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Geometric properties of Anosov representations

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Geometric properties of Anosov representations

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Content GEOMETRIC PROPERTIES OF ANOSOV REPRESENTATIONS by Guillaume Dreyer A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (MATHEMATICS) August 2012 Copyright 2012 Guillaume Dreyer \Vous vous trompez; je le connais fort bien; il est po ete et math ematicien. Comme po ete et math ematicien, il a d^ u raisonner juste; comme simple math ematicien, il n'aurait pas raisonn e du tout [...]" Edgar Allan Poe, La Lettre Vol ee, par Baudelaire. ii Acknowledgments \Come on, Guillaume! You should not forget about what research is about. We do not solve a problem, we analyze a problem." There are plenty of things that I could say about my advisor. There are certainly plenty of things that I would like to say about Francis. The above words though perhaps illustrate at best both the clever scientist and the great person that he is. They remain to me as the only true lesson that someone has ever taught me. But more essentially, that day when Francis pronounced them, they were to me very kind, supportive, full of optimism and wisdom. So, let me say it once: Thank you. I want to warmly thank Anna Weinhard and Dick Canary for inviting me many times and spending hours answering my questions. Many thanks also to Bill Goldman, Olivier Guichard, Francois Labourie, Martin Bridgeman, Ken Bromberg, for all the support and interest that they constantly showed for this work. As well, a special thank to Ko, Aravind, Nicolai Haydn and Werner D appen for being members of my committee. At last a HUGE thank to all the sta: Amy (you rock!), Arnold, Alma and Fatima. A big thought to all my friends: you guys mean so much to me: Mihaela, Adam, Russell, Julien, Ibrahim, Noelle, Alexander, Kate, Steve, Nick, Beyer, Rapp, Rodrigue, iii L ea, John, Loech, Gendro d, Vince, Manon, Zog, Tatiana, C eline, Alice, Ted, T2, Jo, r, Marc, Julien, Elise, and all of the others, the list is just endless.. I am someone extremely lucky in life: I feel love.. Thank you ve so much: Anne, Robert, Juliane, Thibaut and Tapas. iv Table of Contents Dedication ii Acknowledgments iii Abstract vii Chapter 1: Introduction 1 1.1 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.1 Length functions for Anosov representations. . . . . . . . . . . . . 2 1.2.2 Cataclysm deformations for Anosov representations. . . . . . . . . 4 Chapter 2: Background 6 2.1 Hitchin representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Anosov representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.1 Bundle description . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.2 Flag curve description . . . . . . . . . . . . . . . . . . . . . . . . . 9 Chapter 3: Length functions 11 3.1 Motivations and main results . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Construction of the length functions . . . . . . . . . . . . . . . . . . . . . 13 3.2.1 H older geodesic currents . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2.2 Lengths of closed curves . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2.3 The lengths of H older geodesic currents . . . . . . . . . . . . . . . 16 3.2.4 Properties of length functions . . . . . . . . . . . . . . . . . . . . . 22 3.3 An application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Chapter 4: Cataclysm deformations 28 4.1 Motivations and main results . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.2.1 Some estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.2.2 Transverse cocycles for geodesic laminations . . . . . . . . . . . . . 33 4.3 Cataclysms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.4 Cataclysm deformations and ag curves . . . . . . . . . . . . . . . . . . . 42 4.5 Injectivity of cataclysms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 v 4.5.1 A summation description of the shear "2C twist ( b ) . . . . . . . . . 47 4.5.2 1{forms 0 i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.5.3 Thurston's intersection form . . . . . . . . . . . . . . . . . . . . . . 59 4.5.4 Variation of length functions ` i . . . . . . . . . . . . . . . . . . . . 60 Bibliography 63 vi Abstract Let S be a connected, closed, oriented surface of negative Euler characteristic, we con- sider the PSL n (R){character variety Rep PSLn(R) (S). An interesting connected compo- nent of the latter space is the Hitchin spaceH PSLn(R) (S): it contains a copy of the Teichm uller spaceT (S), and hence is regarded as the higher rank Teichm uller space in the case of PSL n (R). In order to study the elements in the Hitchin spaceH PSLn(R) (S), F. Labourie introduced the notion of Anosov representation. In particular, he proved that every Hitchin representation is discrete and injective, some properties already shared by Teichm uller representations. In this dissertation, we extend to Anosov representations several classic tools from hyperbolic geometry designed to study Teichm uller representa- tions: we generalize Thurston's length function and cataclysm deformation, and analyze how these two notions relate to each other. We then explain how these techniques illu- minate the geometry of Anosov representations, and provide crucial information about a new system of coordinates on the Hitchin spaceH PSLn(R) (S). vii Chapter 1 Introduction 1.1 Context Let S be a closed, connected, oriented surface of negative Euler characteristic. The G{character variety Rep G (S) is the space of conjugacy classes of homomorphisms from the fundamental group 1 (S) to a Lie group G. Character varieties are objects of great interest in mathematics and theoretical physics, as they occur as spaces of geometric structures on surfaces. As an example, in gauge theory, the space Rep G (S) classies equivalence classes of at connections on G{bundles over S, which occur as critical points of the Chern-Simons and Yang-Mills functionals. A classic and fundamental example of character variety is the PSL 2 (R){character variety Rep PSL 2 (R) (S), where PSL 2 (R) = SL 2 (R)=fIdg. Such a surface admits hyper- bolic structures, namely complete nite area Riemannian metrics of constant curvature equal to1. Because PSL 2 (R) is also the group of positive isometries of the hyperbolic plan, hyperbolic structures correspond to the subset of discrete faithful representations of Rep PSL 2 (R) (S); this subset is called the Teichm uller spaceT (S) of the surface S. An important property due to results of Weil [36], Kazhdan, Margulis, Zassenhaus [32, 31] is that the Teichm uller spaceT (S) is a connected component of the space Rep PSL 2 (R) (S). About 20 years ago, Hitchin [26, 18] extended some properties of the Teichm uller space T (S) Rep PSL 2 (R) (S) to more general Lie groups. More precisely, when G is a split real Lie group (e.g. G = PSL n (R), PSp 2n (R), PO n;n+1 or PO n;n ), Hitchin identied a preferred component, called the Hitchin spaceH G (S), which contains a copy of the Teichm uller spaceT (S). In particular, the Hitchin spaceH G (S) and the Teichm uller spaceT (S) coincide when G = PSL 2 (R). 1 Teichm uller theory has been a topic of active mathematical research over the last cen- tury, which has lead to the development of deep and fruitful interactions between analysis and geometry. More recently, fundamental work of Choi and Goldman [11], Labourie [28, 29, 30], Guichard and Wienhard [21, 22, 23, 24, 25], Fock and Goncharov [14, 15], has revealed beautiful geometric and dynamic properties for the elements ofH G (S), inau- gurating the development of higher Teichm uller theory. Following this perspective, this work proposes to extend to higher Teichm uller theory several tools and techniques from classical hyperbolic geometry. To do so, we will mainly follow the dynamical approach to higher Teichm uller theory via Anosov representations, which were introduced by F. Labourie. 1.2 Results 1.2.1 Length functions for Anosov representations. In [12], we develop the construction of length functions for Anosov representations when G = PSL n (R). Our construction generalizes, and is motivated by the length function introduced by Thurston [34, 35] to study elements of the Teichm uller spaceT (S). More precisely, let 2T (S) be a hyperbolic structure on a surface S. An essential invariant for are the lengths of the simple closed geodesics in S. A beautiful idea of Thurston consists of introducing the space of measured laminationsML(S), which is a certain completion of the set of all simple closed geodesics on the hyperbolic surface S. The main property is that the \lengths of closed geodesics" extend to a continuous function` :ML(S)!R + called Thurston's length function; the latter function constitutes a very useful invariant of the Teichm uller representation . We extend Thurston's length function to the context of Anosov representations. Given an Anosov representation : 1 (S)! PSL n (R), Labourie [28] showed that the 2 image ( )2 PSL n (R) of any non-trivial 2 1 (S) is diagonalizable, with distinct real eigenvalues i ( ) that we canonically index so that j 1 ( )j>j 2 ( )j>>j n ( )j: This property is well-known for Teichm uller representations. Consider the (real) vector space of H older geodesic currentsC H (S); this space was introduced by Bonahon [1, 5, 2, 6]. In particular, it contains the space of measured laminationsML(S). Theorem 1 ([12]). Let : 1 (S)! PSL n (R) be an Anosov representation. For every i = 1, 2, . . . , n, there exists a continuous linear function ` i :C H (S)!R such that, for every oriented closed curve S, ` i ( ) = logj i ( )j. In the special case when n = 2, the functions ` 1 , ` 2 are related to Thurston's length function ` :C H (S)!R via the properties that ` 1 =` and ` 2 =` . We also prove the following identities. Let R : T 1 S ! T 1 S be the orientation- reversal involution of the unit tangent bundleT 1 S; for every geodesic current2C H (S), R 2C H (S) denotes the pullback of under R. Proposition 2 ([12]). For every H older geodesic current 2C H (S), 1. P n i=1 ` i () = 0; 2. ` i (R ) =` ni+1 (). The continuity of the length functions ` i :C H (S)!R is a crucial property. As an application of this continuity, we obtain for instance the following stability estimate for eigenvalues in the image of an Anosov representation. 3 Corollary 3 ([12]). Let : 1 (S)! PSL n (R) be an Anosov representation, and let , 2 1 (S). Recall that i () and i ( k ) denote the i{th eigenvalues of () and ( k ), respectively. Then the ratio i ( k ) i () k has a nite limit as k tends to1. This result uses the full force of H older geodesic currents, and can be interpreted as a dierentiability property. Indeed, the above limit is equal to e ` i (_ ) for some H older geodesic current _ = lim k!1 k k2C H (S). 1.2.2 Cataclysm deformations for Anosov representations. In [13], we extend to Anosov representations the notion of cataclysm deformation intro- duced by Thurston [35, 4], and we study their action on the length functions ` i . Let 0 : 1 (S)! PSL 2 (R)2T (S) be a hyperbolic structure on a surface S. We begin with cutting the surface S into nitely many thin triangles called ideal triangles; this can be done with xing a maximal geodesic lamination S, namely is a certain partial foliation of S whose leaves are simple geodesics, and such that the complement S is made of ideal triangles. A cataclysm deformation of the hyperbolic structure 0 then consists of shearing the ideal triangles of S along the leaves of the geodesic lamination. Such an operation modies the \gluing" of the triangles along the leaves of , resulting in a deformation of the initial hyperbolic structure 0 onS towards another hyperbolic structure 1 . 4 Theorem 4 ([13]). Let be an Anosov representation. Then there exist an open neigh- borhood 02U C twist ( b ) along with a continuous, injective, cataclysm deformation map :U ! A(S) " 7! " such that 0 =. An interesting feature of the length functions ` i lies in the fact that they detect cataclysm deformations. More precisely, they keep track of the shearing variation of the triangles along the leaves of . Consider the orientation cover b of the geodesic lamination. Let :C H ( b )C H ( b )!R be Thurston's intersection form [7, 4], which is a certain symplectic pairing dened over the vector space ofC H ( b ) of transverse cocycles for the geodesic lamination b . The space of transverse cocyclesC H ( b ) identies [7] with a subspace of the vector space of H older geodesic currentsC H (S). Theorem5 ([13]). Let` i and` 0 i be the length functions associated with and 0 = " , respectively. For every transverse cocycle 2C H ( b ) for the geodesic lamination b , ` 0 i () =` i () +(;" i ): 5 Chapter 2 Background Throughout,S denotes a closed, connected, oriented surface of negative Euler character- istic, endowed with a hyperbolic metric m 0 . We x an universal cover e S and denote by @ 1 e S its ideal boundary. A well-known fact is that @ 1 e S is a compact H older manifold which is homeomorphic to the circle S 1 [19, 9, 16]. 2.1 Hitchin representations We consider preferred components of the PSL n (R){character variety Rep PSLn(R) (S) = Hom 1 (S); PSL n (R) ==PSL n (R) identied by N. Hitchin [26]. When n = 2, a Teichm uller representation r : 1 (S)! PSL 2 (R) is an injective homomorphism with discrete image. W. Goldman [17] showed that Rep PSLn(R) (S) has 4g 3 components. The Teichm uller representations occupy 2 of these components, and each of these Teichm uller components is homeomorphic toR 6g6 . When n 3, a Fuchsian representation is a representation : 1 (S)! PSL n (R) of the form =r where r : 1 (S)! PSL 2 (R) is a Teichm uller representation and where : PSL 2 (R)! PSL n (R) is the n{dimensional irreducible representation of PSL 2 (R). A Hitchin repre- sentation is a homomorphism : 1 (S)! PSL n (R) that lies in the same component of Rep PSLn(R) (S) as some Fuchsian representations. 6 In his foundational paper [26], using tools from Higgs bundles theory, Hitchin proved that, when n 3, the character variety Rep PSLn(R) (S) has 3 components if n is odd, and 6 components ifn is even. In addition, the Hitchin representations constitute 1 or 2 components of Rep PSLn(R) (S), according to whethern is odd or even, and these Hitchin components are all homeomorphic toR (2g2)(n 2 1) . 2.2 Anosov representations 2.2.1 Bundle description As observed by Hitchin [26], the complex geometric framework of Higgs bundles oers little information on the geometry of the Hitchin representations. F. Labourie [28, 29, 30] (see also [21]) introduced the concept of Anosov representation as a dynamical approach to Hitchin representations. This dynamical framework will be the key tool for our study, and we now review some of Labourie's construction. The choice of a hyperbolic metricm 0 onS induces a geodesic ow (g t ) t2R on the unit tangent bundle T 1 S; we refer to the associated orbit space as the m 0 {geodesic foliation F of T 1 S. Let M be the space of n{tuplets of lines (V 1 ;V 2 ;:::;V n )2 (RP n1 ) n which form a direct decomposition ofR n , i.e. R n = V 1 V n . Let : 1 (R)! PSL n (R) be a representation. Consider the at twisted M{bundle T 1 S M =T 1 e SM= 1 (S) where e S is the universal cover of S and where the action of 1 (S) is dened by the property that (u;v) = ( u;( )v) for every 2 1 (S) and (u;v)2T 1 e SM. Besides, the geodesic ow (g t ) t2R ofT 1 S lifts via the at connection to a ow (G t ) t2R on T 1 S M. 7 The representation : 1 (S) ! PSL n (R) is said to be Anosov if there exists a continous section :T 1 S!T 1 S M, u7! (V 1 (u);V 2 (u);:::;V n (u)), such that 1. is at, namely, each V i is invariant under the action of the ow (G t ) t2R ; 2. Consider the at twisted End(R n ){bundleT 1 S End(R n )!T 1 S, where( 1 (S)) acts on the second factor End(R n ) by conjugation. Let (G t ) t2R be the lift to T 1 S End(R n ) of the geodesic ow (g t ) t2R . The above section (u) = (V 1 (u);V 2 (u);:::;V n (u)) enables us to denen 2 n line sub-bundlesV i V j !T 1 S of the bundle T 1 S End(R n ). Note that, since is at, each line sub-bundle V i V j is invariant under the action of the lifted ow (G t ) t2R . Furthermore, we require the restricted ow (G t ) t2R to each lineV i V j to be Anosov, namely, there exists a Riemannian metrickk on the line bundle V i V j , and a constant a> 0 such that,8u2T 1 S,8X u 2V i V j (u),8t 0, if i>j, kG t X u k gt(u) e at kX u k u ; if i<j, kG t X u k g t (u) e at kX u k u : Remark 6. When the representation : 1 (S)! PSL n (R) lifts to a representation : 1 (S)! SL n (R) (this is the case when is Hitchin for instance), there is a convenient description of Anosov representations. Let : 1 (S)! SL n (R) be a lift; in particular, we can consider the at twistedR n {bundle T 1 S R n !T 1 S. Then the representation : 1 (S)! SL n (R) is Anosov if there exists a line sub-bundle decompositionV 1 V n of the bundleT 1 S R n with the above properties 1. and 2. For the sake of the exposition, we shall often implicitly assume that the representation : 1 (S)! PSL n (R) lifts to : 1 (S)! SL n (R). In particular, we shall refer to the line sub-bundle decomposition V 1 V n !T 1 S of the bundle T 1 S R n !T 1 S as Labourie's line decomposition. The Anosov dynamics on the line sub-bundles V i V j ! T 1 S has several major consequences, that we now state. 8 Theorem 7 (Labourie [28]). Let : 1 (S)! PSL n (R) be an Anosov representation. Then is injective and discrete. In addition, the image ( )2 PSL n (R) of any non- trivial 2 1 (S) is diagonalizable, and its eigenvalues have distinct absolute values. Futhermore, let us denote byA(S) Rep PSLn(R) (S) the set of Anosov representa- tions. Theorem 8 (Labourie [28]). The set of Anosov representationsA(S) is open in the character variety Rep PSLn(R) (S). Moreover, every Hitchin representation is Anosov, i.e. H(S)A(S): For more general results regarding surface group representations in a semisimple Lie group, see [25]. 2.2.2 Flag curve description A fundamental property of Anosov representations is the existence of an associated equivariant boundary map. Recall that a (complete) ag F ofR n consists of a nested sequence of vector subspaces F =F 1 F 2 F n1 where each F i is a subspace ofR n dimension i. Let Flag(R n ) be the ag variety ofR n . Theorem 9. Let : 1 (S)! PSL n (R) be an Anosov representation. There exists a unique {equivariant ag curve :@ 1 e S! Flag(R n ) such that 1. :@ 1 e S! Flag(R n ) is H older continuous; 2. For every x6=y2@ 1 e S, i (x) M ni (y) =R n : 9 The existence and the uniqueness, as well as the regularity of the ag curve, are consequences of the Anosov dynamics [28, 25, 21]. Indeed, the {equivariant ag curve :@ 1 e S! Flag(R n ) and the associated at twisted bundleT 1 S R n !T 1 S are related as follows. Consider Labourie's line sub-bundle decomposition V 1 V n ! T 1 S of T 1 S R n that lifts to e V 1 e V n ! T 1 e S. For every e u2 T 1 e S, let g e S be the oriented geodesic directed bye u with positive and negative endpointsx + g andx g 2@ 1 e S, respectively. Then, for every i = 1, ::: , n, e V i (e u) = i (x + g )\ ni+1 (x g )R n : Reversely, let : 1 (S) ! PSL n (R) be a representation admitting a continuous { equivariant curve : @ 1 e S ! Flag(R n ) such that, for every x 6= y 2 @ 1 e S, i (x)\ ni+1 (y)6=?. Making use of the above relation, we can dene a at line sub-bundle decomposition V 1 V n ! T 1 S of the at twisted bundle T 1 S R n . Then, the representation : 1 (S)! PSL n (R) is Anosov if and only if the latter line decompo- sition satises the conditions 1. and 2. stated inx2.2.1. In particular, we obtain a characterization of Anosov representations in terms of equivariant ag curves. To close this short review, we mention a more direct and intuitive description of the {equivariant ag curve : @ 1 e S! Flag(R n ), as the limit set for the action of the fundamental group 1 (S) on the ag variety Flag(R n ) via the representation . Proposition 10 (Labourie [28]). Let : 1 (S)! PSL n (R) be an Anosov representation along with its associated ag curve :@ 1 e S! Flag(R n ). Then the image (@ 1 e S) is the limit set for the action of the group ( 1 (S)) PSL ( R), namely, it is the intersection of all ( 1 (S)){invariant closed subsets in the ag variety Flag(R n ). 10 Chapter 3 Length functions Throughout, S denotes a closed, connected, oriented surface of negative Euler charac- teristic, endowed with a hyperbolic metric m 0 . We x an universal cover e S and denote by @ 1 e S its ideal boundary. 3.1 Motivations and main results When n = 2, given a Teichm uller representation r : 1 (S)! PSL 2 (R), W. Thurston introduced a very useful invariant associated with r called the length function ` r . More precisely, Thurston considers in [34] the space of measured laminationsML(S), which is a certain completion of the set of all isotopy classes of unoriented simple closed curves on S. The length function ` r is then dened as a certain continuous homogeneous function ` r :ML(S)!R such that, for every simple closed curve S, ` r ( ) = logj( )j wherej( )j is the largest absolute value of the eigenvalues of r( )2 PSL 2 (R). Geo- metrically, r constitutes the monodromy 1 (S)! PSL 2 (R) of a hyperbolic metric m on S; the number ` r ( ) is then half the length of the unique closed m{geodesic that is homotopic to . This length function ` r was later extended by F. Bonahon [1, 2] to the spaceC(S) of measure geodesic currents, which appears as a completion of the set of all homotopy classes of closed curves on S. Bonahon later developed in [4] a dierential calculus of measured laminations that is based on H older geodesic currents, namely on transverse H older distributions for the geodesic foliation of the unit tangent bundle T 1 S of S. He 11 then obtained dierentiability properties for Thurston's original function ` r :ML(S)! R by continuously extending ` r to the spaceC H (S) of H older geodesic currents. We generalize this construction to Anosov representations. Since a matrix in PSL n (R) hasn eigenvalues, we now haven length functions. Theorem 8 shows that, if a representa- tion is Anosov and if 2 1 (S) is non-trivial, the eigenvalues i ( ) of( )2 PSL n (R) can be indexed so that j 1 ( )j>j 2 ( )j>>j n ( )j: We will refer to i ( ) as the i{th eigenvalue of ( ). The main result of this chapter is the following. Theorem 11. Let be an Anosov representation and letC H (S) be the space of H older geodesic currents on S. Then, for each i = 1, 2, . . . , n, there exists a continuous linear function ` i :C H (S)!R such that, for every oriented closed curve S, ` i ( ) = logj i ( )j. Since ( ) 2 PSL n (R), P n i=1 logj i ( )j = 0. Likewise, as a consequence of our indexing conventions, we have i ( 1 ) = 1= ni+1 ( ), and thus logj i ( 1 )j = logj ni+1 ( )j. These two facts suggest some symetry properties for the length func- tions ` i . Let R : T 1 S! T 1 S be the involution dened by R(u) =u, where u2 T 1 x S. For every current 2C H (S), R denotes the pullback current of the current under R. The length functions ` i satises the following identities. Theorem 12. For every current 2C H (S), 1. P n i=1 ` i () = 0; 2. ` i (R ) =` ni+1 (). 12 Our main motivation for dening length functions associated with an Anosov repre- sentation is to obtain information about a possible system of coordinates \ a la Thurston" on the Hitchin spaceH(S) via the shearing coordinates. Indeed, when n = 2, Thurston showed that the length function ` r constitutes a full invariant of the Teichm uller repre- sentation r, and used length functions to parametrize the Teichm uller spaceT (S). In the case of a Hitchin representation , we shall discuss later how the lengths ` i capture crucial geometric information regarding . Finally, as a direct application of our work, and to illustrate the power of the fame- work of H older geodesic currents, we prove the following result. Theorem 13. Let : 1 (S)! PSL n (R) be a Hitchin representation, and let , 2 1 (S). Then the ratio i ( k ) i () k has a nite limit as k tends to1. This limit is equal to e ` i (_ ) for some H older geodesic current _ = lim k!1 k k2 C H (S). 3.2 Construction of the length functions 3.2.1 H older geodesic currents Before constructing the length functions ` i :C H (S)!R, we need to remind the reader of the denition of H older geodesic currents. See [2, 4, 5, 1] for details. In the following, even if it is not explicitly mentioned, closed curves are always supposed to be oriented. In a metric space (X;d), a H older distribution is a continuous linear functional on the space of H older continuous functions : X!R with compact support. A special case of H older distributions are positive Radon measures, which are linear functionals on 13 the space of continuous functions with compact support and associate a non-negative number with a non-negative function. The unit tangent bundle T 1 S is a 3{dimensional manifold, and the orbits of the geodesic ow (g t ) t2R dene a 1{dimensional foliation F of T 1 M, called its geodesic foliation. It turns out that, whereas the geodesic ow depends of the auxiliary metric m that we have chosen on S, the geodesic foliation does not. More precisely, if another negatively curved metricm 0 denes a geodesic foliationF 0 , there is a homeomorphism of T 1 S that sendsF toF 0 . In addition, this homeomorphism can be chosen to be isotopic to the identity, and H older bi-continuous. We refer the reader to [9, 16, 19, 20] for further details. A H older geodesic current onS is a transverse H older distribution for the geodesic foliationF. Namely, assigns a H older distribution D on each surface D T 1 S transverse toF, and this assignment is invariant under restriction of D to a subsurface D 0 D, and under homotopy of D to another transverse surface D 00 by a homotopy preservingF. When the H older distribution D is actually a measure D for every transverse surfaceD, the corresponding H older geodesic current is a measure geodesic current. Let C H (S) andC(S) denote the space of H older geodesic currents and the space of measure geodesic currents, respectively. Note thatC H (S) is a (real) vector space andC(S) is stable under positive scalar multiplication. The spaceC H (S) is endowed with the weak-* topology, which is the weakest topology for which the functions7!('), as' ranges over all H older continuous functions with compact support dened on a surface D transverse toF, are continuous. A typical example of measure geodesic current is provided by a closed curve S. Let k 0 be the largest integer such that is homotopic to a k{multiple k 1 of a closed curve 1 . The homotopically primitive curve 1 is then homotopic to a unique closed m{geodesic, which itself corresponds to a closed orbit of the geodesic ow or, equivalently, to a closed leaf 1 of the geodesic foliationF. We can then associate with 14 1 the transverse 1{weighted Dirac measure forF dened by this closed orbit, which with a surface D transverse toF associates the counting measure at the points D\ 1 . Finally, we associate with k{times the transverse 1{weighted Dirac measure dened by 1 . In this way, we have an embedding foriented closed curves in Sg=homotopyC(S)C H (S): The positive real multiples of homotopy classes of closed curves are dense inC(S). The spaceML(S) of measured laminations can be dened [1, 2] as the closure inC(S) of the set of positive real multiples of homotopy classes of simple closed curves in S. 3.2.2 Lengths of closed curves Let : 1 (S)! PSL n (R) be a Hitchin representation. For a non-trivial 2 1 (S), Theorem 8 shows that the eigenvalues i ( ) of the matrix ( ) are real and can be indexed so that j 1 ( )j>j 2 ( )j>>j n ( )j: Set ` i ( ) = logj i ( )j. Note that i ( ) and ` i ( ) depend only on the conjugacy class of 2 1 (S), and therefore depend only on the free homotopy class of the closed curve S. This denes n maps ` i :foriented closed curves in Sg=homotopy!R: 15 However, the latter length functions` i are not very convenient, since they are dened on a space which does not carry any interesting structure. Our goal is to extend these lengths to continuous linear functions ` i :C H (S)!R: Besides being continuous and dened over a much more suitable space, the length functions` i will also contain much more intrinsic information about the associated rep- resentation . Note that the restrictions` i :C(S)!R are positively homogeneous, in the sense that ` i (c) =c` i () for everyc2R + and2C(S). As a result, since positive real multiples of closed curves are dense in the spaceC(S) of measure geodesic currents, the restriction of this continuous extension toC(S) is unique. The same property holds in the closure of the linear span ofC(S) inC H (S), which contains all the H older geodesic currents that have arisen in geometric applications so far. 3.2.3 The lengths of H older geodesic currents We now introduce our construction of the length ` i () of a H older geodesic current . In order to do so, for everyi, we rst dene a 1{form! i along the leaves of the geodesic foliationF. Given a lifted Anosov representation : 1 (S)! SL n (R), consider the associated at twisted bundle T 1 S R n !T 1 S. We begin with a version of main Labourie's line subbundle decomposition theorem containing all the properties that we will need in the following construction. Theorem 14 (Labourie [28]). The at twisted bundle p :T 1 S R n !T 1 S splits as a direct sum of n line sub-bundles V 1 V n such that 1. Each bundle V i is at, namely it is invariant under the ow (G t ) t2R lifting the geodesic ow (g t ) t2R of T 1 S; 16 2. If u2 T 1 S is xed by g t : T 1 S! T 1 S for some t, and if 2 1 (S) represents the corresponding closed orbit of the geodesic ow, then the lift G t acts on the bre p 1 (u) by multiplication by 1= i ( ) on the line V i (u)p 1 (u); 3. The breV i (u) is a H older continuous function ofu2T 1 S in the following sense: if we lift the bundle V i to a line sub-bundle e V i of the trivial bundle T 1 e SR n !T 1 e S, the map T 1 e S ! RP n which with e u 2 T 1 e S associates the line e V i (e u) is H older continuous. We can make the second property more precise as follows. Let (e g t ) t2R be the lift of the geodesic ow on the universal cover e S. Lift u2 T 1 S to e u2 T 1 e S; this determines an identication of the bre p 1 (u) withfe ugR n . The choice of e u also species in a unique way the element 2 1 (S) by the property thate g t (e u) = e u. Because of the at connection, the two lines e V i (e u) and e V i (g t (e u)) = e V i ( e u) = ( ) e V i (e u) identify with the same subspace ofR n . Therefore, via the identicationp 1 (u) =fe ugR n =R n , the line e V i (e u) corresponds to the eigenspace of ( ) associated with the i{th eigenvalue i ( ). An easy consequence of this observation is the following property, that we state as a lemma for future reference. Lemma 15. For e u2 T 1 e S projecting to u2 T 1 S, let e V i (e u)fe ugR n and e V j (e u) fe ugR n lift the lines V i (u) and V j (u), respectively. Then these two lines e V i (e u) and e V ni+1 (e u) are equal as subsets ofR n . Proof. When e u is in a closed orbit of the geodesic ow, the property immediately fol- lows from the condition (2) in Theorem 14, and from our ordering conventions for the eigenvalues of ( )2 PSL n (R), when 2 1 (S). The general case then follows from the latter since closed orbits are dense in T 1 S. Endow the bres of the vector bundle p : T 1 S R n ! T 1 S with a Riemannian metrickk. In particular, the normkk u that it induces on the bre p 1 (u) depends smoothly on u2T 1 S. 17 Let L be a leaf of the geodesic foliationF. We dene a dierential 1{form ! i along L as follows. Pick a pointu 0 2LT 1 S and a vectorX i in the breV i (u 0 ) of the line sub-bundle V i ! T 1 S. Consider the geodesic ow (g t ) t2R of T 1 S together with its lift (G t ) t2R on the at bundle T 1 S R n . We can dene a function f u 0 ;X i on a neighborhood I of u 0 in the leaf L by the property that f u 0 ;X i g t (u 0 ) = logkG t (X i )k gt(u 0 ) for every t in a neighborhood of 0 in R. Dened as above, it is immediate that the function f u 0 ;X i is smooth along the leaf L. Then set ! i =df u 0 ;X i on the same neighborhood I of u 0 in the leaf L. Lemma 16. The 1{form ! i does not depend on the choices of the point u 0 2I and of the vector X i 2V i (u 0 ). Proof. Since the bre V i (u 0 ) has dimension 1, any other choice X 0 i for X i is of the form X 0 i =cX i for some c2R. Then f u 0 ;X 0 i =f u 0 ;X i + logjcj and df u 0 ;X 0 i =df u 0 ;X i . This proves the independence of ! i regarding the choice of X i . Choose another point u 0 0 =g t 0 (u 0 )2I, and the vector X 0 i =G t 0 (X i ) in the bre of V i (u 0 0 ). Then the functions f u 0 0 ;X 0 i =f u 0 ;X i coincide on I because G t (X 0 i ) =G t+t 0 (X i ), and hence have the same dierential ! i . 18 By Lemma 16, it follows that ! i is a well-dened dierential 1{form ! i along the leaves ofF. Let us also make the following observation regarding the global regularity of the ! i . Lemma 17. The 1{form ! i is smooth along the leaves of the geodesic foliationF, and is transversally H older continuous. Proof. This is an immediate consequence of the regularity condition (3) in Theorem 14. We now use this dierential form ! i to dene the length ` i () of a H older geodesic current 2C H (S). SinceS is compact, we can coverT 1 S by a nite familyfU j g j=1;:::;m of foliated open subsetsU j . By foliated, we mean that there exists a dieomorphismU j = D j (0; 1), where D j is an open subset of R 2 and where, for every x 2 D j , the open segment fxg (0; 1) corresponds an arc contained in a leaf ofF. Letf j g j=1;:::;m be a partition of unity subordinate to the open coveringfU j g j=1;:::;m . Integrating the dierential form j ! i over the leaves ofU j , we obtain a function h j :D j !R dened by h j (x) = Z fxg(0;1) j ! i : The H older geodesic current induces a H older distribution on D j , that we shall still denote by . By Lemma 17, the above function h j :D j !R is H older continuous with compact support. We shall denote the evaluation of at the function h j by (h j ) = Z U j j ! i d where the integral notation is suggested by the case where is a transverse measure for F. 19 Finally, put ` i () = Z T 1 S ! i d = m X j=1 Z U j j ! i d: By the usual linearity arguments, ` i () is independent of the choice of the open cover fU j g j=1;:::;m and of the partition of unityf j g j=1;:::;m . Lemma 18. The length function ` i () is independent of the choice of the Riemannian metrickk on T 1 S R n . Proof. Letkk 0 be another Riemannian metric onT 1 S R n , dening another 1{form! 0 i along the leaves of the geodesic foliationF. Then, there exists a positive dierentiable function f :T 1 S!R such thatkX i k =f(u)kX i k 0 for every X i 2V i (u), so that ! 0 i =! i d logf on each leaf L ofF. As a consequence, Z T 1 S ! 0 i d = Z T 1 S ! i d Z T 1 S d logfd: Note that d logf = P m j=1 d( j logf) since P m j=1 j = 1 and P m j=1 d j = 0. Therefore, Z T 1 S d logfd = m X j=1 Z U j d( j logf)d = 0 by Stokes's. This proves that R T 1 S ! 0 i d = R T 1 S ! i d as requested. Remark 19. Note that the proof of Lemma 18 does not use the full regularity of the Riemannian metrickk. It holds under the weaker assumption that the Riemannian metric is smooth along the leaves of the geodesic ow and H older continuous transversally for that ow. See the proof of Proposition 22 below. 20 We now have a well-dened function ` i :C H (S)!R: Proposition 20. The function ` i :C H (S)!R is linear and continuous. Its restriction ` i :C(S)!R is positively homogeneous. Proof. By construction, ` i () = P m j=1 (h j ). The linearity and homogeneity is immedi- ate, and the continuity follows from the denition of the weak-* topology ofC H (S). To close the proof of Theorem 11, we need to connect these functions` i to the length functions for closed curves discussed inx3.2.2. We saw inx3.2.1 that a closed curve S denes a measure geodesic current 2C(S)C H (S). It also determines an element 2 1 (S) up to conjugacy, so that the i{th eigenvalue i ( ) of ( ) is well-dened. Proposition 21. For every closed curve S, ` i ( ) = logj i ( )j where` i ( ) is the image of the geodesic current 2C H (S) under the map` i :C H (S)!R and where i ( ) is the i{th eigenvalue of ( ). Proof. We need to return to the denition of 2C H (S). By homogeneity of the function ` i , we can restrict attention to the case where the closed curve is homotopically primitive, namely is not homotopic to a multiple k 1 of a closed curve 1 withk 2. Thus determines a simple closedm{geodesic ofS, and a closed leaf of the geodesic foliationF. We then identify with the geodesic current 2C H (S) is the transverse 1{weighted Dirac measure dened by this leaf (seex3.2.1). By denition of the map ` i :C H (S)!R, ` i ( ) = Z T 1 S ! i d = Z ! i : 21 To calculate this integral, pick a point u 0 2 T 1 S and a non-zero vector X i 2 V i (u 0 ) in the bre of the line bundle V i . By denition of the 1{form ! i , Z ! i = Z d logkG t (X i )k = logkG 0 (X i )k log G t (X i ) wheret denotes the necessary time to go around by the geodesic ow, namely where t is the smallest t> 0 such that g t (u 0 ) =u 0 . By the condition (2) of Theorem 14, G t (X i ) = 1= i ( )X i , whereas G 0 (X i ) = X i since (G t ) t2R is a ow. The result immediately follows. 3.2.4 Properties of length functions We now prove the dierent properties stated in Theorem 12. Proposition 22. For every 2C H (S), n X i=1 ` i () = 0: Proof. Endow each brefe ugR n of the trivial bundle T 1 e SR n with the canonical volume form =dx 1 ^dx 2 ^:::^dx n ofR n . Recall that 1 (S) acts on T 1 e SR n via the diagonal action. Since ( )2 PSL n (R), the form is invariant under the action of 1 (S). In addition, the lift e G t of the geodesic ow acts trivially on the factor R n of T 1 e SR n , and consequently preserves . As a result, descends to a well-dened G t {invariant volume form on the bres of the bundle T 1 S R n . The construction of the length functions ` i uses a Riemannian metrickk on the bres of the bundle T 1 S R n . Without loss of generality, we can arrange that the sub-bundles V i are orthogonal forkk, and that the volume form of g coincides with the volume form . Observe that as a consequence of the regularity of the line bundles V i , the latter Riemannian metrickk is smooth along the leaves of the geodesic foliationF, and is transversally H older continuous; see Remark 19. 22 By denition of the ! i , for an arbitrary choice of vectors X i (u 0 )2V i (u 0 ), n X i=1 ! i = n X i=1 d ds log kG s X i (u 0 )k g gs(u 0 ) = d ds log n Y i=1 kG s X i (u 0 )k g gs(u 0 ) ! = d ds log gs(u 0 ) (G s X 1 (u 0 );G s X 2 (u 0 );:::;G s X n (u 0 )) = d ds log ( u 0 (X 1 (u 0 );X 2 (u 0 );:::;X n (u 0 ))) = 0 at the point g s (u). By integration, it follows that P n i=1 ` i () = 0 for every 2C H (S). The unit tangent space T 1 S comes with a natural involution R :T 1 S!T 1 S R n , which tou2T 1 S associatesR(u) =u. In particular, R respects the geodesic foliation F and induces an involutionR :C H (S)!C H (S) which with2C H (S) associatesR . R denotes the pullback current of the current under R, namely, if ' : D!R is H older continuous with compact support dened on a transverse surface DT 1 S, then (R )(') =('R). Proposition 23. For every 2C H (S), ` i (R ) =` ni+1 (): Proof. The involutionR acts freely on the unit tangent bundleT 1 S with quotient b T 1 S = T 1 S=R. As a result, it also acts freely on the bundle T 1 S R n with quotient a bundle b T 1 S R n . If we construct the Riemannian metric kk on T 1 S R n by lifting a Riemannian metric on b T 1 S R n , we can therefore arrange that the Riemannian metric kk is invariant under the involution R. 23 On the other hand, by denition ofR,G t R =RG t . In particular, by Lemma 15, R sends the sub-bundle V i to V ni+1 . It immediately follows that R (! i ) =! ni+1 . By integration, this proves that ` i (R ) =` ni+1 () for every 2C H (S). 3.3 An application The analytical framework of H older geodesic currents can be used to obtain rst order estimates. Bonahon developed in [1] a dierential calculus for measured geodesic lami- nations on a surface S based on H older geodesic currents. The extension of the length functions` i to H older geodesic currents provides regularity properties for these functions. As an application of this property, we now prove the following asymptotic identity. Theorem 24. Let : 1 (S)! PSL n (R) be a Hitchin representation, and let , 2 1 (S). Then the ratio i ( m ) i () m has a nite limit as m!1. Proof. Without loss of generality, we can assume that is primitive in 1 (S). As explained inx3.2.1, let us identify the closed curves and m with the corresponding closed orbits of the geodesic ow. Endowing these closed orbits with the transverse Dirac measures that they dene, we can thus also consider and m as geodesic currents. For m large enough, the closed orbit m is made up of one piece of uniformly bounded length (\representing "), and of another piece that wraps m times around . Asm tends to1, the closed orbit m converges to the union of the closed orbit and of an innite orbit 1 of the geodesic ow whose two ends spiral around . Figure 1 shows the situation on the surface S. Note that both orbits projects onto two disjoint geodesics in S. More precisely, let D T 1 S be a small surface transverse to the geodesic foliation F that intersects the closed orbit in one pointx 1 1 , as shown on Figure 2. The innite 24 orbit 1 intersectsD in two sequences of pointsx 1 1 ,x 1 2 , . . . andy 1 1 ,y 1 2 , . . . , in such a way that x 1 1 , x 1 2 , . . . converges in this order towards one end of 1 , and y 1 1 , y 1 2 , . . . converges in this order towards the other end. In addition, the two sequences x 1 1 , x 1 2 , . . . and y 1 1 , y 1 2 , . . . both converge to the point x 1 1 . Then m intersects D in points x m 1 , x m 2 , . . . , x m km , y m 1 , y m 2 , . . . , y m lm (see Figure 2), in such a way that, as m tends to1, each x m k converges to x 1 k , and each y m l converges to y 1 l . Furthermore, the total number k m +l m of points is of the order of m and, more precisely, the dierence m (k m +l m ) is equal to a constant c D for m large enough. As a consequence, if ' is a continuous function dened on D, lim m!1 1 m m (')(') = lim m!1 1 m km X i=1 '(x m i ) + lm X j=1 '(y m j ) (k m +l m +c D )'(x 1 1 ) ! = 0: If, in addition, ' is H older continuous, the estimate can be improved because the tech- niques of [4] can be used to show that the limit lim m!+1 ( m m) (') = lim m!+1 km X i=1 '(x m i )'(x 1 1 ) + lm X j=1 '(y m j )'(x 1 1 ) c D '(x 1 1 ) = 1 X i=1 '(x 1 i )'(x 1 1 ) + 1 X j=1 '(y 1 j )'(x 1 1 ) c D '(x 1 1 ) 25 exists and is nite. In other words, the above calculation shows that there exists a limit lim m!1 m m = _ in the spaceC H (S) of H older geodesic currents. The limit geodesic current _ is supported on the union of the closed orbit and of the innite orbit 1 , whose two ends spiral around . By linearity and continuity of the length functions ` i , lim m!1 ` i ( m )m` i () exists and is equal to ` i ( _ ). Taking the exponential on both sides, we conclude that i ( m ) i () m converges to e ` i (_ ) , which proves the desired result. We close this section with one last observation. _ = lim m!1 m m = lim m!1 1 m m 1 m is very reminiscent of the expression of a derivative. In fact, there exists a 1{parameter family of geodesic currents t 2C H (S), t2 [0;"], such that 0 = , 1 m = 1 m m , and d dt + t jt=0 = _ . In other words, the distribution _ must be understood as a tangent vector at the point , and the above estimate appears as a direct consequence of the rst order expansion ` i ( t )` i ( 0 ) +t d dt + ` i ( t ) jt=0 26 where d dt + ` i ( t ) jt=0 = ` i ( _ ) by the previous facts. This should be compared with the dierentiability property of the length function when n = 2 developed in [4]. 27 Chapter 4 Cataclysm deformations Throughout, S denotes a closed, connected, oriented surface of negative Euler charac- teristic, endowed with a hyperbolic metric m 0 . We x an universal cover e S and denote by @ 1 e S its ideal boundary. A classical fact is that @ 1 e S is a H older manifold, which is homeomorphic to the circle S 1 . Besides, @ 1 e S inherits a natural orientation from S. 4.1 Motivations and main results The purpose of this chapter is to extend to Anosov representations a classical type of deformation in hyperbolic geometry called cataclysm. Cataclysm deformations were introduced by Thurston [35, 4], and generalize (left) earthquake deformations [34, 3, 27]. Given a measured lamination andt2R, an earthquakeE t :T (S)!T (S) is a certain 1{parameter family of transformations of the Teichm uller space T (S). Earthquakes constitute essential deformations in Teichm uller theory, which illuminate the geometry ofT (S). For instance, Thurston [34, 35] showed that two points in the Teichm uller space T (S) can always be joined by an unique earthquake path. Fix a maximal geodesic lamination S [5, 4], namely, a union of disjoint, simple, complete geodesics, so that the complementS is made of nitely many ideal triangles. A cataclysm [35, 4] consists of a deformation that shears the ideal triangles of S along the leaves of . It is therefore very similar to an earthquake, except that the shear is allowed to simultaneously occur to the left and to the right. Such an operation modies the gluing of the ideal triangles along the leaves of the geodesic lamination , which results in a deformation of the initial hyperbolic structure on the surface S towards another hyperbolic structure on S. At last, as with an earthquake, a cataclysm 28 deformation is parametrized by a transverse cocycle " for the geodesic lamination [6, 4], which determines the shift of the triangles along the leaves of , namely, the magnitude of the cataclysm. Theorem 25 ([13]). Let be an Anosov representation. Then there exist an open neighborhood 02U C twist ( b ) and a continuous, injective, cataclysm deformation map :U ! A(S) " 7! " such that 0 =. In Chapter 3, given an Anosov representation : 1 (S)! PSL n (R), we developed the construction of associated length functions ` i , which generalize Thurston's length function` r of a Teichm uller representationr : 1 (S)! PSL 2 (R). Similarly as in classical hyperbolic geometry [4, 27, 3], we show that the length functions` i constitute interesting invariants for cataclysm deformations, as they detect some information about how the Anosov representation varies under such deformations. Theorem26 ([13]). Let` i and` 0 i be the length functions associated with and 0 = " , respectively. For every transverse cocycle 2C H ( b ) for the geodesic lamination b , ` 0 i () =` i () +(;" i ): where :C H ( b )C H ( b )!R is Thurston's intersection form. 29 4.2 Preliminaries 4.2.1 Some estimates LetS be a maximal geodesic lamination. We shall say that an arck is transverse to ifk a simple arc which is transverse to the leaves of and which does not backtrack, so thatk intersects each leave of at most once. Given such an arck, letd 0 be a component of k which does not contain an endpoint of k. Denote by g d 0 and g + d 0 the two geodesic leaves passing through the end points of d 0 . Note that, since the lamination S is maximal, the geodesics g d 0 and g + d 0 are asymptotic. Consider all components dk which are bounded by g d 0 and g + d 0 , namely every component d such that g d 0 and g + d 0 are both passing by the endpoints of d. In particular, as shown on Figure ..., on each side of d 0 , we obtain a set of components d k. The fact that the two asymptotic geodesicsg d 0 andg + d 0 spread out in the opposite direction guarantees that at least one of two latter sets is nite. We dene the divergence radius r(d 0 )2N as the smallest cardinal of these two sets. Lemma27. Letr be a nonnegative integer and letD r be the set of componentsdk such that r(d) =r. There is a number N > 0 independent of k such that 8r2N;jD r jN: Lemma 28. There exist some constants A > 0 and B 0, both depending on the geodesic arck, such that the length of each componentsdk is bounded byBe Ar(d) . Proof. Lemma 27 is an immediate consequence of the fact that the complement S is made of nitely ideal triangles. Lemma 28 follows from an usual hyperbolic geometry estimate. Let e e S be the lift of the maximal geodesic laminationS. Letk be a transverse arc to e . For every componentdk e , we shall denote byz d 2@ 1 e S the common ideal endpoint of the asymptotic geodesics g d and g + d bounding d, and by x d and x + d 2@ 1 e S 30 the other ideal endpoints ofg d andg + d , respectively. Consider the associated equivariant ag curve :@ 1 e S! Flag(R n ) of some Anosov representation : 1 (S)! PSL n (R). Proposition 29. There exist some constants K > 0 and M 0, both depending on k, such that, for every component d k e , the distance dist (x d );(x + d ) is bounded above by Me Kr(d) . Proof. Recall that a chart of the H older manifold @ 1 e S is dened as follows. Fix a point u 0 2 e S and consider all geodesics rays based at u 0 , the ideal boundary @ 1 e S is then paprametrized via the oriented angle between two rays. Since it is always possible to homotop k onto a transverse geodesic arc to e , we may assume k to be geodesic. k being compact, there exists a constant C 0 (depending on k and u 0 ) such that, for every component dk e , the angle between the two geodesic rays u 0 x d and u 0 x + d is bounded above by C length(d). In addition, is H older continuous, which implies that dist (x d );(x + d ) is bounded above by C 0 length(d) with C 0 0 and 0 < 1. At last, an application of Lemma 28 yields the desired estimate. Recall the following important property of the equivariant ag curve : @ 1 e S ! Flag(R n ) associated with an Anosov representation : 1 (S)! PSL n (R). For every pair of distinct points x, y2 @ 1 e S, the ags (x) and (y) are transverse. Let g e S be a geodesic with endpoints x g and x + g . Suppose g to be oriented from x g to x + g . For i = 1, ::: , n, set e V i (g) = i (x + g )\ ni+1 (x g ): By transversality, e V i (g)R n is a 1{dimensional subspace and e V 1 (g) e V 2 (g) e V n (g) forms a direct summand inR n . As a result, with every oriented geodesic g e S, we associate a line decomposition e V 1 (g) e V 2 (g) e V n (g) = R n . Finally, given " = (" 1 ;" 2 ;:::;" n )2R n , T " g :R n !R n shall denote the linear map which acts on each line e V i (g) by multiplication by e " i . Let k be a transverse oriented arc k to e . Orient the leaves of e intersecting k so that the angle between k and every leaf of e is positively oriented. 31 Lemma 30. For every component d k e and for every "2R n , there exists some constant K 0 depending on k such that T " g d T " g + d Id = O e 2k"kKr(d) : Proof. Observe that, for every component dk e , the oriented asymptotic geodesics g d = z d x d and g + d = z d x + d bounding d either both converge to or diverge from their common endpoint z d . Thus, since e V i (g d ) = i (z d )\ ni+1 (x g ) and k is compact, for every component dk e , we have dist e V i (g d ); e V i (g + d ) M dist (x d );(x + d ) for some constant M 0 (depending on k). Then, applying Lemma 28, we obtain that, for every dk e , dist e V i (g d ); e V i (g + d ) M 0 e Kr(d) for some constant M 0 0 (depending on k). Now, letB d be a basis of unit vectors adapted to the line decomposition e V 1 (g d ) e V 2 (g d ) e V n (g d ) =R n . By an easy calculation, it follows from the above inequality that, for every dk e , Mat B d (T " g d T " g + d Id) Matn(R) M 00 e Kr(d) for some constantM 00 0 (depending onk). Again,k being compact, the set of adapted basisfB d g dk e made of unit vectors is contained in a compact subset of (R n ) n . As a result, for every dk e , T " g d T " g + d Id End(R n ) = Mat Bcan (T " g d T " g + d Id) Matn(R) M 000 e Kr(d) for some constant M 000 0 (depending on k). 32 4.2.2 Transverse cocycles for geodesic laminations We need to remind the reader of the denition of transverse cocycles for geodesic lam- inations, along with their main properties. We refer the reader to [6, 4, 5] for further details. Let S be a geodesic lamination. A transverse cocycle for can be thought as a transverse signed measure for which is nitely additive. More precisely, assigns to every transverse arck to a number(k), with the property that(k) =(k 1 ) +(k 2 ), where k 1 and k 2 are two subarcs of k with disjoint interior and such that k = k 1 [k 2 . Moreover, is homotopy invariant, i.e. (k) = (k 0 ) whenever one transverse arc can be mapped onto the other via a homotopy preserving the leaves of . We shall denote byC H () the (real) vector space of transverse cocycles for . For the purpose of our work, we will mostly consider transverse cocycles for the orientation cover b of a maximal geodesic lamination S. Let U S be an open neighborhood of obtained after puncturing the interior of each ideal triangle in S; the orientation cover b ! can be extended to a 2{cover b U ! U. Note that, since the geodesic lamination b is oriented and the surface b U is oriented, every transverse arc k b U admits a positive transverse orientation, so that the angle between k and every leaf of b is positively oriented. Therefore, by a transverse cocycle for b , we shall always mean a transverse cocycle for the geodesic lamination b contained in some open surface b U dened as above. However, for the sake of the exposition, while working with the oriented cover b , we will often omit to mention the surface b U. LetC H ( b ) be the vector space of transverse cocycles for b . It follows from the fact that is maximal that the dimension ofC H ( b ) is equal to 12g11 [6]. At last, observe that the oriented geodesic lamination b can be viewed as a subset of T 1 S. In particular, transverse cocycles for b and H older geodesic currents of S are related as follows: there is a bijection between the space of transverse cocyclesC H ( b ) and the subspace of H older geodesic currents of C H (S), which are supported on b T 1 S [6]. 33 Let k b U be a transverse arc to b . For every component d k b , let k d be the subarc of k joining the negative endpoint u k of k to any point u contained in d. Fix a normkk on the vector spaceC H ( b ). Lemma 31. There exists a constant C (depending on k) such that, for anyC H ( b ) and for every component dk b , j(k d )jCkk (r(d) + 1) where r(d) is the divergence radius of d. Proof. We refer the reader to [6] for a detailed proof of this fact. Let R : b ! b be the orientation reversal involution. For any 2C H ( b ), R denotes the pull back transverse cocycle of by R. We dene the vector space of transverse n{twisted cocycles for b as C twist ( b ) = n " = (" 1 ;" 2 ;:::;" n )2C H ( b ) n = X " i = 0, R " i =" ni+1 o : Lemma 32. The dimension of the vector spaceC twist ( b ) is equal to (n 1)(6g 6) + n1 2 . Proof. Set E =C H ( b ), and let : E! E, 7! R be the pull back endomorphism. Since is an involution, the spaceE splits as a direct sumE + E , whereE denotes to the1{eigenspace. Observe that the subspaceE + corresponds to the space of transverse cocycles for the geodesic lamination, whose dimension is 6g 6 since is maximal [6]. The dimension of E is thus equal to 6g 5. Given 2 E, write = + + , with 2E . For every i = 1, ::: , n, we have R " i =" ni+1 34 which is equivalent to 8 > > < > > : " + i =" + ni+1 " i =" ni+1 By an easy calculation, it thus follows that the dimension of the vector space n " = (" 1 ;" 2 ;:::;" n )2C H ( b ) n =R " i =" ni+1 o is equal to (n 1)(6g 6) + n1 2 + (6g 5). Finally, we see that the second condition P " i = 0 is in fact a condition on the components " i only; this decreases the latter dimension by 6g 5. 4.3 Cataclysms We now detail the construction of cataclysm deformations for Anosov representations. This construction mostly will take place in the universal cover e S. Indeed, as we shall see, a cataclysm deformation formally consists of a certain equivariant family of linear maps " =f' " P g P e S e SL n (R), which is parametrized by a transverse n{twisted cocycle " = (" 1 ;" 2 ;:::;" n )2C twist ( b ). Consider an Anosov representation : 1 (S)! PSL n (R) along with its associated equivariant ag curve : @ 1 e S! Flag(R n ). Fix a maximal geodesic lamination in S that lifts to e e S. Let P and Q be two ideal triangles in the complement e S e , and letP PQ be the set of ideal triangles lying between P and Q. Let k be a transverse oriented geodesic arc joining a point in the interior of P to a point in the interior of Q. In particular, since k does not backtrack, for every triangle R2P PQ , the oriented arc k intersects exactly two egdes g R and g + R , where g R and g + R denote the closest leaves to the triangleP andQ, respectively. Orient the leaves of e intersectingk according to the transverse orientation determined by the oriented arc k. 35 Let " = (" 1 ;" 2 ;:::;" n )2C twist ( b ) be a transverse n{twisted cocycle. As inx4.2.1, for every triangle R2P PQ , the ag curve :@ 1 e S! Flag(R n ) enables us to associate with the two oriented geodesicsg R andg + R intersecting the arck two linear mapsT "(P;R) g R and T "(P;R) g + R , respectively. In the latter expression, the term "(P;R)2R n denotes the n{twisted measure "( \ p(k PR ))2R n , where \ p(k PR ) is the transverse oriented arc to b dened as follows. Consider a transverse oriented arc k PR e S (to e ) joining a point in the interior of the triangle P to a point in the interior of the triangle R, which projects onto the transverse oriented arc p(k PR )S (to ). As inx4.2.2, let US be an open neighborhood of the maximal geodesic lamination together with its associated 2{cover b U b . Since it is always possible to homotop the transverse arc p(k PR ) S onto a transverse arck 0 US, we may assume without lost of generality that the transverse arc p(k PR ) is contained in U. The arc p(k PR ) being oriented, it admits a preferred lift \ p(k PR ) b U among its two possible lifts, i.e. \ p(k PR ) is the unique lift which is positively oriented for the transverse orientation of b . Given a nite subsetP =fR 1 ;R 2 ;:::;R m gP PQ , where the indexj ofR j increases as one goes from P to Q, we consider the following linear map ' P =T "(P;R 1 ) g 1 T "(P;R 1 ) g + 1 T "(P;R 2 ) g 2 T "(P;R 2 ) g + 2 T "(P;Rm) g m T "(P;Rm) g + m T "(P;Q) g Q where g j denote the two leaves g R j bounding the triangle R i as above. Endow the vector spaceC twist ( b ) with the normk"k = maxk" i k, where " = (" 1 ;:::;" n )2C twist ( b ) and wherekk is itself a norm on the vector spaceC H ( b ). Likewise,kk R n andkk End(R n ) denote a norm onR n and on End(R n ), respectively. Proposition 33. For every "2C twist ( b ) small enough, lim P!P P;Q ' P exists and is an element of SL n (R). 36 Proof. Set P =T "(P;R 1 ) g 1 T "(P;R 1 ) g + 1 T "(P;R 2 ) g 2 T "(P;R 2 ) g + 2 T "(P;Rm) g m T "(P;Rm) g + m . We begin with showing that P is uniformely bounded. We have k P k End(R n ) T "(P;R 1 ) g 1 T "(P;R 1 ) g + 1 End(R n ) T "(P;Rm) g m T "(P;Rm) g + m End(R n ) : Lemma 30 implies that for every j = 1, ::: , m, T "(P;R j ) g j T "(P;R j ) g + j Id End(R n ) = O e 2k"(P;R j )k R n Kr(k\R j ) for some constantK (depending onk). Hence, there exists a constantM (depending on k) such that k P k End(R n ) m Y j=1 1 +Me 2k"(P;R j )k R n Kr(k\R j ) : The convergence of the right hand side term is guaranteed when the series P m j e 2k"(P;R j )k R n e Ar(k\R j ) is convergent. By Lemma 31, k"(P;R j )k R n = "( \ p(k PR j )) R n Ck"k r( d p(k)\ b R j ) + 1 : where b R j b U is the lift of the (punctured) triangle R j U. Since b U ! U is a 2{ cover, it is immediate that r( d p(k)\ b R j ) r(k\R j ). It then follows from Lemma 27 that the series P m j=1 e 2k"(P;R j )k R n e Kr(k\R j ) is bounded by nitely many series of the form P 1 r=0 e 2Ck"k(r+1) e Kr . As a result, ifk"k<K=2C, thenk P k End(R n ) is uniformly bounded. We now prove thatk P k End(R n ) converges asP goes toP PQ . LetP m be an increasing sequence of nite plaques converging to P PQ , with the cardinal of P m equal to m. Consider the maps Pm and P m+1 . SinceP m+1 contains one more triangleR thanP m , Pm = P P 0 and P m+1 = P T "(P;R) g R T "(P;R) g + R P 0 37 whereP m =P[P 0 . Thus, by Lemma 30, P m+1 Pm End(R n ) k P k End(R n ) T "(P;R) g R T "(P;R) g + R Id End(R n ) P 0 End(R n ) M 0 e 2k"(P;R)k R nKr(k\R) for some constant M 0 0, since k P k End(R n ) is uniformely bounded. In addition, Lemma 27 implies that lim m!1; R2Pm r(k\R) =1. Hence, ifk"k<K=2C, the sequence Pm is Cauchy, thus convergent. We conclude that P , and so ' P = P T "(P;Q) g Q , are convergent for " small enough. We also have the following corollary, which will come handy for the following. Corollary 34. Under the hypotheses of Propostion 33, there exists a constant B > 0, depending on k and ", such that ' PQ = PQ T "(P;Q) g Q with PQ = Id + O P R2P PQ e Br(k\R) . The map ' PQ carries the following properties. Proposition 35. For " small enough, for every triangles P , Q, R of e S e meeting the arc k, ' QP =' 1 PQ and ' PR =' PQ ' QR . In order to prove the above assertion, we now propose an alternative description of the map' PQ . Letk be a transverse oriented geodesic arc joing the trianglesP andQ as previously. For every integerr> 0, we denote byP r PQ the nite set of trianglesR2P PQ such thatr(k\R)r. Index the elements ofP r PQ asR 1 ,R 2 ,::: ,R m so that the index j ofR j increases as one goes fromP toQ. For everyj = 1,::: ,m, we choose a geodesic h j separating the interior of R j from the interior of R j+1 . We also chooseh 0 (resp. h m ) between P and R 1 (resp. R n and Q). Finally, we orient the h j positively according to the oriented arc k. We set ' r PQ =T "(P;R 1 ) h 0 T "(R 1 ;R 2 ) h 1 T "(R 2 ;R 3 ) h 2 T "(Rm;Q) hm : 38 In particular, Proposition 35 will be an immediate corollary of the following result. Proposition 36. Under the hypotheses of Proposition 33, for " small enough, ' r PQ is convergent as r tends to1 and lim r!1 ' r PQ =' PQ : Proof. We shall rst estimate the dierence between P r PQ and r PQ :=' r PQ T "(PQ) hm . Rewriting the expression of r PQ , we have r PQ =T "(P;R 1 ) h 0 T "(P;R 1 ) h 1 T "(P;R 2 ) h 1 T "(P;R 2 ) h 2 :::T "(P;Rm) h m1 T "(P;Rm) hm and as previously, P r PQ =T "(P;R 1 ) g 1 T "(P;R 1 ) g + 1 T "(P;R 2 ) g 2 T "(P;R 2 ) g + 2 T "(P;Rm) g m T "(P;Rm) g + m : The map r PQ is obtained from P r PQ by replacing each term T "(P;R j ) g j T "(P;R j ) g + j by T "(P;R j ) h j1 T "(P;R j ) h j . Consider the train track associated with the arc k. Observe that the two geodesics g + j and g j+1 follow the same edge path of length 2r; otherwise, there would be another R2P r PQ between R j and R j+1 . The geodesic h j being between g + j and g j+1 , it also follows the same egde path. As a result, an easy hyperbolic estimation shows that the distance between any of these two geodesics is bounded by O(e Ar ), where A 0 si a constant depending on k. In particular, for every j, the distance between g + j and h j and the distance between g j+1 and h j are both O(e Ar ). Similarly, the distance between g + j and g j+1 is a O(e Ar(k\P i ) ). Since r(k\R j ) r, it follows that the distance between h j and h j+1 39 is also a O(e Ar(k\R j ) ). Thus, following the proof of Lemma 30, the above inequalities yield, for every j, T "(P;R j ) h j T "(P;R j ) h j+1 Id End(R n ) Me 2Ck"k(r(k\R j )+1) e K 0 r(k\R j ) for some constants M 0 and K 0 (both depending on k). In particular, if is any map obtained from P r PQ by replacing some of the m termes T "(P;R j ) g j T "(P;R j ) g + j by T "(P;R j ) h j1 T "(P;R j ) h j or by the identity, it follows as in the proof of Proposition 33 that logk k End(R n ) = O 0 @ m X j=1 e 2Ck"k(r(k\R j )+1) e K 0 r(k\R j ) 1 A = O 1 X r=0 e 2Ck"k(r+1) e K 0 r ! : Consequently, wheneverk"kK 0 =2C, the norm of such is uniformely bounded. Let l be obtained from P r PQ by replacing each T "(P;R j ) g j T "(P;R j ) g + j with j l by T "(P;R j ) h j1 T "(P;R j ) h j , so that 0 = P r PQ and m = r PQ . As in the proof of Proposition 33, we estimate the dierence bewteen l1 and l . Let us rewrite l1 = T "(P;R l ) g l T "(P;R l ) g + l 0 and l = T "(P;R l ) h l1 T "(P;R l ) h l 0 , where and 0 are obtained from replacing some T "(P;R j ) g j T "(P;R j ) g + j by T "(P;R j ) h j1 T "(P;R j ) h j or the identity. As observed above,k k End(R n ) and 0 End(R n ) are uniformely bounded. It follows that k l1 l k End(R n ) k k End(R n ) 0 End(R n ) T "(P;R l ) g l T "(P;R l ) g + l T "(P;R l ) h l1 T "(P;R l ) h l End(R n ) = O e 4Ck"k(r+1)K 0 r As a result, r PQ P r PQ End(R n ) =k m 0 k End(R n ) m O e 4Ck"k(r+1)K 0 r = O re 4Ck"k(r+1)K 0 r : 40 since m = Card(P r PQ ) = O(r) by Lemma 27. Consequently, r PQ and P r PQ have the same limit as r tends to1, ifk"k<K 0 =4C. At last, observe that h m converges to g Q , which implies that both ' r PQ = r PQ T "(P;Q) hm and ' P r PQ = P r PQ T "(P;Q) g Q converge to the same limit, namely ' PQ . In each of the previous statements, " depends on the arc k. The following lemma enables us to nd a uniform ", and hence to free ourselves from this dependence. Lemma 37. For "2C twist ( b ) small enough (depending on ), for every plaques P , Q, R of e S e , the map ' P converges to a linear map ' PQ 2 SL n (R), asP tends toP PQ . In addition, ' QP =' 1 PQ and ' PR =' PQ ' QR . Proof. We can choose in the surface S nitely many geodesic arcs k 1 ,::: ,k N tranverse to , and such that each component of S meets at least one of the k i . Let P and Q be two components of e S e . There is a nite sequence of plaquesR 0 =P ,R 1 ,::: ,R N , R N+1 = Q such that each R j separates R j1 from R j+1 , and such that R j and R j+1 meets the same lift e k i j . Choose " small enough so that the convergence of the ' R j ;R j+1 is guaranteed for every j = 1, ::: , N. As a result, ' PQ = lim P!P PQ ' P exists and is equal to ' R 0 R 1 ' R 1 R 2 :::' R N R N+1 . Now, given a transverse n{twisted cocycle "2C twist ( b ) suciently small, x a tri- angle P 0 e S e , and set, for every 2 1 (S), " ( ) = 0 ( ) =' P 0 P 0 ( ): 41 We must show that 0 : 1 (S)! PSL n (R) denes a group homomorphism. By denition of the linear map' PQ , we easily see that, for every 2 1 (S) and for everyP ,Q e S e , ' P Q =( )' PQ ( ) 1 . Thus, for every , 2 1 (S), 0 ( ) = ' P 0 P 0 ( ) = ' P 0 P 0 ' P 0 P 0 ()( ) = ' P 0 P 0 ()' P 0 P 0 ( ) = 0 () 0 ( ): Note that a dierent choice of triangleP 0 e S e yields another representation 00 which is conjugate to the previous 0 . Therefore, 0 = " denes with no ambiguity a point in the character variety Rep 1 (S); PSL n (R) ; we shall refer to 0 = " as the"{cataclysm deformation of the Anosov representation along the maximal geodesic lamination . Recall that the setA(S) of Anosov representations is open in the character variety Rep 1 (S); PSL n (R) [28, 25]. We can now state the essential result of this section. Theorem38. Let be an Anosov representation. Then there exist an open neighborhood 02U C twist ( b ) and a continuous, injective, cataclysm deformation map :U ! A(S) " 7! " such that 0 =. The injectivity of the map will be established in section 4.5. 4.4 Cataclysm deformations and ag curves In this section, we analyze the eect of a cataclysm deformation on the associated equiv- ariant ag curve : @ 1 e S! Flag(R n ) of some Anosov representation . In particular, 42 we obtain a geometric description of cataclysm deformations for Anosov representations in terms of deformations of the associated ag curve. When n = 2, a cataclysm deformation " r of a Teichm uller representation r : 1 (S) ! PSL 2 (R) 2 T (S) consists of deforming the corresponding hyperbolic met- ric m r on the surface S via a shearing operation of the ideal triangles of S along the leaves of a maximal geodesic lamination . In the case of an Anosov representa- tion : 1 (S)! PSL n (R), the attentive reader may have noticed that a cataclysm deforms the representation via a deformation of its associated equivariant ag curve :@ 1 e S! Flag(R n ). We now make the latter fact more precise. Let be an Anosov representation along with its associated equivariant ag curve : @ 1 e S ! Flag(R n ), and let 0 = " be a cataclysm deformation of for some "2C twist ( b ) small enough. Fix a triangle P 0 e S e , and consider the equivariant family of linear mapsf' P 0 P g P e S e SL n (R) as inx4.3. LetV @ 1 e S be the set of vertices of the ideal triangles of e S e ; note thatV @ 1 e S is 1 (S)-invariant. For every triangle P e S e , the ag curve assigns to the three vertices (x;y;z) a ag decoration, so thatP can be seen as a triplet of ags(P ) = ((x);(y);(z)). Therefore, the set of imagesf' P 0 P ((P ))g P e S e enables to dene a ag map 0 :V ! Flag(R n ). By the equivariance property of the ag curve : @ 1 e S! Flag(R n ) and of the family f' P 0 P g P e S e , it easily follows that 0 is 0 {equivariant. LetW @ 1 e S be the set of ideal endpoints of all geodesics contained in e e S; note thatW V . We wish to extend the above ag map 0 to a ag map 0 :W ! Flag(R n ). To this end, we slightly modify the way to dene the ag map 0 :V ! Flag(R n ). Let g e be a leaf, and let k be a transverse arc to e intersecting g and joining a point in the interior of the triangle P 0 to a point in the interior of another triangle of 43 e S e . Similarly as inx4.3, for every geodesic leaf g e , letP P 0 g be the set of ideal triangles of e S e intersecting k and lying between P 0 and g. Set P =T "(P 0 ;R 1 ) g 1 T "(P 0 ;R 1 ) g + 1 T "(P 0 ;R 2 ) g 2 T "(P 0 ;R 2 ) g + 2 T "(P 0 ;Rm) g m T "(P 0 ;Rm) g + m whereP =fR 1 ;R 2 ;:::;R m gP P 0 g is a nite subset, where the index j ofR j increases as one goes fromP 0 tog, and where"(P 0 ;R) ="( \ p(k P 0 R ))2R n . Then, let us dene the linear map P 0 g 2 SL n (R) as P 0 g = lim P!P P 0 g P asP increases toP P 0 g . In the light of the arguments developed in the proves of Proposition 33 and Propo- sition 36, we easily see that there exists a transverse n{twisted cocycle "2C twist ( b ) small enough (depending on k) such that, for every leaf g e \k, the linear map P 0 g is well dened, i.e., lim P!P P 0 g P exists. On the other hand, recall that by Lemma 37, there exists "2C twist ( b ) small enough such that the linear map P 0 P (which, with our notation, is equal to P 0 g P , whereg P is the edge ofP intersecting the arck joiningP 0 to P ) is well dened for every P e S e . Therefore, it follows that for some "2C twist ( b ) small enough, the linear map P 0 g is well dened for every leaf g. In addition, as a consequence of Corollary 34 and Lemma 37, we have the following estimate: there exists some constant B > 0 such that, for every transverse arc k, for every leaf g e \k, P 0 g = O 0 @ X R2P P 0 g e Br(k\R) 1 A : (4.1) Lemma 39. The ag map 0 previously dened extends to a 0 {equivariant ag map 0 :W ! Flag(R n ). Proof. For every x2W , set 0 (x) = P 0 g ((x)) 44 whereg e is a geodesic whosex is an endpoint. We must check that the above property denes with no ambiguity a curve 0 :W ! Flag(R n ), which coincides with the one previously dened. Observe that when x 2 W V , there is an unique geodesic g e with x as an endpoint. Thus, the above procedure assigns to such an x an unique ag 0 (x)2 Flag(R n ). Suppose thatx2V , namely,g is one of two edgesg P andg + P bounding some triangle P e S e . We need to verify that P 0 g P ((x)) =' P 0 P ((x)): If g =g P , then P 0 g P ((x)) = P 0 P ((x)) = P 0 P T "(P 0 ;P) g P ((x)) =' P 0 P ((x)) since the ag (x) is xed by the linear map T "(P 0 ;P) g P . If g =g + P , P 0 ;g + P ((x)) = P 0 P T "(P 0 ;P) g P T "(P 0 ;P) g + P ((x)) = P 0 P T "(P 0 ;P) g P ((x)) = ' P 0 P ((x g )) since the ag (x) is also xed by T "(P 0 ;P) g + P . As a result, 0 :W ! Flag(R n ) is a well-dened extension. We already know that the restriction 0 jV :V ! Flag(R n ) is 0 {equivariant. Let x 2 W V be an endpoint of the leaf g e , and consider a sequence of leaves (g n ) n e converging to g, where each g n bounds some triangle R n 2 e S e . Since lim n!1 Pgn = Pg and :@ 1 e S! Flag(R n ) is continuous, we have lim n!1 Pgn ((x gn )) = Pg ((x)) 45 where (x n ) n @ 1 e S is a sequence of endpoints of (g n ) n converging to x. Therefore, the 0 {equivariance property extends to the ag map 0 :W ! Flag(R n ) by taking limits. Let 0 :@ 1 e S! Flag(R n ) be the equivariant ag curve associated with 0 = " . We can now state the main result of this section. Theorem 40. The restriction 0 jW : W ! Flag(R n ) coincides with the ag map 0 :W ! Flag(R n ). In particular, the above result conrms what the intuition suggests: the cataclysm " deforms the initial{equivariant ag curve :@ 1 e S! Flag(R n ) onto the 0 {equivariant ag curve 0 :@ 1 e S! Flag(R n ). Proof. As inx2.2.2, consider the at bundle T 1 S 0M!T 1 S associated with the rep- resentation 0 = " , where M denotes the space of n{tuplets of lines (V 1 ;V 2 ;:::;V n )2 (RP n1 ) n such thatR n =V 1 V n . By Theorem 38, the representation 0 is Anosov. Therefore, the at bundle T 1 S 0 M! T 1 S admits a unique continuous at section 0 : T 1 S! T 1 S 0 M! T 1 S, the Anosov section. The restriction 0 j b determines a continuous section of the at bundle b 0M! b . Observe that, since the geodesic lam- ination b T 1 S is compact and invariant under the action of the geodesic ow (g t ) t , the at bundle b 0M! b also inherits the Anosov property; in particular, the restriction 0 j b : b ! b 0M is the Anosov section of the at bundle b 0 Flag(R n )! b . Similarly as inx2.2.2, making use of the 0 {equivariant ag map 0 :W ! Flag(R n ), we can construct a at section 0 : b ! b 0 M of the at bundle b 0 M! b . In addition, the estimate 4.1 on the dierent mapsf P 0 g g g e SL n (R) easily shows that the at section 0 : b ! b 0M is H older continuous. Hence 0 = 0 j b . Since 0 and 0 j b determine the ag maps 0 :W ! Flag(R n ) and 0 jW :W ! Flag(R n ), respectively, we conclude that 0 = 0 jW . 46 4.5 Injectivity of cataclysms In this section, we establish some geometric properties of the cataclysms. In particular, the main goal is to relate cataclysm deformations to the length functions ` i . 4.5.1 A summation description of the shear "2C twist ( b ) Given a cataclysm deformation 0 = " , we now give another description of the shear " = (" 1 ;:::;" n )2C twist ( b ) in terms of certain summations. Fix a maximal laminationS, and letk be a transverse arc to. Given an Anosov representation , let V 1 , ::: , V n !T 1 S be the associated line bundles. Recall that the line sub-bundles V i are smooth along the leaves of geodesic foliationF of T 1 S, and are transversally H older continuous. Besides, let : @ 1 e S! Flag(R n ) be the equivariant ag curve associated with , and let e V 1 , , e V n ! T 1 e S be the lifts of the ( at) line sub-bundles V 1 , , V n !T 1 S. The ag curve and the lines e V i R n are related as follows: ife u2T 1 e S lies on the leaf g of endpoints x + g , x g , then for every i, e V i (e u) = i (x + g )\ ni+1 (x g )R n : (4.2) Now, let 0 = " 2A(S) be a cataclysm deformation of the Anosov representation , along with the corresponding equivariant familyf' P g P e S e SL n (R). Let V 0 1 , , V 0 n ! T 1 S be the line bundle associated with 0 , and let e V 0 1 , , e V 0 n ! T 1 e S be their lifts. Let k be a transverse oriented arc to e , and let us positively orient the leaves of e intersecting k (in fact, k should be seen as a transverse arc to the oriented geodesic lamination e b T 1 e S). For every component dk e , let us denote by u + d and u d the endpoints of a component d k e , and by R d e S e the ideal triangle containing the subarc d. Again, since k is oriented, the endpoints u + d and u d can be seen as points in e b T 1 e S. It follows from Theorem 40 and the above relation 4.2 that ' R d ( e V i (u d )) = e V 0 i (u d ): 47 Endow the bundle b R n ! b ( b S 0R n ! b resp.) with a Riemannian metrickk (kk 0 resp.). LetX i be a local unitary section of the (restricted) line bundle V i ! b that lifts to a section e X i of the bundle e V i ! e b . For every i = 1, ::: , n, set 0 i (k) := X dk e , d6=d log ' d e X i (u d ) 0 u d ' d e X i (u + d ) 0 u + d log ' d e X i (u + d ) 0 u + d + log ' d + e X i (u d + ) 0 u d + : where d and d + are the two components containing the positive and the negative endpoints of the oriented arc k, respectively, and where ' d denotes the linear map ' R d to alleviate the notation. Lemma 41. For " small enough, for every i = 1, ::: , n, the above series is absolutely convergent. Proof. In the light of the arguments used inx4.3, to show the (absolute) convergence of the above series, it is sucient to show that lim r!1 X dk e ;d6=d ;r(d)r log ' d e X i (u d ) 0 u d ' d e X i (u + d ) 0 u + d exists for"2C twist ( b ) small. Note that, sincek\ e is compact, the (lifted) Riemannian metrickk 0 jk\ e b is equivalent to the canonical metrickk R n onR n . Thus, it suces to prove that the following limit lim X dk e ;d6=d ;r(d)r log ' d e X i (u d ) R n ' d e X i (u + d ) R n exists. 48 The regularity of the line bundle V i ! b implies that the lifted section e X i is locally H older continuous for the canonical normkk R n. Consequently, by Lemma 28, when the radius of convergence r(d) is large enough (i.e. the subarc d is small enough), e X i (u d ) e X i (u + d ) R n Ke Ar(d) for some constants K and A > 0 (both depending on k and ). Besides, we have ' d = d T "(P 0 ;R d ) g R d , whereP 0 e S e is xed as inx4.3. Therefore, by Corollary 34 and Lemma 41, ' d e X i (u d )' d e X i (u + d ) R n k d k R n T "(P 0 ;R d ) g R d R n e X i (u d ) e X i (u + d ) R n Me Ck"k(r(d)+1) e Ar(d) for some constants M and C 0 (depending on k and ). Since there are nitely many ideal triangles in S, the above series is bounded above by nitely many series of the form X dk e ;dT log ' d e X i (u d ) R n ' d e X i (u + d ) R n where T S is some triangle intersecting k (by d T , we mean that the subarc d e S projects onto a subarc contained in T ). To establish the convergence of the latter series, we need to bound the terms log k' d e X i (u d )k R n k' d e X i (u + d )k R n from below and from above. The previous inequality shows that ' d e X i (u + d ) R n ' d e X i (x d ) R n 1 + ' d Y i (u d ) ' d e X i (u d ) R n ' d e X i (u + d ) ' d e X i (u d ) R n R n 1 + 1 ' d e X i (u d ) R n ' d e X i (u d )' d e X i (u + d ) R n 49 By Corollary 34, it is straightforward that the sequence ( d ) dT , where d = P 0 R d , is convergent in SL n (R), and hence bounded below by some constant > 0 depending on k. Thus, when r(d) is large enough, for every X2R n , k' d (X)k T "(P 0 ;R d ) g R d (X) R n Combining the above inequality with the one above, we obtain ' d e X i (u + d ) R n ' d e X i (u d ) R n 1 + 1 1 T "(P 0 ;R d ) g R d ( e X i (u d )) R n ' d e X i (u d )' d e X i (u + d ) R n 1 + M e " i (P 0 ;R d ) e Ck"k(r(d)+1) e Ar(d) 1 +M 0 e 2Ck"k(r(d)+1)Ar(d) for some constant M 0 depending on k. Therefore, for every component dT log ' d e X i (u + d ) R n ' d e X i (u d ) R n M 0 e 2Ck"k(r(d)+1)Ar(d) : (4.3) Likewise, a similar caculation leads to ' d e X i (u d ) R n ' d e X i (u + d ) R n 1 + 1 m 1 T "(P 0 ;R d ) g R d ( e X i (u + d )) R n ' d e X i (u d )' d e X i (u + d ) R n Note that T "(P 0 ;R d ) g R d ( e X i (u d ))T "(P 0 ;R d ) g R d ( e X i (u + d )) R n T "(P 0 ;R d ) g R d R n Ke Ar(d) 50 which implies T "(P 0 ;R d ) g R d ( e X i (u + d )) R n e " i (P 0 ;R d ) K 0 e Ck"k(r(d)+1) e Ar(d) e Ck"k(r(d)+1) K 0 e Ck"k(r(d)+1)Ar(d) for some K 0 depending on k. Supposek"k < A=2C, then for r(d) large enough, the right-hand side is nonnegative. Therefore ' d e X i (u d ) R n ' d e X i (u + d ) R n 1 + 1 m Me Ck"k(r(d)+1)Ar(d) e Ck"k(r(d)+1) K 0 e Ck"k(r(d)+1)Ar(d) 1 + 1 Me 2Ck"k(r(d)+1)Ar(d) 1K 0 e 2Ck"k(r(d)+1)Ar(d) 1 + 2M e 2Ck"k(r(d)+1)Ar(d) when r(d) is large enough. Finally log ' d e X i (u d ) R n ' d e X i (u + d ) R n 2M e 2Ck"k(r(d)+1)Ar(d) : (4.4) By inequalities 4.3 and 4.4, it follows that lim r!1 P dk e ;r(d)r log k' d e X i (u d )k R n k' d e X i (u + d )k R n exists, and thus, that the number 0 i (k) is well dened, wheneverk"k<A=2C. Proposition42. Letk be a transverse arc for e b . For"2C twist ( b ) small enough, then{ tuplet 0 (k) = ( 0 1 (k);:::; 0 n (k)) is equal to the n{uplet "(k) = (" 1 (k);:::;" n (k))2 R n . Note that the transversen{twisted cocycle"2C twist ( b ) is here regarded as a 1 (S){ invariant n{twisted cocycle for the lift e b . 51 Proof. Set S r = X dk e ;d6=d ;r(d)r log ' d e X i (u d ) 0 u d ' d e X i (u + d ) 0 u + d log ' d e X i (u + d ) 0 u + d + log ' d + e X i (u d + ) 0 u d + : With the same notation as inx4.3, let P and Qk e be the triangles joined by the transverse oriented arc k to e . Let m = Card(P r PQ ). Index the elements ofP r PQ as R r 1 , R r 2 ,::: ,R r m so that the indexi ofR r i increases as one goes fromP toQ. For convenience, let us also set R r 0 =P and R r m+1 =Q. Let d j =k\R r j . Then S r = m X j=1 log ' d j e X i (u j ) 0 u j ' d j e X i (u + j ) 0 u + j log ' d 0 e X i (u + 0 ) 0 u + 0 + log ' d m+1 e X i (u m+1 ) 0 u m+1 : Note that the endpoints u j of the subarcs d j depend on r, but we make the choice to alleviate the notation. Reordering the above summation, we obtain S r = m X j=0 log ' d j+1 e X i (u j+1 ) 0 u j+1 log ' d j e X i (u + j ) 0 u + j = m X j=0 log ' d j+1 e X i (u j+1 ) 0 u j+1 ' d j e X i (u + j ) 0 u + j : In the rst place, we need to prove that lim r!1 S r exists. As in the previous proof,k\ e b being compact, the (lifted) Riemannian metrickk 0 jk\ e b is equivalent to the canonical metrickk R n onR n . Therefore, it is enough to show that the series F r = m X j=0 log ' d j+1 e X i (u j+1 ) 0 R n ' d j e X i (u + j ) 0 R n 52 is convergent for "2C twist ( b ) small enough. We begin with nding an estimate for each term of the above series. According to Corollary 34, ' d j+1 ( e X i (u j+1 )) = ' d j P j P j+1 T "(P j ;P j+1 ) g P j+1 ( e X i (u j+1 )) = e " i (P j ;P j+1 ) ' d j P j P j+1 ( e X i (u j+1 )) = e " i (P j ;P j+1 ) ' d j e X i (u j+1 ) +h j ( e X i (u j+1 ) where h j is linear map such thatkh j k R n = O P RP R r j R r j+1 e Br(k\R) . Put Z j = e X i (u j+1 ) e X i (u + j ), then ' d j+1 ( e X i (u j+1 )) = e " i (R r j ;R r j+1 ) ' d j e X i (u + j ) +Z j +h j ( e X i (u j+1 ) = e " i (R r j ;R r j+1 ) ' d j e X i (u + j ) +e " i (R r j ;R r j+1 ) ' d j Z j + e " i (R r j ;R r j+1 ) ' d j h j ( e X i (u j+1 )) Consequently, ' d j+1 ( e X i (u j+1 )) ' d j e X i (u + j ) R n e " i (R r j ;R r j+1 ) 2 4 1 + ' d j R n kZ j k R n ' d j e X i (u + j ) R n + ' d j R n h j ( e X i (u + j )) R n ' d j e X i (u + j ) R n 3 5 ; and thus, log ' d j+1 ( e X i (u j+1 )) R n ' d j e X i (u + j ) R n " i (R r j ;R r j+1 ) + ' d j R n kZ j k R n ' d j e X i (u + j ) R n + ' d j R n h j ( e X i (u + j )) R n ' d j e X i (u + j ) R n : Observe that e X i (u) R n is uniformly bounded on the compact arc k. Therefore, the term ' d j R n ' d j e X i (u + j ) R n remains bounded ( ' d j e X i (u + j ) R n 6= 0 since' d j is an isomorphism). Besides, the lift e X i being locally H older continuous, forr large enough, we havekZ j k R n = e X i (u j+1 ) e X i (u + j ) R n = O(dist(u j+1 ;u + j ) ) for some 2 (0; 1]. Recall that, when 53 d k S d j (d j = k\R r j ), we have r(d) r. Thus, by Lemma 28, it follows that, for every j, O(dist(u j+1 ;u + j ) ) = O(e Ar ) for some A> 0 (depending on k). Likewise, kh j k R n = O(e Br ) for some B > 0 depending on k. As a result, log ' d j+1 ( e X i (u j+1 )) R n ' d j e X i (u + j ) R n = " i (R r j ;R r j+1 ) + O(e Ar(d) ) + O(e Br(d) ) Therefore, F r = m X j=0 log ' d j+1 ( e X i (u j+1 )) R n ' d j e X i (u + j ) R n = m X j=0 " i (R r j ;R r j+1 ) + m X j=0 O(e Ar(d) ) + m X j=0 O(e Br(d) ): Recall that, by Lemma 27;m = Card(P r PQ ) = O(r), which implies m X j=0 O(e Ar(d) )m O(e Ar(d) )! r!1 0: In ne, lim r!1 F r exists, so does lim r!1 S r = 0 i (k). Moreover, we just proved that lim r!1 F r =" i (k): At last, since the (lifted) Riemannian metrickk 0 u depends in a H older continuous way on the point u2 k\ e b , it follows easily that lim r!1 S r = lim r!1 F r , hence 0 i (k) = " i (k). As a corollary of Proposition 42, we obtain. Theorem 43. Let be an Anosov representation and letU 3 0C twist ( b ) an open neighborhood small enough. Then the cataclysm map :U ! A(S) " 7! " 54 is injective. Proof. If " and " 0 2C twist ( b ) are such that " = " 0 , then " = 0 =" 0 . 4.5.2 1{forms 0 i Let : 1 (S)! PSL n (R) be an Anosov representation. In 3.2.3, we constructed n associated 1{forms! i , fori = 1,::: ,n, dened along the leaves of the geodesic foliation F of the unit tangent bundle T 1 S. Let us brie y recall the construction. Recall that the line sub-bundles V 1 , ::: , V n !T 1 S of Labourie's decomposition of the ( at) Anosov bundle T 1 S R n ! T 1 S are invariant under the action of the lift (G t ) t of the geodesic ow (g t ) t . Endow the bundleT 1 S R n !T 1 S with a Riemannian metrickk. Given a point u 0 2 T 1 S and a vector X i (u 0 )2 V i (u 0 ), for every u2 T 1 S, put ! i (u) = d dt logkG t X i (u 0 )k gt(u 0 ) dt jt=tu where u = g tu (u 0 ) for some t u 2R. The above expression denes a 1{form ! i on T 1 S along the leaves ofF. In particular, the denition of ! i does not depend on the choices of u 0 2 T 1 S and X i (u 0 )2 V i (u 0 ). Finally, due the regularity of the line bundles V i , the 1{forms! i have the following properties: they are smooth, closed 1{forms along the leaves of the geodesic foliationF, and are transversally H older continuous. We refer the reader to [12] for further details. Given an Anosov representation and a cataclysm deformation 0 = " along a xed maximal geodesic lamination S, let 0 i = ! 0 i ! i ; the 1{form 0 i is smooth, closed along the leaves of the geodesic foliationF of T 1 S, and transversally H older continuous. As in 4.2.2, consider an open subset US containing the maximal geodesic lamination , along with its 2{cover b U b . Viewed as a subset of T 1 S, the geodesic lamination b b U inherits n smooth, closed 1{forms 0 i along its leaves, and transversally H older continuous. 55 We begin with showing that the 0 i denes a cohomology class inH 1 ( b U). We then relate that cohomology class to the homology class inH 1 ( b U) dened by to the transverse cocycle " i 2C H ( b ). The following observation is the key point to connect the 1{form 0 i to the shear cocycle " i 2C twist ( b ). Recall that the complement b U b is made of one-holed ideal hexagons b T , where each b T is the lift of an ideal (one-holed) triangle TU. Lemma 44. Let h b be a leaf bording some hexagon b T b U b . For every u2 h, choose a lift e u 2 e h T 1 e S. Likewise, let e T e S be a lift of T U, and let ' e T 2 SL n (R) be the linear map associated with the cataclysm deformation 0 = " . Consider a local unit section X i of the line bundle V i !T 1 S, that lifts to a section e X i of e V i !T 1 e S. Then 0 i (u) =d e u log ' e T e X i (e u) 0 e u where the dierential is taken along the leaf e h. Proof. The equivariance of the dierent objects occurring in the above expression guar- antees thatd e u log ' T e X i (e u) 0 e u descends to a well-dened 1{form along the leaf h b . Recall that by denition, for every u2h, 0 i (u) =! 0 i ! i (u) = d dt log e G t e X i (e u) e gt(e u) log e G 0 t e X 0 i (e u) 0 e gt(e u) dt jt=0 : 56 Note that' T ( e X i (e u))2 e V 0 j (e u), thus' T ( e X i (e u)) = e X 0 i (e u) for some constant2R. Since the connection is at and' T is linear, e G 0 t e X 0 i (e u) =' T ( e G t e X i (e u)). Therefore, for everyt, log e G t e X i (e u) e gt(e u) log e G 0 t e X 0 i (u) 0 e gt(e u) = log e G 0 t e X 0 i (e u) 0 e gt(e u) e G t e X i (e u) e gt(e u) = log jj ' e T ( e G t e X i (e u)) 0 e gt(e u) e G t e X i (e u) e gt(e u) = log ' e T 0 B @ e G t e X i (e u) e G t e X i (e u) e gt(e u) 1 C A 0 e gt(e u) logjj = log ' e T e X i (e g t (e u)) 0 e gt(e u) logjj since X i is a local unit section of the line bundle V i . Hence 0 i (u) = d dt log ' e T e X i (e g t (e u)) 0 e gt(e u) logjj jt=0 =d e u log ' e T e X i (e u) 0 e u : Consider an one-holed hexagon b T b U b , we now extend each 0 i to a piecewise smooth, exact 1{form in the interior of b T . Denote by h j b , for j = 1, ::: , 6, the 6 (oriented) geodesic edges of b T . Let alsou j 2h j be the junction points of the six adjacent horocycle arcs as shown on Figure XXX. There are six wedge regions of the hexagon b T delimited by these six adjacent horocycle arcs. Foliate each wedge region, vertically with horocycle arcs, and horizontally with geodesic rays asymptotic to the end of the wedge and stopping at the horocycle arc bounding the wedge region. Observe that the two foliations are naturally oriented: the orientation of the horizontal foliation is determined by the oriented edgesh j , and since the surface b U is oriented, the vertical foliation inherits a transverse orientation. In each wedge region, we have local coordinates (s;t) dened as follows: s is a coordinate function for the vertical foliation, such that every (oriented) 57 horocycle arc is parametrized by s2 [0; 1], and t2R is the time coordinate along each oriented geodesic rays such that every point on the horocycle bounding the wedge region has time coordinatet = 0. Note thats is constant along a geodesic ray, andt is constant along a horocycle arc. Then, in each wedge bounded by the asymptotic (oriented) edges h j and h j 0, put F i (s;t) =(1s) log ' e T e X i ( ^ g t (u j )) 0 ^ gt(u j ) s log ' e T e X i ( ^ g t (u j )) 0 ^ gt(u j ) where the coordinate s2 [0; 1] is supposed to go from h j to h j 0, and t either belongs to [0;1] or [1; 0] depending on wether the asymptotic edges h j to h j 0 are both oriented in the ingoing direction to the spike or in the outgoing direction. Since the function F i is smooth in the interior of each wedge region, thus the 1{form dF i provides a 1{form in the interior of each wedge region. In addition, it is easy to see that dF i extends the 1{form 0 i in the wedge region of b T : indeed, dF i restricted to the the tangent space along the edges h j coincides with 0 i , in other words, dF i j@ b T = 0 i . By construction, the function F i is continuous along the six main horocycles. As a result, in the interior of the nonwedge region part of b T , the function F i extends to a smooth function, which provides an extension of the 1{form 0 i to the 1{holed hexagon b T . Note that this extension is not continuous in the interior of b T , but it is smooth, exact in each wedge region and in the nonwedge region. In addition, we easily check that given any path b T , the integral R 0 i is well dened, and depends only on the endpoints of . Let k b U be a transverse oriented arc to b . Set Z k 0 i := X dk b Z d 0 i : Lemma 45. For "2C twist ( b ) is small enough, the path integral R k 0 i is well dened. 58 Proof. We must prove that the above summation is convergent. Let e k e b U be a lift of the arc k. By denition of 0 i , for every subarc dk b , d6=d , Z d 0 i = log ' e d e X i (u + e d ) 0 u + e d + log ' e d e X i (u e d ) 0 u e d where u e d and u + e d are the endpoints of the lift e d e k e b , and where ' e d SL n (R) is as inx4.5.1. Therefore, Z k 0 i = X dk b Z d 0 i = X e d e k e b , e d6= e d log ' e d e X i (u e d ) 0 u e d ' e d e X i (u + e d ) 0 u + e d + Z e d 0 i + Z e d + 0 i where the latter series is absolutely convergent by Lemma 41, provided that"2C twist ( b ) is small enough. Let F i be the function as above. Corollary 46. When " 2 C twist ( b ) is small enough, for every oriented arc k b U transverse to b , " i (k) = Z k 0 i F i (u + k ) +F i (u k ): where u k denote the endpoints of the arc k. 4.5.3 Thurston's intersection form The vector spaceC H ( b ) admits a natural symplectic form :C H ( b )C H ( b )!R, which is a version of Thurston's intersection number. Let us explain how this pairing is dened and how to calculate it. Consider an open surface b U b as inx4.2.2. Let k 1 , ::: , k m b U be a nite family of disjoint transverse arcs to the geodesic lamination b such that every leaf intersects 59 at least one k j . Thus, b S k j is made of oriented arcs which can be regrouped into nitely many parallel classes: two oriented arcs belong to the same parallel class when the negative (positive, resp.) endpoints belong to the same k j (k j 0 resp.). Collapse each k j to a point x j , and each parallel class to an oriented edge joining the endpoint x j and x j 0. We obtain an oriented graphG with weights assigned on the edges as follows: if k is a transverse arc intersecting (exactly) the leaves of a given parallel class, the corresponding edgeG is assigned with the weight (k). Given and 2C H ( b ), the pairing (;) is the self-intersection number between the two weighted oriented graphsG andG , which is dened as follows. Perturb a little the graphG onto a graphG 0 so that the latter is in transverse position with G . Then assign to each intersection point the product of the weights multiplied by +1 or1 depending on if the angle between the two oriented edges is positively or negatively oriented, and take the sum of all of these numbers (note that, by denition of transversality, the number of intersection points is nite). It is easy to see that the resulting sum does not depend onG andG 0 , which guarantees that the pairing (;) is well-dened. In fact, (;) can be related to the classical self-intersection pairing in homology. Indeed, by the additivity property of the transverse cocycle , it results that the oriented weighted graphG is a 1{cycle. Hence 2C H ( b ) denes a homology class []2H 1 ( b U): Thurston's intersection number onC H ( b ) coincides with the classical homology intersection pairing dened on H 1 ( b U) (up to a nonzero scalar multiplication). 4.5.4 Variation of length functions ` i We now relate the length functions ` i to cataclysm deformations. Let be an Anosov representation along with its associated length functions ` i : C H (S)!R. Fix a maximal lamination S, with orientation cover b . Recall that the space of transverse cocyclesC H ( b ) is in bijective correspondence with the subspace of H older geodesic currents of S which are supported in the geodesic lamination b T 1 S. We consider the restrictions ` ijC H ( b ) :C H ( b )!R, that we shall simply denote by ` i . 60 There exists a simple identity describing how the length functions ` i varies when we deform the representation via a cataclysm along the leaves of a maximal lamination. Let 0 = " be a cataclysm deformation, and let ` 0 i be its associated length functions. Recall that " = (" 1 ;. . .;" n )2C twist ( b ) is a transverse n{twisted cocycle. Theorem 47. For every transverse cocycle 2C H ( b ) for the geodesic lamination b , ` 0 i () =` i () +(;" i ): where :C H ( b )C H ( b )!R is Thurston's intersection form. Proof. Let 2C H ( b ). 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Abstract (if available)

Abstract Let S be a connected, closed, oriented surface of negative Euler characteristic, we consider the PSLn(R)-character variety Rep_{PSLn(R)}(S). An interesting connected component of the latter space is the Hitchin space H_{PSLn(R)}(S): it contains a copy of the Teichm uller space T(S), and hence is regarded as the higher rank Teichm uller space in the case of PSLn(R). In order to study the elements in the Hitchin space H_{PSLn(R)}(S), F.Labourie introduced the notion of Anosov representation. In particular, he proved that every Hitchin representation is discrete and injective, some properties already shared by Teichm uller representations. In this dissertation, we extend to Anosov representations several classic tools from hyperbolic geometry designed to study Teichm uller representations: we generalize Thurston's length function and cataclysm deformation, and analyze how these two notions relate to each other. We then explain how these techniques illuminate the geometry of Anosov representations, and provide crucial information about a new system of coordinates on the Hitchin space H_{PSLn(R)}(S).

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Asset Metadata

Creator Dreyer, Guillaume (author)

Core Title Geometric properties of Anosov representations

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Mathematics

Publication Date 07/30/2012

Defense Date 05/17/2012

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag Anosov flow,Anosov representation,cataclysm deformation,geodesic lamination,Hitchin component,Hitchin representation,Hölder geodesic current,length function,OAI-PMH Harvest,surface group representation,Thurston's intersection number

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Bonahon, Francis (committee chair), Asok, Aravind (committee member), Daeppen, Werner (committee member), Dappen, Werner (committee member), Haydn, Nicolai T. A. (committee member), Honda, Ko (committee member)

Creator Email dreyfactor@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-74253

Unique identifier UC11290088

Identifier usctheses-c3-74253 (legacy record id)

Legacy Identifier etd-DreyerGuil-1055-0.pdf

Dmrecord 74253

Document Type Dissertation

Rights Dreyer, Guillaume

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

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