Flat structure of meromorphic connections on Riemann surfaces
aa r X i v : . [ m a t h . C V ] N ov Flat structure of meromorphic connections on Riemannsurfaces
Karim Rakhimov
Abstract.
In the first part of the paper we study relation among meromor-phic k -differentials, singular flat metrics and meromorphic connections. In thesecond part we define the notion of meromorphic G -differential for a multiplica-tive group G ⊂ C ∗ and we show a relation between meromorphic G -differentialsand meromorphic connection. Moreover we prove a Poincar´e-Bendixson the-orem for infinite self-intersecting geodesics of meromorphic connections mon-odromy in G , with arg G k = { } for some k ∈ N .
1. Introduction
Meromorphic connections on Riemann surfaces have been well studied by manyauthors (see for example [ ]). A meromorphic connection on a Riemann surface S is a C − linear operator ∇ : M T S → M S ⊗ M T S , where M T S is the sheaf ofgerms of meromorphic sections of the tangent bundle
T S and M S is the space ofmeromorphic 1-forms on S , satisfying the Leibniz rule ∇ ( f s ) = d f ⊗ s + f ∇ s forall s ∈ M T S and f ∈ M S . A geodesic for a meromorphic connection ∇ is a realsmooth curve σ : I → S o , where I ⊆ R is an interval and S o is the complementof the poles of ∇ in S , satisfying the geodesic equation ∇ σ ′ σ ′ ≡
0. To the bestof our knowledge, geodesics for meromorphic connections in this sense were firstintroduced in [ ]. In [ ], the authors discovered that there is a strong relationshipbetween the local dynamics of the time-1 map of homogeneous vector fields and thedynamics of geodesics for meromorphic connections on Riemann surfaces. Hencethe study of geodesics for meromorphic connections on a Riemann surface S hassome applications to local complex dynamical systems in C n .In [ ], Abate and Tovena studied the ω -limit sets of the geodesics of meromor-phic connections on P ( C ). They gave complete classification of the ω -limit sets ofsimple geodesics. Later, in [ ], Abate and Bianchi studied the same problem forany compact Riemann surface S and they proved Poincar´e-Bendixson theorems forsimple geodesics. These works, however, left open the following question: “whathappens if a geodesic intersects itself infinitely many times?”. When S = P ( C )Abate conjectured the following Mathematics Subject Classification.
Key words and phrases.
Meromorphic connections, Quadratic differentials, Meromorphic flatsurfaces, G − differentials, Poincar´e-Bendixson theorem. Conjecture . Let σ : [0 , ε ) → S o be a maximal geodesic for a meromorphicconnection ∇ on P ( C ) , where S o = P ( C ) \ { p , p , . . . , p r } and p , p , . . . , p r arethe poles of ∇ . If σ intersects itself infinitely many times then the ω -limit set of σ is P ( C ) . In the first part of this work we study relation between meromorphic connec-tions and meromorphic k − differentials. In the second part of this work we studyinfinitely self intersecting geodesics of a class of meromorphic connections. Beforestating the main results of the paper we need to introduce a few notations anddefinitions.Let ∇ be a meromorphic connection on a Riemann surface S . Let { ( U α , z α ) } be an atlas for S . It is not difficult to see that there exists η α ∈ M S ( U α ) on U α such that ∇ (cid:16) ∂∂z α (cid:17) = η α ⊗ ∂∂z α , where ∂∂z α is the induced local generator of T S over U α . We shall say η α is the local representation of ∇ on U α . The residue Res p ∇ of ∇ at a point p ∈ S is the residue of any 1-form η α representing ∇ on a local chart( U α , z α ) at p . A pole p is said to be Fuchsian if the local representation of ∇ onone (and hence any) chart ( U, z ) around p has a simple pole at p . If all poles of ∇ are Fuchsian then we shall say ∇ is Fuchsian . Let S o be the complement of thepoles of ∇ and G a multiplicative subgroup of C ∗ . We say ∇ has monodromy in G if there exists an atlas { ( U α , z α ) } for S o such that the representations of ∇ on U α ’sare identically zero and the transition functions are of the form z β = a αβ z α + c αβ on U α ∩ U β , with a αβ ∈ G and c αβ ∈ C . We say ∇ has real periods if G ⊆ S .To state the first main result we need to introduce the notion of meromor-phic k -differentials. A meromorphic k -differential q on a Riemann surface S is ameromorphic section of the k -th power of the canonical line bundle. The zerosand the poles of q constitute the set Σ of critical points of q . It is not difficultto see that a k -differential q is given locally as q = q ( z ) dz k (for more details seeDefinition 4.1). Then there is a metric g locally given as g = | q ( z ) | k | dz | . Inparticular, g is a flat metric on S o := S \ Σ (see Proposition 4.5). Finally, a smoothcurve σ : [0 , ε ) → S o is a geodesic for q if it is a geodesic for g . When k = 1 weget the meromorphic Abelian differential which have been thoroughly studied bothfrom a geometrical and a dynamical point of view. Theory of translation surfaces(corresponding to Abelian differentials) provides new insights in the study of thedynamics of billiards through the methods of algebraic geometry and renormaliza-tion theory. When k = 2 we get the meromorphic quadratic differential which are awell studied subject in Teichm¨uller theory. Extracting the k -th root, one can thinkof an arbitrary k -differential as a multi-valued meromorphic Abelian differential on S . In general, differentials of order k > k -differentials. We say a meromorphic k -differential q and a meromorphicconnection ∇ are adapted to each other if they have the same geodesics and thesame critical points. Our first main theorem (see Theorem 4.12) describes an exactrelation between meromorphic k -differentials and meromorphic connections. Theorem . Let ∇ be a Fuchsian meromorphic connection on a Riemannsurface S . If ∇ has monodromy in Z k and residues in k Z then there is a meromor-phic k -differential q adapted to ∇ (here we identify Z k with the multiplicative groupof k -th roots of unity). Moreover, q is unique up to a non-zero constant multiple. LAT STRUCTURE OF MEROMORPHIC CONNECTIONS ON RIEMANN SURFACES 3
On the other hand, if q is a meromorphic k -differential on a Riemann surface S then there exists a unique meromorphic connection ∇ adapted to q . Moreover, ∇ is Fuchsian and it has monodromy in Z k and residues in k Z . The next main result (Theorem 6.4; see also Subsection 2.3 for the terminology)describes the possible classifications of the ω -limit sets of infinite self-intersectinggeodesics of meromorphic connections having monodromy in G , with arg G k = { } . Theorem . Let ∇ be a meromorphic connection on a compact Riemannsurface S and Σ the set of poles of ∇ . Set S o = S \ Σ . Let σ : [0 , ε ) → S bea maximal geodesic of ∇ . Assume ∇ has monodromy in G with arg G k = { } forsome k ∈ N . If σ intersects itself infinitely many times, then either (1) the ω -limit set of σ in S is given by the support of a (possibly non-simple)closed geodesic; or (2) the ω -limit set of σ in S is a graph of (possibly self-intersecting) saddleconnections; or (3) the ω -limit set of σ has non-empty interior and non-empty boundary, andeach component of its boundary is a graph of (possibly self-intersecting)saddle connections with no spikes and at least one pole; or (4) the ω -limit set of σ in S is all of S . Actually, we do not have any examples for the cases 1-3. Hence to prove Conjecture1.1 for this case we just have to show that the cases 1-3 do not appear.The paper is organized as follows. In Section 2 we repeat some preliminarynotions. Moreover, we introduce the notion of ∇ -chart and ∇ -atlas and we provesome useful properties. In Section 3, we introduce the notion of singular flat met-rics and we present general properties of singular flat metrics. Moreover, we studyrelation between singular flat metrics and meromorphic connections. In Section4, we recall some definitions and fundamental results on the theory of meromor-phic k − differentials and we describe a relation between meromorphic connectionsand meromorphic k -differentials. As a consequence we prove a conjecture proposedby Abate and Tovena in [ ] (see Corollary 4.13). In Section 5, we introduce thenotion of meromorphic G -differential and we study relation between meromorphic G -differentials and meromorphic connections. Furthermore, we introduce the no-tion of argument along geodesics of a meromorphic G -differentials. By using it wefind a subset of the set of meromorphic connections such that all geodesics of themeromorphic connections are simple. Finally, in Section 6, we introduce the notionof canonical covering for a meromorphic connection on a compact Riemann surface S and we prove Theorem 1.3. Acknowledgements.
This work is part of the author’s PhD thesis which wasprepared at the University of Pisa. The author would like to thank his advisorMarco Abate for suggesting this problem and for his patient guidance. The au-thor is currently supported by the Programme Investissement d’Avenir (I-SITEULNE /ANR-16-IDEX-0004 ULNE and LabEx CEMPI /ANR-11-LABX-0007-01)managed by the Agence Nationale de la Recherche.
2. Preliminary notions
In this section we repeat some definitions and theorems from [ ] and [ ]. KARIM RAKHIMOV
Definition . Let
T S be the tangent bundle of a Riemann surface S . Aholomorphic connection on T S is a C − linear map ∇ : T S → Ω S ⊗ T S satisfyingthe Leibniz rule ∇ ( se ) = ds ⊗ e + s ∇ e for all s ∈ O S and e ∈ T S , where T S denotes the sheaf of germs of holomorphicsections of
T S , while O S is the sheaf of germs holomorphic functions on S and Ω S is the sheaf of germs of holomorphic 1-forms on S .Let us see what this condition means in local coordinates. Let ( U α , z α ) be alocal chart for S , and ∇ a holomorphic connection on S . It is not difficult to seethat there exists a holomorphic 1 − form η α ∈ Ω S ( U α ) on U α such that ∇ ( ∂ α ) = η α ⊗ ∂ α , where ∂ α := ∂∂z α is the induced local generator of T S over U α . Definition . We say that η α is the local representation of ∇ on U α .Now, for given local representations { η α } we look for a condition which guar-antees the existence of a holomorphic connection ∇ . Let { ξ αβ } be the cocyclerepresenting the cohomology class ξ ∈ H ( S, O ∗ ) of T S (hence ξ αβ = ∂z α ∂z β ); over U α ∩ U β we have ∂ β = ∂ α ξ αβ and thus ∇ ( ∂ β ) = ∇ ( ∂ α ξ αβ ) ⇔ η β ⊗ ∂ β = ξ αβ η α ⊗ ∂ α + dξ αβ ⊗ ∂ α if and only if(2.1) η β = η α + dξ αβ ξ αβ Recalling the short exact sequence of sheaves0 → C ∗ → O ∗ ∂ log −−−→ Ω S → ∇ is equivalent to the vanishing of the image of ξ under the map ∂ log : H ( S, O ∗ ) → H ( S, Ω S ) induced on cohomology. Hence the class ξ is the image of a class ˆ ξ ∈ H ( S, C ∗ ). We recall how to find a representative ˆ ξ αβ of ˆ ξ. After shrinking the U α ’s,if necessary, we can find holomorphic functions K α ∈ O ( U α ) such that η α = ∂K α on U α . Set(2.2) ˆ ξ αβ = exp( K α )exp( K β ) ξ αβ . in U α ∩ U β . Then (2.1) implies that ˆ ξ αβ is a complex non-zero constant defining acocycle representing ξ . Definition . The homeomorphism ρ : π ( S ) → C ∗ corresponding to theclass ˆ ξ under the canonical isomorphism H ( S, C ∗ ) ∼ = Hom( H ( S, Z ) , C ∗ ) = Hom( π ( S ) , C ∗ )is the monodromy representation of the holomorphic connection ∇ . We shall saythat ∇ has monodromy in G , a multiplicative subgroup of C ∗ , if the image of ρ iscontained in G , that is if ˆ ξ is the image of a class in H ( S, G ) under the natural
LAT STRUCTURE OF MEROMORPHIC CONNECTIONS ON RIEMANN SURFACES 5 inclusion
G ֒ → C ∗ . We say G is a monodromy group of ∇ . We shall say that ∇ has real periods if it has monodromy in S . Remark . In [ ] it is introduced a period map ρ : H ( S, Z ) → C associatedto ∇ and having the following relation(2.3) ρ = exp(2 πiρ )with the monodromy representation ρ . By (2.3) we can see that ∇ has real periodsif and only if the image of the period map is contained in R . ∇− atlas. Let S be a Riemann surface and ∇ a holomorphic connectionon the tangent bundle T S . For simplicity we just say ∇ is a holomorphic connectionon S instead of T S . Let us introduce the notion of ∇ -chart. Definition . Let ∇ be a holomorphic connection on a Riemann surface S .A simply connected chart ( U α , z α ) is said to be a ∇− chart , if the representation of ∇ is identically zero on U α . A ∇− atlas is an atlas { ( U α , z α ) } for S such that allcharts are ∇− charts.A natural question is: is it always possible to find a ∇ -atlas for any Riemannsurface S with a holomorphic connection ∇ ? In the next lemma we give a positiveanswer to this question. Lemma . Let ∇ be a holomorphic connection on a Riemann surface S . If p ∈ S , then there exists a ∇ -chart around p . In particular, there exists a ∇− atlasfor S . Proof.
Let ( U α , z α ) be a simply connected chart centered at p , and η α therepresentation of ∇ on U α . Let K α be a holomorphic primitive of η α . Let J α be aholomorphic primitive of exp( K α ). Then there exists a simply connected domain˜ U α ⊆ U α such that J α : ˜ U α → J α ( ˜ U α ) is one-to-one. By setting w α := J α we definea chart ( ˜ U α , w α ). Let ˜ η α be the representation of ∇ on ˜ U α . By the transformationrule we have η α = ˜ η α + dξξ on ˜ U α , where ξ := dw α dz α = exp( K α ). Consequently, ˜ η α ≡
0. Hence ( ˜ U α , w α ) is a ∇ -chart.As we have seen above, it is always possible to find a ∇ -chart around any pointof S . Hence there exists an atlas for S such that all charts are ∇− charts. (cid:3) Let us recall the notion of Leray atlas.
Definition . Let S be a Riemann surface. A Leray atlas for S is a simplyconnected atlas { ( U α , z α ) } , such that intersection of any two charts of the atlas issimply connected or empty.Let us study relation between monodromy group and ∇ -atlas Lemma . Let G be a multiplicative subgroup of C ∗ . Let ∇ be a holomorphicconnection on a Riemann surface S such that ∇ has monodromy in G . Thenthere exists a Leray ∇− atlas { ( U α , z α ) } for S such that the atlas has the transitionfunctions z β = a αβ z α + c αβ on U α ∩ U β , where a αβ ∈ G and c αβ ∈ C . KARIM RAKHIMOV
On the other hand if ∇ is a holomorphic connection such that there exists aLeray ∇ -atlas { ( U α , z α ) } for S with the transition functions z β = a αβ z α + c αβ on U α ∩ U β , where a αβ ∈ G and c αβ ∈ C , then ∇ has monodromy in G . Proof.
Let ∇ be a holomorphic connection on S such that ∇ has monodromyin G . By Lemma 2.6 there exists a ∇− atlas for S . Take a Leray refinement ofthe ∇− atlas. So we have a Leray ∇ -atlas { ( U α , z α ) } . Since the representationsof ∇ are identically zero on U α we have ξ αβ := dz α dz β ∈ C ∗ on U α ∩ U β . Since ∇ has monodromy in G there exists constants c α ∈ C ∗ such that ξ αβ = c β c α ˆ ξ αβ with ˆ ξ αβ ∈ G . By setting w α = c α z α we define an atlas { ( U α , w α ) } for S . Hence { ( U α , w α ) } is a Leray ∇ -atlas such that dw α dw β ∈ G on U α ∩ U β . So the atlas has thetransition functions w β = a αβ w α + c αβ on U α ∩ U β , where a αβ ∈ G and c αβ ∈ C .Let now ∇ be a holomorphic connection such that there exists a Leray ∇ -atlas { ( U α , z α ) } for S with the transition functions z β = a αβ z α + c αβ on U α ∩ U β , where a αβ ∈ G and c αβ ∈ C . Since ( U α , z α ) is a ∇− chart the representation η α of ∇ on U α is identically zero. Let K α ≡ η α . Thenˆ ξ αβ = a αβ = ξ αβ in (2.2). Consequently, ∇ has monodromy in G . (cid:3) It is well known that to a Hermitian metric g on the tangent bundle over acomplex manifold M can be associated a connection ∇ (not necessarily holomor-phic) such that ∇ g ≡
0, the
Chern connection of g . The converse problem was alsostudied by Abate and Tovena for holomorphic connections: given a holomorphicconnection ∇ , does there exist a Hermitian metric g so that ∇ g ≡ Definition . Let ∇ be a holomorphic connection on a Riemann surface S .We say that a Hermitian metric g on T S is adapted to ∇ if ∇ g ≡
0, that is if X ( g ( R, T )) = g ( ∇ X R, T ) + g ( R, ∇ X T )and X ( g ( R, T )) = g ( ∇ X R, T ) + g ( R, ∇ X T )for all sections R , T of T S , and all vector fields X on S .As usual, let us check the condition in local coordinates. Let { ( U α , z α ) } be anatlas for S . A Hermitian metric g on T S is locally represented by a positive C ∞ function n α ∈ C ∞ ( U α , R + ) given by n α = g ( ∂ α , ∂ α ) . Then we can see that ∇ g ≡ U α if and only if(2.4) ∂n α = n α η α , where η α is the holomorphic 1-form representing ∇ . Proposition , Proposition 1.1]) . Let ∇ be a holomorphic connectionon a Riemann surface S . Let ( U α , z α ) be a local chart, and define η α ∈ Ω S ( U α ) bysetting ∇ ∂ α = η α ⊗ ∂ α . Assume that we have a holomorphic primitive K α of η α on U α . Then (2.5) n α = exp(2Re K α ) is a positive solution of (2.4) . Conversely, if n α is a positive solution of (2.4) thenfor any z ∈ U α and any simply connected neighborhood U ⊆ U α of z there is aholomorphic primitive K α ∈ O ( U ) of η α over U such that n α = exp(2Re K α ) in LAT STRUCTURE OF MEROMORPHIC CONNECTIONS ON RIEMANN SURFACES 7 U . Furthermore, K α is unique up to a purely imaginary additive constant. Finally,two (local) solutions of (2.4) differ (locally) by a positive multiplicative constant. It is not difficult to see that Gaussian curvature of the local metrics (2.5) isidentically zero. The proposition shows that for any holomorphic connection ∇ wecan always associate local flat metrics g adapted to ∇ . A global metric adapted to ∇ might not exist. Theorem , Proposition 1.2]) . Let ∇ be a holomorphic connection ona Riemann surface S . Then there exists a flat metric adapted to ∇ if and only if ∇ has real periods. Definition . A geodesic for a holomorphic connection ∇ on S is a realcurve σ : I → S , with I ⊆ R an interval, such that ∇ σ ′ σ ′ ≡
0, where ∇ u s := ∇ s ( u ).Let { ( U α , z α ) } be an atlas for S and σ : I → U α , with I ⊆ R an interval, asmooth curve. Then σ is a geodesic for a meromorphic connection ∇ if and only if(2.6) ( z α ◦ σ ) ′′ + ( f α ◦ σ )( z α ◦ σ ) ′ ≡ η α = f α dz α is the local representation of ∇ on U α . Proposition . Let ∇ be a holomorphic connection on a Riemann surface S . Let ( U α , z α ) be a ∇ -chart. Let σ : [0 , ε ) → U α be a smooth curve. Then σ isa geodesic for ∇ if and only if the representation z α ◦ σ is a Euclidean segment in z ( U α ) . Proof.
Since ( U α , z α ) is a ∇ -chart the local representation of ∇ on U α isidentically zero. Consequently, by (2.6) we can see that σ is a geodesic if and onlyif ( z α ◦ σ ) ′′ ≡ . Hence z α ◦ σ is a Euclidean segment. (cid:3) We will now define meromorphic connec-tion
Definition . A meromorphic connection on the tangent bundle T S of aRiemann surface S is a C -linear map ∇ : M T S → M S ⊗M T S satisfying the Leibnizrule ∇ ( ˜ f ˜ s ) = d ˜ f ⊗ ˜ s + ˜ f ∇ ˜ s for all ˜ s ∈ M T S and ˜ f ∈ M S , where M T S denotes the sheaf of germs of meromor-phic sections of
T S , while M S is the sheaf of germs of meromorphic functions and M S is the sheaf of meromorphic 1-forms on S .Let ( U α , z α ) be a local chart for S , and ∇ a meromorphic connection on S .Then there exists η α ∈ M S ( U α ), such that ∇ ( ∂ α ) = η α ⊗ ∂ α , where ∂ α := ∂∂z α is the induced local generator of T S over U α . Similarly as forholomorphic connection we have(2.7) η β = η α + 1 ξ αβ ∂ξ αβ KARIM RAKHIMOV on U α ∩ U β , where ξ αβ := ∂z α ∂z β . In particular, if all the representations are holomor-phic then ∇ is a holomorphic connection. We say p ∈ S is a pole for a meromorphicconnection ∇ if p is a pole of η α for some (and hence any) local chart U α at p . IfΣ is the set of poles of ∇ , then ∇ is a holomorphic connection on S o = S \ Σ. Sowe can define notion of geodesic for ∇ on S o as in holomorphic connection. Definition . A geodesic for a meromorphic connection ∇ on T S is a realcurve σ : I → S o , with I ⊆ R an interval, such that ∇ σ ′ σ ′ ≡ . Definition . The residue
Res p ∇ of a meromorphic connection ∇ at a point p ∈ S is the residue of any 1-form η α representing ∇ on a local chart ( U α , z α ) at p .The set of all residues of ∇ is denoted by Res ∇ , i.e., Res ∇ := { Res p ∇ : p ∈ S }\{ } .By condition (2.7) the residue of ∇ does not depend on the choice of charts. Definition . We say that p ∈ S is a Fuchsian pole of a meromorphicconnection ∇ if there exists a (and hence any) chart ( U α , z α ) around p such thatthe representation of ∇ has a simple pole at p . If all poles of ∇ are Fuchsian thenwe say ∇ is a Fuchsian meromorphic connection . In this subsection we recall Poincar´e-Bendixson theorems for meromorphic connections on compact Riemann surfaces,i.e., a classification of the possible ω -limit sets for the geodesics of meromorphicconnections on compact Riemann surfaces. Definition . Let σ : ( ε − , ε + ) → S be a curve in a Riemann surface S .Then the ω -limit set of σ is given by the points p ∈ S such that there exists asequence { t n } , with t n ր ε + , such that σ ( t n ) → p . Similarly, the α -limit set of σ is given by the points p ∈ S such that there exists a sequence { t n } , with t n ց ε − ,such that σ ( t n ) → p . Definition . A geodesic σ : [0 , l ] → S is closed if σ ( l ) = σ (0) and σ ′ ( l ) isa positive multiple of σ ′ (0) ; it is periodic if σ ( l ) = σ (0) and σ ′ ( l ) = σ ′ (0). Definition . A saddle connection for a meromorphic connection ∇ on S is a maximal geodesic σ : ( ε − , ε + ) → S \ { p , ..., p r } (with ε − ∈ [ −∞ ,
0) and ε + ∈ (0 , + ∞ ]) such that σ ( t ) tends to a pole of ∇ both when t ↑ ε + and when t ↓ ε − .A graph of saddle connections is a connected graph in S whose vertices are polesand whose arcs are disjoint saddle connections. A spike is a saddle connection of agraph which does not belong to any cycle of the graph.A boundary graph of saddle connections is a graph of saddle connections whichis also the boundary of a connected open subset of S . A boundary graph is discon-necting if its complement in S is not connected.Next we state the Poincar´e-Bendixson theorem for the meromorphic connec-tions on any compact Riemann surface S which proved in [ , Theorem 4.6] and [ ,Theorem 4.3]. Theorem . Let σ : [0 , ε ) → S o be a maximalgeodesic for a meromorphic connection ∇ on S , where S o = S \ { p , p , . . . , p r } and p , p , . . . , p r are the poles of ∇ . Then either (1) σ ( t ) tends to a pole of ∇ as t → ε ; or (2) σ is closed; or LAT STRUCTURE OF MEROMORPHIC CONNECTIONS ON RIEMANN SURFACES 9 (3) the ω -limit set of σ in S is given by the support of a closed geodesic; or (4) the ω -limit set of σ in S is a boundary graph of saddle connections; or (5) the ω -limit set of σ in S is all of S ; or (6) the ω -limit set of σ has non-empty interior and non-empty boundary, andeach component of its boundary is a graph of saddle connections with nospikes and at least one pole; or (7) σ intersects itself infinitely many times.Furthermore, in cases 2 or 3 the support of σ is contained in only one of thecomponents of the complement of the ω -limit set, which is a part P of S having the ω -limit set as boundary. Definition . Let ∇ be a meromorphic connecton on a Riemann surface S and p a Fuchsian pole for ∇ . A Fuchsian pole p is said to be resonant if − − Res p ∇ ∈ N ∗ , non-resonant othervise.To study the ω -limit set of a geodesic of a meromorphic connection ∇ on aRiemann surface it is useful to know local behavior of geodesics around a pole p .The next theorem gives such an understanding around the non-resonant Fuchsianpoles. Theorem , Proposition 8.4]) . Let ∇ be a meromorphic connection ona Riemann surface S . Let p be a Fuchsian pole for ∇ . Let ρ := Res p ∇ . Suppose − − ρ / ∈ N ∗ . Then there is a neighborhood U of p such that: (1) if Re ρ < − then every geodesic ray which enters into U tends to p . Thetwo rays of any geodesic which stays in U tend to p ; (2) if Re ρ > − then all geodesics but one issuing from any point p ∈ U \ { p } escapes U ; (3) if Re ρ = − but ρ = − then the geodesics not escaping U are eitherclosed or accumulate the support of a closed geodesic in U ; (4) if ρ = − then any maximal geodesic σ : I → U \ { p } , maximal in bothforward and backward time, is either periodic or escapes U in one ray andtends to p in another ray. In [ ] the resonant case − − ρ ∈ N ∗ is also studied and the following conjectureis advanced Conjecture ]) . Let ∇ be a meromorphic connection on a Riemannsurface S . Let p be a Fuchsian pole for ∇ . Let ρ := Res p ∇ . Suppose − − ρ ∈ N ∗ .Then there is a neighborhood U of p such that every geodesic ray which enters into U tends to p . The two rays of any geodesic which stays in U tend to p . We shall prove this conjecture in Corollary 4.13.
3. Singular flat metrics
In this section we define the notion of singular flat metric and we study someof its properties.
Definition . Let S be a Riemann surface and Σ = { p , ..., p r } a finite set.Set S o := S \ Σ. We say that g is a singular flat metric on S , if g is a flat metricon S o and for any p ∈ Σ there exist c p , b p ∈ R with b p > U α , z α ) is a chart centered p , with U α ∩ Σ = { p } , then the flat metric g = e u α | dz α | on U α \ { p } satisfies lim z α → e u α | z α | c p = b p where u α : U α \ { } → R is a harmonic function. We say that c p is the residue of g at the critical point p and Σ is the singular set of g . Remark . The residue c p does not depend on the chosen chart. Let ( U β , z β )be another chart centered p and g = e u β | dz β | on U α \ { p } for some harmonicfunction u β : U α \ { p } → R . Then we have e u β = e u α | ξ αβ | , where ξ αβ = dz α dz β . Consequently,lim z β → e u β | z β | c p = lim z β → e u α | z β | c p | ξ αβ | = lim z β → e u α | z α | c p | z α | c p | z β | c p | ξ αβ | = b p | ξ αβ ( p ) | c p +1 . Since ξ αβ ( p ) = 0, we conclude that c p does not depend on the chosen chart.If the residue of a critical point p is c p = 0, then it is a removable critical point,i.e., the flat metric g extends (remaining flat) to the critical point. The interestingcase is when c p = 0. Lemma . If g is a singular flat metric on a Riemann surface S and p acritical point with residue c p then for any simply connected chart ( U, z ) centered at p with U ∩ Σ = { p } there exists a holomorphic function F : U → C such that theflat metric is given by g = | z | c p | e F ( z ) dz | on U \ { p } . Proof.
By definition of meromorphic flat surface there exists b p > z → e u | z | c p = b p where e u is the representation of g on U \{ p } . Let define a function v : U \{ p } → R as follows v ( z ) := u ( z ) − c p log | z | . The function is harmonic on U \ { p } andlim z → v ( z ) = log b p which means v extends harmonically to U . Since U is simply connected, thereexists a holomorphic function F on U such that Re F = v. Then we have g = e v + c p log | z | | dz | = | z | c p | e F dz | as claimed. (cid:3) LAT STRUCTURE OF MEROMORPHIC CONNECTIONS ON RIEMANN SURFACES 11 C . Let us study singular flat metrics on asimply connected domain D ⊂ C . Proposition . Let D ⊆ C be a simply connected domain. Let g be a singularflat metric on D . Let Σ := { z , ..., z n } ⊂ D be the set of singularities of g and ρ j the residue of g at z j for j = 1 , ..., n . Then there exists a holomorphic function F : D → C such that g = | z − z | ρ · | z − z | ρ ... · | z − z n | ρ n exp(Re F ) | dz | . Proof.
Set D o = D \ Σ. Then there is a harmonic function u : D o → R suchthat g = exp( u ) | dz | . Set v = u − ρ log | z − z | − ρ log | z − z | − ... − ρ n log | z − z n | It is easy to see that v is harmonic on D o . Similarly as Lemma 3.3 we can showthat v has harmonic continuation to any critical point z j , for j = 1 , .., n. Hence v isa harmonic function on D . Since D is simply connected, there exists a holomorphicfunction F : D → C such that Re F = v. Consequently, g = | z − z | ρ · | z − z | ρ ... · | z − z n | ρ n exp(Re F ) | dz | . (cid:3) Definition . Let g = | z − z | ρ · | z − z | ρ ... · | z − z n | ρ n exp(Re F ) | dz | = u ( z ) | dz | be a singular flat metric on a simply connected domain D ⊂ C , where F : D → C is a holomorphic function and ρ , ρ , ..., ρ n the residues of g at the singularities z , ..., z n respectively. We say u is the representation of the singular flat metric g on D .Some properties of singular flat metrics of the form | f | λ | dz | on a simply con-nected domain D ⊆ C are described in [ ], where λ > f : D → C is aholomorphic function. Let ∇ be a holomorphic connection on a Riemann surface S . As wehave seen in Theorem 2.11 there exists a flat (Proposition 2.10) metric g adaptedto ∇ if and only if ∇ has real periods. Note that if g is adapted to ∇ then theyhave the same geodesics. Definition . Let ∇ be a meromorphic connection on a Riemann surface S and let Σ denote the set of poles of ∇ . We say that a singular flat metric g on S is adapted to ∇ if it is adapted to ∇ on S o := S \ Σ and Σ is the set of critical pointsof g . Definition . Let S be a Riemann surface and Σ ⊂ S a discrete set nothaving limit points in S . Set S o = S \ Σ. A
Leray atlas adapted to ( S o , Σ) is aLeray atlas { ( U α , z α ) } ∪ { ( U k , z k ) } of S such that { ( U α , z α ) } is a Leray atlas for S o , each ( U k , z k ) is a simply connected chart centered at p k ∈ Σ and U k ∩ U h = ∅ if k = h . Let us state an analogue of Theorem 2.11
Theorem . Let ∇ be a Fuchsian meromorphic connection on a Riemannsurface S , and Σ the set of poles of ∇ . Set S o = S \ Σ . If ∇ has real periods on S o and Res ∇ ⊂ R then there exists a singular flat metric g adapted to ∇ . Moreover, g is unique up to a positive constant multiple.Conversely, if g is a singular flat metric on S with singular set Σ then thereexists a unique meromorphic connection ∇ with Σ as set of poles such that g isadapted to ∇ . Moreover, ∇ is Fuchsian, it has real periods on S o and Res ∇ ⊂ R .Furthermore, if c p is the residue of a critical point p of g then Res p ∇ = c p andvice versa. Proof.
Let ∇ be a Fuchsian meromorphic connection on a Riemann surface S with real periods and Res ∇ ⊂ R . Let Σ be the set of poles of ∇ . Set S \ Σ. ByTheorem 2.11 there exists a flat metric g adapted to ∇ on S o . Let { ( U α , z α ) } ∪{ ( U k , z k ) } be a Leray atlas adapted to ( S o , Σ). Let η α be the representation of ∇ on U α . By Proposition 2.10 g is defined(3.1) g α = exp(Re F α ) | dz α | on U α for a suitable holomorphic primitive F α of η α . Let p k be a pole of ∇ . Let( U k , z k ) be the chart centered at p . Let η k = (cid:18) c k z k + f k (cid:19) dz k be the representation of ∇ on U k , where f k : U k → C is a holomorphic functionand c k := Res p k ∇ . Let V ⊂ U k \ { p k } be a simply connected open set. Then byProposition 2.10 for a suitable holomorphic primitive K k of η k on V we have g = exp(Re K k ) | dz k | . Then K k = c k log z k + F k for a holomorphic primitive F k of f k on V . Hence g = | z k | c k | e F k dz k | on V . Since F k is a holomorphic primitive of f k on V and f k is a holomorphicfunction on U k there exists a holomorphic primitive ˜ F k of f k on U k such that˜ F k | V = F k . Consequently, we have g = | z k | c k | e ˜ F k dz k | on U k . Hence lim z k → | z k | c k | e ˜ F k || z k | c k = e ˜ F (0) . By definition of singular flat metric we can see that p k is a singular point for g withresidue c k . Hence g has a continuation as a singular flat metric to any pole of ∇ .Since g is adapted to ∇ on S o it is adapted to ∇ on S .Let ˜ g be another singular flat metric adapted to ∇ . Let η α be the representationof ∇ on ( U α , z α ). For suitable holomorphic primitives F α and ˜ F α of η α we have g α = exp(Re F α ) | dz α | and ˜ g α = exp(Re ˜ F α ) | dz α | LAT STRUCTURE OF MEROMORPHIC CONNECTIONS ON RIEMANN SURFACES 13 on U α . Since F α and ˜ F α holomorphic primitives of η α there exists C α ∈ C suchthat F α = ˜ F α + C α . Hence g α = | e C α | ˜ g α on U α . Let ( U β , z β ) be a chart with U α ∩ U β = ∅ . Then wehave g α = g β on U α ∩ U β , and it is equivalent to (cid:12)(cid:12) e C α (cid:12)(cid:12) ˜ g α = (cid:12)(cid:12) e C β (cid:12)(cid:12) ˜ g β . Hence (cid:12)(cid:12) e C α (cid:12)(cid:12) = (cid:12)(cid:12) e C β (cid:12)(cid:12) = r for some r >
0. Consequently, we have g = r ˜ g . Hence g is unique up to positive constant multiple.On the other hand, let g be a singular flat metric on S and Σ its singular set.Set S o = S \ Σ. Let { ( U α , z α ) } ∪ { ( U k , z k ) } be a Leray atlas adapted to ( S o , Σ).Since g is flat on S o for each α there exists a harmonic function u α on U α such that g α = e u α | dz α | on U α . Furthermore, one has(3.2) e u α − u β = | ξ αβ | on U α ∩ U β , where ξ αβ = dz α dz β . Since U α is simply connected, there exists dualharmonic functions v α for u α on U α such that f α = u α + iv α defines a holomorphicfunction on U α . Set(3.3) η α := df α on U α ; we claim that the η α ’s are representatives of a holomorphic connection ∇ on S o . By (3.2) we have u α − u β = log | ξ αβ | . Since ξ αβ : U α ∩ U β → C is a nonzeroholomorphic function then F := f α − f β − log ξ αβ is a holomorphic function on U α ∩ U β with Re F = 0. Hence there exists C ∈ R such that F = iC , i.e., f α − f β = log ξ αβ + iC . After differentiating the last equality we have η α − η β = dξ αβ ξ αβ which is (2.1). We have defined a holomorphic connection ∇ on S o . It is notdifficult to see that g is adapted to ∇ on S o . By Theorem 2.11 it follows that ∇ has real periods on S o .Let p k be a critical point for g with residue c k and ( U k , z k ) the chart of theatlas centered at p . Then by Lemma 3.3 there exists a holomorphic function F k on U k such that the singular flat metric is g = | z k | c k (cid:12)(cid:12) e F k dz k (cid:12)(cid:12) on U k . Set η k := (cid:18) c k z k + F ′ k (cid:19) dz k on U k . We claim that we can extend ∇ to a meromorphic connection representedby η k on U k . We have to check that this definition satisfies condition (2.1). Let( U α , z α ) be a chart with U α ∩ U k = ∅ . Then there exists a holomorphic map F α : U α → C such that g = | e F α dz α | . Let V ⊆ U α ∩ U k be a simply connectedopen set. Then by definition of flat metric we have | z k | c k | e F k | = | e F α ξ αk | on V , where ξ αk := dz α dz k . This is equivalent to c k log | z k | + Re F k = Re F α + log | ξ αk | . Since V is simply connected it is not difficult to see that there exists a constant C ∈ C such that c k log z k + F β − F k − log ξ αk = C on V . Consequently,(3.4) η k = η α + dξ αk ξ αk on V . Since η k , η α and ξ αk are well defined on U α ∩ U k and the equality (3.4) holdson any simply connected subset of U α ∩ U k then (3.4) holds on U α ∩ U k . Hencewe have extended ∇ to a Fuchsian meromorphic connection on S . It is easy to seethat, if p k is a critical point of g with residue c k then Res p k ∇ = c k . Let ˜ ∇ be another meromorphic connection adapted to g . Let η α and ˜ η α be therepresentations of ∇ and ˜ ∇ on a chart ( U α , z α ) respectively. Then by (3.3) thereexists holomorphic primitives F α and ˜ F α of η α and ˜ η α respectively we have g = | e F α dz α | = | e ˜ F α dz α | on U α . Hence there exists a constant C ∈ C such that F α ≡ ˜ F α + C . Consequently, η α ≡ ˜ η α . Hence ∇ = ˜ ∇ on S o . Let now ( U k , z k ) be a chart around a pole p k . Let η k and ˜ η k be the representations of ∇ and ˜ ∇ on U k respectively. Since ∇ = ˜ ∇ on S o we have η k ≡ ˜ η k on U k \ { p k } . Since η k has a unique extension to U k we have η k ≡ ˜ η k on U k . Thus ∇ = ˜ ∇ on S . Hence ∇ is the unique meromorphic connectionadapted to g . (cid:3)
4. Meromorphic k − differentials The main purpose of this section is to describe the relation between meromor-phic k -differentials and meromorphic connections. Meromorphic k -differentials arestudied by many authors (see for example [
3, 8, 10, 11 ]). In this section we recallsome results on the theory of meromorphic k -differentials. Definition . Let k ∈ N . Let { ( U α , z α ) } be a holomorphic atlas for a Rie-mann surface S . A meromorphic k -differential q on S is a set of meromorphicfunction elements q α defined in local charts ( U α , z α ) for which the following trans-formation law holds:(4.1) q β ( p ) = (cid:18) dz α dz β ( p ) (cid:19) k q α ( p ) , p ∈ U α ∩ U β Globally, a k − differential is a global meromorphic section of the line bundle( T ∗ S ) ⊗ k .It makes no sense to speak about the value of a k − differential q at a point p ∈ S , since it depends on the local chart near p , but we can speak of its zeros andpoles. Definition . Let q be a meromorphic k − differential on a Riemann surface S and p a zero (pole) of q . The order of p is the order of z α ( p ) for any q α representing q on a local chart ( U α , z α ) around p .Indeed, dz α dz β is a never vanishing holomorphic function on U α ∩ U β , and hence(4.1) yields that q α and q β have the same order of zero (respectively, pole) at p . LAT STRUCTURE OF MEROMORPHIC CONNECTIONS ON RIEMANN SURFACES 15
Definition . The critical points of a meromorphic k − differential q on aRiemann surface S are its zeroes and poles. All other points of S are regular points of q . A finite critical point of q is either a zero or a pole of order one. Other criticalpoints will be called infinite critical points . If q has no poles then we say that q isa holomorphic k − differential. If all points are regular points for q then we say that q is a regular k − differential. Proposition , Theorem 5.1]) . Let q be a regular k − differential on aRiemann surface S . Then there exists an atlas { ( U α , z α ) } for S such that the rep-resentation of q is identically one on each U α . The atlas have transition functions z β = a αβ z α + c αβ on U α ∩ U β , where a αβ ∈ Z k := { ε ∈ S : ε k = 1 } . The next proposition shows a relation between singular flat metrics and k -differentials. Proposition ]) . Let q be a meromorphic k -differentialon a Riemann surface S and { ( U α , z α ) } an atlas for S . Then there exists a singularflat metric g on S locally given by (4.2) g = | q α | k | dz α | on U α , where q α is the local representation of q on U α . Definition . Let q be a meromorphic k -differential on a Riemann surface S and g is the singular flat metric defined as in (4.2). Set S o := S \ Σ, where Σ isthe set of critical points of q . A smooth curve σ : [0 , ε ) → S o is a geodesic for q ifit is a geodesic for g . Definition . Let q be a meromorphic k -differential, g a singular flat metric.We say q and g are adapted to each other if they have the same geodesics. When k = 2, meromorphic k − differentials arecalled meromorphic quadratic differentials . Meromorphic quadratic differentials area well studied subject in Teichm¨uller theory. Studying quadratic differentials is away to understand interval exchange transformations (see for example [
5, 9, 12,13 ]). Let us begin recalling a few standard facts about meromorphic quadraticdifferentials on a Riemann surface S (see, e.g., [ ]).The next theorem describes a local behavior of geodesics of a quadratic differ-ential q around its critical points. Theorem , Chapter 3]) . Let q be a meromorphic quadratic differentialon a Riemann surface S . Let p be a critical point for q of order k . Then there isa neighborhood U of p such that: (1) if k < − then every geodesic ray which enters into U tends to p . Thetwo rays of any geodesic which stays in U tend to p ; (2) if k ≥ − then all geodesics but one issuing from any point p ∈ U \ { p } escape U ; (3) if k = − then any maximal geodesic σ : I → U \ { p } , maximal in bothforward and backward time, is either periodic or escapes U in one ray andtends to p in the other ray. There are some differences between meromorphic quadratic differentials andmeromorphic k -differentials for k >
2. For instance, for any k > k -differential having self-intersecting geodesic, but there are no self-intersecting geodesics for quadratic differentials except periodic geodesics. Proposition , Theorem 5.5]) . Let q be a meromorphic quadratic dif-ferential on a Riemann surface S . Then there are no self-intersecting geodesics of q except periodic geodesics. k -differentials. In this section we study relations between meromorphic quadraticdifferentials and meromorphic connections on a Riemann surface S . Definition . Let ∇ be a meromorpic connection and q a meromorphic k -differential on a Riemann surface S . We say that ∇ and q are adapted to eachother if there exists a singular flat metric g such that g is adapted to ∇ and q .First of all, we describe the relation between regular quadratic differentials andholomorphic connections on a Riemann surface S . Theorem . Let q be a regular k − differential on a Riemann surface S .Then there exists a unique holomorphic connection ∇ on S with monodromy in Z k such that ∇ is adapted to q , where Z k is the multiplicative group of k -th roots ofunity.On the other hand, if ∇ is a holomorphic connection with monodromy in Z k then there exists a regular k − differential q adapted to ∇ . Moreover, k − differentialis unique up to a non-zero constant multiple. Proof.
Let q be a regular quadratic differential on S and g the flat metricadapted to q . Due to Theorem 3.8 there exists unique holomorphic connection ∇ on S such that g is adapted to ∇ . Hence q is adapted to ∇ . It is also possible towrite the relation between the local representations of ∇ and q . Let { ( U α , z α ) } bean atlas for S . By Proposition 4.5 we have g = | q α | k | dz α | on U α , where q α is the representation of q on U α . Then by (3.3) we can see thatthe local representation of ∇ is given by(4.3) η α = dq α kq α on U α , where q α is the representation of q on U α .Thanks to Proposition 4.4 there exists an atlas { ( U α , z α ) } such that the repre-sentations of q are identically 1 and the transition functions are given by(4.4) z β = a αβ z α + c αβ on U α ∩ U β , where a αβ ∈ Z k and c αβ is a complex number. By (4.3) we can see thatthe representations η α of ∇ on those charts are identically zero. Hence { ( U α , z α ) } is a ∇ -atlas. Note that transition functions are given by (4.4), i.e., ξ αβ = dz α dz β ∈ Z k .Consequently, by Lemma 2.8 ∇ has monodromy in Z k .On the other hand, let now ∇ be a holomorphic connection with monodromyin Z k . Then we can find an atlas { ( U α , z α ) } for S , holomorphic primitives K α of η α , numbers ˜ ξ αβ ∈ Z k and constants c α ∈ C ∗ such that setting ξ αβ = dz α dz β we have e K α e K β ξ αβ = c β c α ˜ ξ αβ . Hence we have(4.5) c kβ e kK β = c α e kK α ξ αβ ; LAT STRUCTURE OF MEROMORPHIC CONNECTIONS ON RIEMANN SURFACES 17 set(4.6) q α = c kα e kK α on U α . Take any charts U α and U β such that U α ∩ U β = ∅ . Then (4.5) implies q β = q α (cid:18) dz α dz β (cid:19) k on U α ∩ U β . Hence (4.6) defines a meromorphic k − differential q . It is not difficultto see that q is adapted to ∇ .Let q and q be meromorphic k − differentials adapted to ∇ . Let g and g be the singular flat metrics adapted to q and q respectively. Then g and g areadapted to ∇ . By Theorem 3.8 there exists a constant c > g = cg .Hence q is unique up to a non-zero constant multiple. (cid:3) Let us state one of the main results of this section
Theorem . Let ∇ be a Fuchsian meromorphic connection on a Riemannsurface S . If ∇ has monodromy in Z k and residues in k Z then there is a meromor-phic k -differential q adapted to ∇ . Moreover, q is unique up to a non-zero constantmultiple.On the other hand, if q is a meromorphic k -differential on a Riemann surface S then there exists a unique meromorphic connection ∇ adapted to q . Moreover, ∇ is Fuchsian and it has monodromy in Z k and residues in k Z . Proof.
Let q be a meromorphic k − differential on S and Σ its singular set.By the previous theorem there exists a holomorphic connection ∇ on S o := S \ Σwhich has monodromy in Z k such that q is adapted to ∇ .Let { ( U α , z α ) } ∪ { ( U h , z h ) } be a Leray atlas adapted to ( S o , Σ). Let p be acritical point of the k − differential q with order m ∈ Z and ( U h , z h ) the chart of theatlas centered at p . Since the order of the critical point p is m up to shrinking U h there exists a never vanishing holomorphic function H h : U h → C such that q h = z mh H h where q h is the representation of q on U h . Let η h be the representation of ∇ on U h \ { p } . By (4.3) we have(4.7) η h = mz m − h H h + z mh H ′ h kz mh H h dz h = (cid:18) mkz h + H ′ h kH h (cid:19) dz h . Hence η h has a meromorphic extention to p with the residue mk at p . Let ( U α , z α )be a chart of S o such that U α ∩ U h = ∅ . Indeed, the condition (2.7) holds on U α ∩ U h because ∇ is a holomorphic connection on S o . By extending local representationsof the holomorphic connection ∇ we can extend ∇ to whole surface S as a mero-morphic connection ˜ ∇ . By (4.7) we can see that resulting meromorphic connection˜ ∇ is Fuchsian with residues in k Z . Since the extension is unique ˜ ∇ is a uniquemeromorphic connection adapted to q .Now, suppose ∇ is a Fuchsian meromorphic connection with monodromy in Z k and residues in k Z . Let Σ be the set of poles of ∇ . Set S o = S \ Σ. By the previoustheorem there exists a regular k − differential q on S o adapted to ∇ . Let { ( U α , z α ) } ∪ { ( U h , z h ) } be a Leray atlas adapted to ( S o , Σ). Let p be a polefor ∇ and Res p ∇ = mk for some m ∈ Z . Let ( U h , z h ) be the chart centered at p .Then the local representation η h on U α has the form η h = (cid:18) mkz h + f h (cid:19) dz h , where f h : U h → C is a holomorphic function. Let q h be the representation of q on U h \ { p } and V α ⊂ U h \ { p } a simply connected open set. By (4.6) we have q h | V α = e m log z h + kF α ( z h ) = z mh e kF α ( z h ) , where F α is a suitable holomorphic primitive of f h on V α . Since U h is simplyconnected and f h is a holomorphic function on U h we can see that F α has uniqueholomorphic extension to U h , i.e, there exists a holomorphic primitive F h of f h on U h such that F h | V α = F α . Consequently, z mh e kF α has a unique holomorphicextension to U h \ { p } and we have q h = z mh e kF h . It is easy to see that q h can be extended to U h as a meromorphic (or holomorphic if m ∈ N ) function ˜ q h . Let ( U α , z α ) be a simply connected chart of S o with U α ∩ U h = ∅ . Then the transformation rule (4.1) holds because q is a regular k − differentialon S and ˜ q h coincides with the representation q h on U h \ { p } . Consequently, byextending the local representations of q we can extend q to S . Since the extensionis unique q is unique up to a non-zero constant multiple. (cid:3) The first important consequence of this result is
Corollary . Let ∇ be a meromorphic connection on a Riemann surface S . Let p be a Fuchsian pole for ∇ and ρ := Res p ∇ . Assume − − ρ ∈ N ∗ . Thenthere is a neighborhood U of p such that every geodesic ray which enters into U tends to p . The two rays of any geodesic which stays in U tend to p . Proof.
Let Σ be the set of poles of ∇ . Let ( U, z ) be a simply connected chartaround p such that U ∩ Σ = { p } . Then there exists a quadratic differential q on U adapted to ∇ . Hence Theorem 4.8 implies the assertion. (cid:3) Note that Corollary 4.13 is a proof of Conjecture 2.24.
Corollary . Let ∇ be a Fuchsian meromorphic connection on a Riemannsurface S with monodromy in Z and residues in Z . Then there are no self-intersecting geodesics of ∇ except periodic geodesics. Proof.
It follows from Theorem 4.12 and Proposition 4.9. (cid:3)
Another consequence of Theorem 4.12 is
Theorem . Let ∇ be a meromorphic connection on a Riemann surface S .Set S o := S \ Σ where Σ is the set of poles for ∇ . Let σ : [0 , ε ) → S o be a maximalgeodesic of ∇ and W its ω -limit set. Let p be a Fuchsian pole with Re Res p ∇ ≤ − .If p ∈ W then W = { p } . Proof.
It follows from Theorem 2.23 for non-resonant case and from Corollary4.13 for resonant case. (cid:3)
LAT STRUCTURE OF MEROMORPHIC CONNECTIONS ON RIEMANN SURFACES 19
5. Meromorphic G -differentials In this section we introduce the notion of meromorphic G -differential and westudy relations between meromorphic G -differentials and meromorphic connections.Furthermore, we introduce the notion of argument along geodesics of a meromorphic G -differential and we prove an analogue of Corollary 4.14. Let us introduce thenotion of G -differential. Definition . Let S be a Riemann surface and Σ ⊂ S a discrete set nothaving limit point in S . Set S o = S \ Σ. Let G be a multiplicative subgroup of C ∗ .A meromorphic G -differential is given by:(1) a Leray atlas adapted to ( S o , Σ), i.e., an atlas { ( U α , z α ) } ∪ { ( U k , z k ) } ,where { ( U α , z α ) } is a Leray atlas of S o = S \ Σ, each ( U k , z k ) is a simplyconnected chart centered at p k with U k ∩ U α is either empty or simplyconnected and U k ∩ U h = ∅ if k = h ;(2) the atlas { ( U α , z α ) } has the transition functions z β = a αβ z α + c αβ on U α ∩ U β , where a αβ ∈ G and c αβ ∈ C ;(3) for each k there is a meromorphic function v k defined on U k with p k asunique pole and ρ k ∈ C so that when U α ∩ U k = ∅ we have dz α dz k = b αk e ρ k log z k e v k on U α ∩ U k for some complex number b αk ∈ C .We say • { ( U α , z α ) } ∪ { ( U k , z k ) } is an atlas adapted to ∇ G ; • v k is a representation of ∇ G on U k ; • p k is a pole and Σ is the critical set of ∇ G ; • ρ k is the residue of ∇ G at p k and we denote it by Res p k ∇ G = ρ k . Definition . A holomorphic G -differential is a meromorphic G -differentialwithout poles.Note that a meromorphic G -differential is holomorphic outside of its criticalset. Definition . Let S be a Riemann surface. Let ∇ G be a meromorphic G -differential, Σ its critical set, and set S o := S \ Σ. Let { ( U α , z α ) } be an atlasrepresenting ∇ G on S o . A geodesic σ : [0 , ǫ ) → S o for ∇ G is a smooth curve in S o such that dz α ( σ ′ ( t )) is constant on each connected component of σ − ( U α ). Remark . Let ∇ G be a meromorphic G -differential on a Riemann surface S . Let { ( U α , z α ) } ∪ { ( U k , z k ) } be an atlas adapted to ∇ G . Let c ∈ C ∗ . By setting w α = cz α on U α and w k = cz k on U k we define an atlas { ( U α , w α ) } ∪ { ( U k , w k ) } .It is not difficult to see that the atlas defines a G -differential and we denote it by c ∇ G . By construction it is easy to see that ∇ G and c ∇ G have the same geodesicsand the same critical points with the same residues. G -differentials. In this subsection we study relation between meromorphic G -differentials and meromorphic connections on a Riemann surface S . Definition . Let ∇ be a meromorphic connection on a Riemann surface S .We say that a G -differential ∇ G on S is adapted to ∇ if ∇ G has the same criticalpoints, residues and the same geodesics as ∇ . The next theorem shows that meromorphic G − differentials and meromorphicconnections on a Riemann surface S are kind of equivalent notions. Theorem . Let ∇ G be a meromorphic G -differential on a Riemann surface S . Then there exists a meromorphic connection ∇ such that ∇ G is adapted to ∇ .Moreover, ∇ has monodromy in G .On the other hand, if ∇ is a meromorphic connection on a Riemann surface S such that ∇ has monodromy in G , then there exists a meromorphic G -differential ∇ G adapted to ∇ . Proof.
Let ∇ G be a meromorphic G -differential on a Riemann surface S and { ( U α , z α ) } ∪ { ( U k , z k ) } an atlas adapted to ∇ G . Set(5.1) η α : ≡ U α . Since the atlas { ( U α , z α ) } has transition functions z β = a αβ z α + c αβ on U α ∩ U β , where a αβ ∈ G and c αβ ∈ C , by Lemma 2.8 there exists a holomorphicconnection ∇ on S o , having the local representation η α on U α . By Lemma 2.8 ∇ has monodromy in G .Let ( U k , z k ) be the chart centered at p k and v k a representation of ∇ G on U k .Set(5.2) η k := (cid:18) ρ k z k + v ′ k (cid:19) dz k . on U k , where ρ k := Res p k ∇ G . Since v k is a meromorphic function we know that v ′ k is a meromorphic function with zero residue at the origin. Let U α be a chart with U α ∩ U k = ∅ . By definition of ∇ G there exists b αk ∈ C ∗ such that ξ αk = b αk e ρ k log z k e v k on U α ∩ U k , where ξ αk := dz α dz k . Since ξ αk has no zeroes we have dξ αk ξ αk = (cid:18) ρ k z k + v ′ k (cid:19) dz k . Consequently, η α + dξ αk ξ αk = (cid:18) ρ k z k + v ′ k (cid:19) dz k = η k on U α ∩ U k . Hence we have defined a meromorphic connection ∇ on S . By (5.2)we can see that the residue of ∇ at p k is ρ k . So ∇ has the same critical points andthe same residues as ∇ G .Let σ : [0 , ǫ ] → U α be a geodesic for ∇ G . Then there exists a ∈ C ∗ and b ∈ C such that z α ( σ ( t )) = at + b. By using Proposition 2.13 we can see that σ is a geodesic for ∇ .Let now ∇ be a meromorphic connection on a Riemann surface S with mon-odromy in G , where G is a multiplicative subgroup of C ∗ . Set S o := S \ Σ, whereΣ is the set of poles of ∇ .By Lemma 2.8 there exists a Leray ∇− atlas { ( U α , z α ) } for S o such that ξ αβ := dz α dz β ∈ G on U α ∩ U β . Hence we have a holomorphic G -differential ∇ G on S o withthe representative atlas { ( U α , z α ) } .By adding to the Leray atlas at most countably many charts { ( U k , z k ) } aroundpoles of ∇ after possibly passing to a refinement we can get a Leray atlas { ( U α , z α ) }∪ LAT STRUCTURE OF MEROMORPHIC CONNECTIONS ON RIEMANN SURFACES 21 { ( U k , z k ) } adapted to ( S o , Σ). Let ( U k , z k ) be the chart centered at p k ∈ Σ and η k the representation of ∇ on U k . Let η k = (cid:18) ρ k z k + f k (cid:19) dz k , where ρ k := Res p k ∇ and f k is a meromorphic function on U k having as uniquepole p k with zero residue. Let v k be a meromorphic primitive of f k on U k . Bysetting v k as a representation of ∇ G on ( U k , z k ) we claim that ∇ G extends to S as a meromorphic G − differential. Let U α be a chart with U α ∩ U k = ∅ . Then bytransformation rule (2.2) we have η k = η α + dξ αk ξ αk = dξ αk ξ αk , on U α ∩ U k , where ξ αk = dz α dz k . Since U α ∩ U k is a simply connected domain and ξ αk has no zeroes, there exists a constant c αk such thatlog ξ αk = ρ k log z k + v k + c αk for a branch of log ξ αk . Consequently, ξ αk = e c αk e ρ k log z k e v k on U α ∩ U k . Hence ∇ G extends to S as a meromorphic G − differential. It is obviousthat ∇ G has the same critical points and residues as ∇ . Again by using Proposition2.13 we can show that a smooth curve σ : [0 , ε ) → S o is a geodesic for ∇ if andonly if it is a geodesic for ∇ G . (cid:3) Remark . Let ∇ be a meromorphic connection on a Riemann surface S .Let ∇ G be a meromorphic G -differential adapted to ∇ . Then c ∇ G is also adaptedto ∇ for any c ∈ C ∗ . G -differentials. In this section we introducethe notion of argument for a holomorphic G -differential, an analogue of the notionof argument for meromorphic quadratic differentials (see for example [ ]).Let ∇ G be a holomorphic G -differential on a Riemann surface S . Let { ( U α , z α ) } be an atlas adapted to ∇ G . If σ : [0 , ε ) → U α is a geodesic for ∇ G thenarg dz α ( σ ′ ( t )) = constbecause dz α ( σ ′ ( t )) is a non-zero constant. Setarg ασ ′ ∇ G := e i arg σ ′ dz α , where arg σ ′ dz α := arg dz α ( σ ′ ( t )). As we have seen above this number is a constantdepending on U α and σ .Let now σ : [0 , ε ) → S be any geodesic for ∇ G . Let U α and U β be differentcharts such that U α ∩ U β ∩ supp( σ ) = ∅ . By definition of ∇ G there exists a αβ ∈ G such that dz β ( σ ′ ( t )) = a αβ dz α ( σ ′ ( t )) . for σ ( t ) ∈ U α ∩ U β . Consequently,(5.3) arg ασ ′ ∇ G = exp( i arg a αβ ) arg βσ ′ ∇ G . Definition . Set arg σ ′ ∇ G := θ, where θ ∈ S /G and G = (cid:26) a | a | : a ∈ G (cid:27) . By (5.3) we can see that arg σ ′ ∇ G is well defined. Definition . Fix θ ∈ S /G . We say that a geodesic σ : [0 , ε ) → S has θ -trajectory (or that θ is the trajectory of σ ) ifarg σ ′ ∇ G = θ. We say that σ is a horizontal geodesic of ∇ G if θ = 1. Lemma . Let ∇ be a holomorphic connection and σ : [0 , ε ) → S a geodesicfor ∇ . Then there exists a holomorphic G -differential ∇ G adapted to ∇ such that σ is a horizontal geodesic for ∇ G . Proof.
Let ∇ G be a holomorphic G -differential adapted to ∇ . Let θ ∈ S /G be the trajectory of σ with respect to ∇ G . Let θ ∈ S such that [ θ ] = θ . Let { ( U α , z α ) } be an atlas adapted to ∇ G . By setting w α = θ z α we define an atlas { ( U α , w α ) } . Then the atlas is adapted to a holomorphic G -differential and wedenote it by θ ∇ G . It is easy to see that θ ∇ G is adapted to ∇ . By the definitionof argument we can see that σ is a horizontal geodesic for θ ∇ G . (cid:3) Definition . Let ∇ be a holomorphic connection on a Riemann surface S . Let σ : [0 , ε ) → S o be a geodesic for ∇ . Assume σ ( t ) = σ ( t ) = p for some t , t ∈ [0 , ε ). Let ( U, z ) be a ∇ -chart centered at p . The intersection angle of σ at p (with respect to t and t ) is | arg dz ( σ ′ ( t )) − arg dz ( σ ′ ( t )) | mod 2 π. Let G ⊂ C ∗ . Set arg G := { φ ∈ [0 , π ) : φ = arg a, a ∈ G } and G k := { a k : a ∈ G } . Let us state an analogue of Corollary 4.14.
Theorem . Let ∇ be a holomorphic connection on a Riemann surface S and G a multiplicative subgroup of C ∗ . Assume ∇ has monodromy in G . Let σ : [0 , ε ) → S o be a geodesic for ∇ . If σ intersects itself then the intersection angleis in arg G . In particular, if arg G = { } then every self-intersecting geodesic of ∇ is closed. Proof.
By Lemma 5.10 there exists a holomorphic G − differential ∇ G adaptedto ∇ such that σ is a horizontal geodesic for ∇ G . Let { ( U α , z α ) } be an atlasrepresenting ∇ G on S o . Note that the representations of geodesics of ∇ G on U α are Euclidean segments. Pick a point p ∈ σ . Assume σ intersect itself at p . Let U α be a chart around p . Let denote by F α the collection of geodesic segments˜ σ such that supp(˜ σ ) ⊂ ¯ U α ∩ supp( σ ) with the property that if σ , σ ∈ F α thensupp( σ ) ∩ supp( σ ) = { p } .Take two different σ , σ ∈ F α . We have LAT STRUCTURE OF MEROMORPHIC CONNECTIONS ON RIEMANN SURFACES 23 arg σ ′ ∇ G = arg σ ′ ∇ G = 1that is arg σ ′ j dz α ∈ arg G for j = 1 ,
2. Now the intersection angle between σ and σ is | arg σ ′ dz α − arg σ ′ dz α | mod 2 π. Since G is a multiplicative group the intersection angle is in arg G .Suppose now arg G = { } . If σ intersects itself then the intersection angle isin arg G = { , π } . This is possible only if σ is closed. (cid:3) Corollary . Let ∇ be a holomorphic connection on a Riemann surface S . Let G be the monodromy group of ∇ and assume that arg G k = { } for some k ∈ N . If σ : [0 , ε ) → S is not a closed geodesic for ∇ and p ∈ supp( σ ) then σ intersects itself at p at most k times. Proof.
Assume σ intersect itself at p more than k times. Then the number ofintersection angles of σ at p are greater than k . Since arg G has at most k elementsit is a contradiction to the previous theorem. (cid:3) In [ ], it wascomputed the monodromy group when S ⊆ C , that is when S is covered by a singlechart. The most interesting case will be when S is the complement in P ( C ) of afinite set of points. Proposition , Proposition 3.6]) . Let S ⊆ C be a (multiply connected)domain, ∇ a holomorphic connection on T S , and η the holomorphic 1-form repre-senting ∇ . Then the monodromy representation ρ : H ( S, Z ) → C ∗ is given by ρ ( γ ) = exp (cid:18)Z γ η (cid:19) for all γ ∈ H ( S, Z ) . By using the previous Proposition we will prove the following
Lemma . Let ∇ be a meromorphic connection on a Riemann surface S .Let G be the monodromy group of ∇ . Let p j be a pole of ∇ and ρ j := Res p j ∇ .Then G j ⊆ G , where G j := (cid:8) e πikρ j : k ∈ Z (cid:9) . Proof.
Let ( U j , z j ) be a simply connected chart around p j such that U j ∩ Σ = { p j } . Then by Proposition 5.14 ∇ has monodromy in G j on U j . Hence G j ⊆ G . (cid:3) Corollary . Let ∇ be a holomorphic connection on a Riemann surface S with monodromy in G , so that arg G k = { } . Then Res ∇ ⊂ k Z + i R .
6. The canonical covering
Let ∇ be a meromorphic connection on a compact Riemann surface S . In thissection we introduce the notion of canonical covering induced by ∇ (see [ , Section2] for quadratic differentials and [ , Section 2.1] for k -differentials). Theorem . Let S be a compact Riemann surface and Σ a finite set. Let ∇ be a holomorphic connection on S o := S \ Σ . Assume that ∇ has monodromy in G and that there exists k ∈ N ∗ so that arg G k = { } . Then there exists a compactRiemann surface ˆ S , a possibly ramified covering π : ˆ S → S , and a holomorphicconnection ˆ ∇ on ˆ S = ˆ S \ ˆΣ , where ˆΣ = π − (Σ) , such that (1) a smooth curve ˆ σ : [0 , ε ) → ˆ S o is a geodesic for ˆ ∇ if and only if σ = π ◦ ˆ σ is a geodesic for ∇ ; (2) any self-intersecting geodesic of ˆ ∇ is closed. Proof.
Let k ∈ N ∗ be the minimum integer satisfying arg G k = { } . Let { ( U α , z α ) } be a Leray ∇− atlas for S o with the transition functions z β = a αβ z α + c αβ on U α ∩ U β , where a αβ ∈ G and c αβ ∈ C . Note that the representation of ∇ on U α is identically zero.Let ζ be a fixed primitive k -th root of unity. Since a αβ ∈ G there exists l αβ ∈ { , ..., k − } such that(6.1) a αβ = ζ l αβ | a αβ | . Take k copies of U α and denote them by U α j ∼ = U α for j = 0 , ..., k −
1. We set z α j := ζ j z α on U α j . Whenever U α ∩ U β = ∅ , we glue the copies U α i and U β j forthe indices j = i − l αβ mod k. Then we have z β j = ζ j z β = ζ j ( a αβ z α + c αβ )= ζ j + l αβ | a αβ | z α + ζ j c αβ = ζ j + l αβ − i | a αβ | z α i + ζ j c αβ = | a αβ | z α i + ζ j c αβ Hence z β j = a α i β j z α j + c α i β j for some a α i β j ∈ R + and c α i β j ∈ C . Consequently, we get a Riemann surface ˆ S o .Take a connected component of ˆ S o and denote it by ˆ S o . Hence we have a coveringmap π ′ : ˆ S o → S o . By Theorem 8.4 from [ ] there exists a compact Riemannsurface ˆ S and a (possibly ramified) covering π : ˆ S → S such that π | ˆ S o = π ′ and π − (Σ) is a finite set. Hence we have a (possibly ramified) covering π : ˆ S → S .Moreover, by the construction there exists a Leray atlas { ( V γ , w γ ) } on ˆ S o withthe transition functions w θ = a γθ w γ + c γθ on U γ ∩ U θ , where a γθ ∈ R + and c γθ ∈ C .Consequently, by setting ˆ η γ ≡ V γ we can get a holomorphicconnection ˆ ∇ on ˆ S o with the local representations { ˆ η γ } .Let ˆ σ : [0 , ε ) → ˆ S o be a smooth curve and σ := π ◦ ˆ σ . Let ( V γ , w γ ) be a chartwith V γ ∩ supp(ˆ σ ) = ∅ . Then there exists a chart ( U α , z α ) and an integer j suchthat V γ ∼ = U α and w γ = ζ j z α . We can see that z α ( σ ( t )) is a Euclidean segment if LAT STRUCTURE OF MEROMORPHIC CONNECTIONS ON RIEMANN SURFACES 25 and only if w γ (ˆ σ ( t )) is a Euclidean segment. Hence σ is a geodesic for ∇ if andonly if ˆ σ ( t ) is a geodesic for ˆ ∇ .Let ˆ G be the monodromy group of ˆ ∇ on ˆ S o . Since the atlas { ( V γ , w γ ) } has thetransition functions w θ = a γθ w γ + c γθ on U γ ∩ U θ , where a γθ ∈ R + and c γθ ∈ C ,we can see that arg ˆ G = { } . Hence any self-intersecting geodesic of ˆ ∇ is closed. (cid:3) Definition . The covering π : ˆ S → S from the last theorem is said to be the canonical covering induced by ∇ . The next lemma shows a relation between the ω -limit sets of geodesics ∇ andˆ ∇ . Lemma . Let ∇ be a meromorphic connection on a compact Riemann surface S and Σ the set of poles of ∇ . Set S o = S \ Σ . Let σ : [0 , ε ) → S be a maximalgeodesic of ∇ . Assume ∇ has monodromy in G with arg G k = { } for some k ∈ N ∗ .Let π : ˆ S → S be a canonical covering of S induced by ∇ . Let ˆ σ be a lift of σ . Let ˆ W be the ω -limit set of ˆ σ . Then π ( ˆ W ) is the ω -limit set of σ . Proof.
Let W be the ω -limit set of σ . It is not difficult to see that π ( ˆ W ) ⊆ W .Let p ∈ W . Then there exists a sequence of positive numbers t n ր ε such thatlim n →∞ σ ( t n ) = p . Take the sequence { ˆ σ ( t n ) } . Since ˆ S is compact { ˆ σ ( t n ) } has a limitset A ⊂ W . It is enough to show that there exists ˆ p ∈ A such that π (ˆ p ) = p .Since A is not empty there exists a point ˆ q ∈ A . Let π (ˆ q ) = q for some q ∈ S .Then there exists a subsequence { ˆ σ ( t n l ) } ∞ l =1 such that lim l →∞ σ ( t n l ) = q . Since thesequence { σ ( t n ) } has single limit point we can see that q = p . We are done. (cid:3) By using the previous lemma and Theorem 6.1 we give a classification of the ω -limit sets of infinite self-intersecting geodesics of meromorphic connections havingmonodromy group G with arg G k = { } for some k ∈ N ∗ . Theorem . Let ∇ be a meromorphic connection on a compact Riemannsurface S and Σ the set of poles of ∇ . Set S o = S \ Σ . Let σ : [0 , ε ) → S bea maximal geodesic of ∇ . Assume ∇ has monodromy in G with arg G k = { } forsome k ∈ N . If σ intersects itself infinitely many times, then either (1) the ω -limit set of σ in S is given by the support of a (possibly non-simple)closed geodesic; or (2) the ω -limit set of σ in S is a graph of (possibly self-intersecting) saddleconnections; or (3) the ω -limit set of σ in S is all of S ; or (4) the ω -limit set of σ has non-empty interior and non-empty boundary, andeach component of its boundary is a graph of (possibly self-intersecting)saddle connections with no spikes and at least one pole. Proof.
Since σ intersect itself infinitely many times the ω -limit set of σ cannot be a single pole. Let π : ˆ S → S be the canonical covering induced by ∇ andˆ ∇ = π ∗ ∇ . Let ˆ σ be a geodesic for ˆ ∇ such that π ◦ ˆ σ = σ . Thanks to Theorem 6.1we can see that ˆ σ does not intersect itself. Consequently, by using Theorem 2.21we have a list of possible ω -limit sets of the corresponding lift geodesic ˆ σ . Let ˆ W be the ω -limit set of ˆ σ . By the previous Lemma we have π ( ˆ W ) = W . Let first ˆ W be the support of a closed geodesic. Since π sends a geodesic ofˆ ∇ to a geodesic of ∇ we have W the support of a (possibly non-simple) closedgeodesic.Let now ˆ W be a graph of saddle connections. By the same argument as abovewe can see that W is a graph of (possibly self-intersecting) saddle connections.Finally, if ˆ W has non-empty interior then W has also non-empty interior. As-sume W has non-empty boundary. Then each component of the boundary of ˆ W isa graph of saddle connections with no spikes and at least one pole. It is not difficultto see that each component of ∂W is a graph of (possibly self-intersecting) saddleconnections with no spikes and at least one pole. We are done. (cid:3) References [1] Abate, M., Bianchi,F. : A Poincar´e–Bendixson theorem for meromorphic connections on Rie-mann surfaces Math. Z. (2016) 282: 247. https://doi.org/10.1007/s00209-015-1540-6[2] Abate, M., Tovena, F.: Poincar´e–Bendixson theorems for meromorphic connections and ho-mogeneous vector fields. J. Differ. Equ. 251, 2612–2684 (2011)[3] Bainbridge, M., Chen, D., Gendron, Q., Grushevsky, S., Moeller, M.: Strata of k -differentials.Algebraic Geometry. 6. 10.14231/AG-2019-011. (2016)[4] Forster O.: Lectures on Riemann Surfaces, Springer-Verlag New York Inc., 1981, pp.51-52[5] Georgios, D., Richard, A.: Wentworth, Harmonic maps and Teichm¨uller theory, Handbook ofTeichm¨uller theory. Vol. I, IRMA Lect. Math. Theor. Phys., vol. 11, Eur. Math. Soc., Zurich,2007, pp. 33–109[6] Ilyashenko, Y., Yakovenko, S.: Lectures on Analytic Differential Equations, Grad. Stud. Math.,vol. 86, American Mathematical Society, Providence, RI, 2008.[7] Lanneau, E.: Hyperelliptic components of the moduli spaces of quadratic differentials withprescribed singularities, Comment. Math. Helv. 79 (2004), no. 3, 471–501; doi:10.1007/s00014-004-0806-0.[8] Schmitt, J.: Dimension theory of the moduli space of twisted k -differentials, Doc. Math. 23(2018), 871–894[9] Strebel, K.: Quadratic differentials, Ergebnisse der Mathematik und ihrer Grenzgebiete (3)[Results in Mathematics and Related Areas (3)], vol. 5, Springer-Verlag, Berlin, 1984. MR86a:30072[10] Smith, R.A., Thomas, E.S.: Transitive flows on two-dimensional manifolds. J. Lond. Math.Soc. 2(37), 569–576 (1988)[11] Tahar, G.: Counting saddle connections in flat surfaces with poles of higher order, GeomDedicata (2018) 196: 145. https://doi.org/10.1007/s10711-017-0313-2[12] Yoccoz, J.–C.: Continued fraction algorithms for interval exchange maps: an introduction,in Frontiers in Number Theory, Physics and Geometry I, Cartier P., Julia B., Moussa P. andVanhove P. editors, Springer–Verlag (2006)[13] Yoccoz, J.–C.: Echanges d’intervalles, Cours College de France (2005) Department of Mathematics, University of Pisa, Pisa, Italy, 56127
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