies on Kifer’s Theorem and the directional critical exponent since we wantthat the length of the geodesics on the second factor to be almost proportional to the length onthe first.Let λ n be the slope for which the directional critical exponent δ ( λ n ) between S and S n ismaximal. Let u > and A n ( u ) := n c ∈ C , (cid:12)(cid:12)(cid:12) ℓ n ( c ) ℓ ( c ) − λ n (cid:12)(cid:12)(cid:12) < u o . By Theorem 3.15 and Corollary3.27 for any u > the critical exponent of P c ∈ A n ( u ) e − s ( ℓ ( c )+ ℓ n ( c )) is equal to the criticalexponent of the whole Poincaré series P ,n , that is to say δ ( S , S n ) . Hence δ ( S , S n ) is equal tothe maximum of the critical exponent of the two following series : • P c ∈C U ∩ A n ( u ) e − s ( ℓ ( c )+ ℓ n ( c )) • P c ∈C c U ∩ A n ( u ) e − s ( ℓ ( c )+ ℓ n ( c )) We saw in the previous claim that the critical exponent of the first one goes to , in particularit is strictly less that δ ( S , S n ) for n sufficiently large, since we suppose that δ ( S , S n ) → / .35ence δ ( S , S n ) is equal to the critical exponent of P c ∈C c U ∩ A n ( u ) e − s ( ℓ ( c )+ ℓ n ( c )) . The end of theproof consists to show that this exponent cannot tends to / .For c ∈ A n ( u ) , ℓ n ( c ) ≥ ℓ ( c )( λ n − u ) , we then have X c ∈C c U ∩ A n ( u ) e − s ( ℓ ( c )+ ℓ n ( c )) ≤ X c ∈C c U ∩ A n ( u ) e − sℓ ( c )(1+ λ n − u ) . By Theorem 4.32, the complementary set of C U is "small" for the metric m , that is thereexists η > and M > such that Card( C c U ∩ C ( T )) ≤ M Card ( C ( T )) e − ηT ≤ M ′ e (1 − η ) T . Let C c U ( k ) = { c ∈ C c U and ℓ ( c ) ∈ [ k, k + 1) } . X c ∈C c U ∩ A n ( u ) e − sℓ ( c )(1+ λ n − u ) ≤ X k ∈ N X c ∈C c U ( k ) e − sℓ ( c )(1+ λ n − u ) ≤ X k ∈ N X c ∈C c U ( k ) e − sk (1+ λ n − u ) . And since C c U ( k ) ⊂ C c U ∩ C ( k ) , X c ∈C c U ∩ A n ( u ) e − sℓ ( c )(1+ λ n − u ) ≤ X k ∈ N X c ∈C c U ∩C ( k ) e − sk (1+ λ n − u ) ≤ X k ∈ N Card { c ∈ C c U ∩ C ( k ) } e − sk (1+ λ n − u ) ≤ X k ∈ N M ′ e (1 − η ) k e − sk (1+ λ n − u ) ≤ X k ∈ N M ′ e k (1 − η − s (1+ λ n − u )) . This finally implies that P c ∈C c U ∩ A n ( u ) e − sℓ ( c )(1+ λ n − u ) has critical exponent less than − η λ n − u .Combined to the fact that the critical exponent of P ,n is less or equal to this last one, and takingthe limit in u → , we get : δ ( S , S n ) ≤ − η λ n . Suppose that δ ( S , S n ) → / , then by Corollary 3.11 we deduce λ n → . Taking the limit n → ∞ , we get / ≤ − η which is absurd. This concludes the proof.Next we look at what happens if both surfaces change. Recall that Teich ǫ ( S ) is the thick partof Teichmüller space: surfaces for which no closed geodesic has length less than ǫ . The mappingclass group preserves the length spectrum, hence acts on the thick part of Teichmüller. Recallthe Mumford compactness theorem. Theorem 5.2. [Mum71] For any ǫ > , Teich ǫ ( S ) /M CG is compact. So if a surface stays in
Teich ǫ ( S ) , we can send it in a fixed compact set by the mapping classgroup. This remark with the previous Theorem allows us to show : Corollary 5.3.
Let S n and S ′ n be two sequences of hyperbolic surfaces. Suppose that at leastone the sequences stays in Teich ǫ ( S ) for some ǫ . Then lim n →∞ δ ( S n , S ′ n ) = 1 / if and only if lim n →∞ d ( S n , S ′ n ) = 0 .
36r in the Lorentzian language
Theorem 5.4.
Let ( M n ) be a sequence of GHMC manifolds, parametrized by ( S n , S ′ n ) . If oneof the sequences stays in the thick part of Teich( S ) then lim n →∞ δ Lor ( M n ) = 1 iff lim n →∞ d T h ( S n , S ′ n ) = 0 . Proof.
We are going to prove that if lim n →∞ δ ( S n , S ′ n ) = 1 / then lim n →∞ d ( S n , S ′ n ) = 0 , theother implication is again a consequence of the continuity of the critical exponent for the Thurstonmetric. By hypothesis we can suppose that S n stays in Teich ǫ ( S ) . Hence there exists a compact K in Teich( S ) and D n in the mapping class group, such that D n ( S n ) ∈ K for every n . Hencewe can suppose that D n ( S n ) converges to S ∞ ∈ K .Let u > for every n sufficiently large, − u ≤ ℓ D n ( S n ) ( c ) ℓ ∞ ( c ) ≤ u Hence the critical exponents satisfies : (1 − u ) δ S ∞ ,D n ( S ′ n ) ≤ δ D n ( S n ) ,D n ( S ′ n ) ≤ (1 + u ) δ S ∞ ,D n ( S ′ n ) . Now since the mapping class group doesn’t change the length spectrum, the critical expo-nent of ( S n , S ′ n ) is equal to the critical exponent of ( D n ( S n ) , D n ( S ′ n )) . If lim n →∞ δ ( S n , S ′ n ) =1 / then lim n →∞ δ ( D n ( S n ) , D n ( S ′ n )) = 1 / and by the previous inequalities it follows that lim n →∞ δ ( S ∞ , D n ( S ′ n )) = 1 / since u is arbitrary small. By Theorem 5.1 this implies that lim n →∞ d ( D n ( S ′ n ) , S ∞ ) = 0 , which finally implies that lim n →∞ d ( S n , S ′ n ) = 0 . References [BB09] Riccardo Benedetti and Francesco Bonsante.
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