Labeled Rauzy classes and framed translation surfaces
aa r X i v : . [ m a t h . G T ] O c t LABELED RAUZY CLASSES AND FRAMED TRANSLATIONSURFACES
CORENTIN BOISSY
Abstract.
In this paper, we compare two definitions of Rauzy classes. The first one wasintroduced by Rauzy and was in particular used by Veech to prove the ergodicity of theTeichmüller flow. The second one is more recent and uses a “labeling” of the underlyingintervals, and was used in the proof of some recent major results about the Teichmüllerflow.The Rauzy diagrams obtained from the second definition are coverings of the initial ones.In this paper, we give a formula that gives the degree of this covering.This formula is related to moduli spaces of framed translation surfaces, which correspondsto surfaces where we label horizontal separatrices on the surface. We compute the number ofconnected component of these natural coverings of the moduli spaces of translation surfaces.Delecroix [Del10] proved a recursive formula for the cardinality of the (reduced) Rauzyclasses. Therefore, we also obtain formula for labeled Rauzy classes.
Contents
1. Introduction 12. Background 33. Rauzy diagrams and framed translation surfaces 74. Topological invariants for framed translation surfaces 105. Number of connected components of C ( F comp ) Introduction
Rauzy induction was introduced in [Rau79] as a tool to study interval exchange maps. It isa renormalization process that associates to an interval exchange map, another one obtainedas a first return map on a well chosen subinterval. After the major works of Veech [Vee82]and Masur [Ma82], the Rauzy induction became a powerful tool to study the Teichmüllergeodesic flow.A slightly different tool was used by Kerckhoff [Ke85], Bufetov [Bu06], and Marmi-Moussa-Yoccoz [MMY05].It is obtained after labeling the intervals, and keeping track of them during the renor-malization process. This small change was a significative improvement, and was used in the
Date : November 13, 2018.2000
Mathematics Subject Classification.
Primary: 37E05. Secondary: 37D40.
Key words and phrases.
Interval exchange maps, Rauzy induction, Abelian differentials, Moduli spaces,Teichmüller flow. ecent past to prove other important results about the Teichmüller geodesic flow, for instancethe simplicity of the Liapunov exponents (Avila-Viana [AV07]), or the exponential decay ofcorrelations (Avila-Gouezel-Yoccoz [AGY06]).An interval exchange map is naturally decomposed into a continuous and a combinatorialdatum. A Rauzy class is a minimal set of such combinatorial data invariant by the com-binatorial Rauzy induction. We will speak of reduced or labeled Rauzy classes dependingwhether we use the definition of Rauzy, or the other one.Let k , . . . , k r be some pairewise distinct nonnegative integers and let n , . . . , n r be pos-itive integers such that P ri =1 n i k i = 2 g − . The stratum of the moduli space of Abeliandifferentials whose corresponding surfaces have precisely n i singularities of degree k i , for all i ∈ { , . . . , r } , is usually denoted as H ( k n , . . . , k n r r ) . In this notation, we implicitely assumethat k i = k j for all i = j . A singularity of degree zero is by convention a regular markedpoint on the surface.The Veech construction naturally associates to a Rauzy class a connected component C of a stratum H ( k n , . . . , k n r r ) of the moduli space of Abelian differentials, and an integer k ∈ { k , . . . , k r } that corresponds to degree of the singularity attached on the left in the Veechconstruction. In the next statement, the pair ( C , k ) will be refered to the data associated toa Rauzy class by the Veech construction. Theorem 1.1.
Let R lab be a labeled Rauzy class and let R be the corresponding reduced one.Let ( C , k ) be data associated to R by the Veech construction, where C is a connected componentof a stratum H ( k n , . . . , k n r r ) of the moduli space of Abelian differentials, and k ∈ { k , . . . , k r } .Let n be number of singularities of degree k for a surface in H ( k n , . . . , k n r r ) . We have: | R lab || R | = Π ri =1 n i !( k i + 1) n i n ( k + 1) ε where ε satisfies: • ε = g if C consists only of hyperelliptic surfaces with two conical singularities ofdegree g − and possibly some regular marked points. • ε = if C contains nonhyperelliptic surfaces and if there exists k ′ ∈ { k , . . . , k r } which is odd. • ε = 1 in all the other cases. Delecroix [Del10] has found a recursive formula for the cardinality of the reduced Rauzyclasses. The previous theorem complete his result for the case of labeled Rauzy classes.Our result is related to moduli spaces of framed translation surfaces. Informally, we willcall a frame on a translation surface X a map F X from a discrete alphabet A , to a set S X of discrete combinatorial data on the surface X such that the moduli space of framedtranslation surfaces is a covering of the corresponding moduli space of translation surface.In our case, we are interested in the case where S X is the set of horizontal outgoingseparatrices . Several different kind of frames will appear in this context. The most importantwill be the space C ( F comp ) . It corresponds to the space were we label exactly one horizontalseparatrix for each singularity (see Section 3.1 for a more precise definition), and the alphabet A is a disjoint union of subalphabets A k , such that the labels in A k correspond to singularitiesof degree k . hoosing α in some A k , one gets a natural covering p α : C ( F comp ) → C ( F k ) , were C ( F k ) isa moduli space of framed translation surfaces that corresponds to translation surfaces witha with a single marked separatrix adjacent to a singularity of degree k .Theorem 1.1 will be a consequence of the two following results, which give a geometricalinterpretation of the formula. Proposition 1.2.
Let R lab be a labeled Rauzy class and let R be the corresponding reducedone. The ratio | R lab | / | R | is equal to the degree of the covering p α , when restricted to aconnected component of C ( F comp ) . Theorem 1.3.
Let C be a connected component of a stratum of the moduli space of Abeliandifferentials. The space C ( F comp ) is not connected in general. More precisely, it has: • g connected components if C is the hyperelliptic connected component of H ( g − , g − ,with possibly some regular marked points. • Two connected components if it does not corresponds to a hyperelliptic connectedcomponent and if there exists odd degree conical singularities. • One connected component in all the other cases.
Acknowledgements.
We thank Vincent Delecroix, Luca Marchese and Erwan Lanneau forremarks and comments on this paper. We also thank [Ste09] for computational help.2.
Background
Labeled and reduced interval exchange transformations.
We give here the twodefinitions of interval exchange transformations that are used in the literature. In order todistinguish them, we will add the terms reduced and labeled .The first one is due to Rauzy [Rau79]
Definition 2.1.
Let I ⊂ R be an open interval and let us choose a finite subset Σ = { x , . . . , x d − } of I . The complement of Σ in I is a disjoint union of d ≥ open subintervals { I j , j = 1 , . . . , d } . An reduced interval exchange transformation is a map T from I \ Σ to I that permutes, by translation, the subintervals I j . It is easy to see that T ( I \ Σ) = I \ Σ ′ ,were Σ ′ ⊂ I is of the same cardinality as Σ , and that T is precisely determined by: • A combinatorial datum : a permutation π ∈ Σ d which expresses that the intervalnumber k , when counted from the left to the right, is sent to the place π ( k ) by themap T . • A continuous datum : a vector λ ∈ R d with positive entries that corresponds to thelengths of the intervals.We will identify an interval exchange transformation with its parameters ( π, λ ) .The second definition of interval exchange transformation was first introduced by Kerck-hoff [Ke85] and later formalized by Bufetov [Bu06] and Marmi, Moussa & Yoccoz [MMY05].As we will see later, it simplifies the description of the Rauzy induction, but we get biggerRauzy classes. Definition 2.2. A labeled interval exchange map is an reduced interval exchange map with apair ( π t , π b ) of one-to-one maps from a finite alphabet A to { , . . . , d } . The interval number k , when counted from the left to the right, is denoted by I π − t ( k ) . Once the intervals areexchanged, the interval number k is I π − b ( k ) . n the previous definition, it is easy to see that the permutation corresponding to theunderlying reduced interval exchange map is π = π b ◦ π − t , and the continuous datum is avector with positive entries λ ∈ R A . We will identify a labeled interval exchange map withthe pair (˜ π, λ ) , were ˜ π = ( π t , π b ) . We will call ˜ π a labeled permutation .We will usually represent a labeled permutation by a table: ˜ π = (cid:18) π − t (1) π − t (2) . . . π − t ( d ) π − b (1) π − b (2) . . . π − b ( d ) (cid:19) . As we can see from this representation, we have “t” for top, “b” for bottom in the notation π t , π b .A renumbering of a labeled permutation is the composition of ( π t , π b ) by a one-to-onemap f from A to A ′ . It just corresponds to changing the labels without changing theunderlying permutation. From the precious definitions, it is clear that a reduced intervalexchange transformation ( resp. a permutation) is an equivalent class of labeled intervalexchange maps ( resp. labeled permutations) up to renumbering. We will sometime identifya permutation with its unique representative ( π t , π b ) with A = { , . . . , d } and π t = Id .2.2. Rauzy-Veech induction.
Let T be a labeled or reduced interval exchange map. TheRauzy–Veech induction R ( T ) of T is defined as the first return map of T to a certainsubinterval J of I (see [Rau79, MMY05] and [Rau79] for details).We recall briefly the construction for the labeled case. Following the terminology of [MMY05]we define the type of T by t if λ π − t ( d ) > λ π − b ( d ) and b if λ π − t ( d ) < λ π − b ( d ) . When T is of type t (respectively, b ) we will say that the label π − t ( d ) (respectively, π − b ( d ) ) is the winner andthat π − b ( d ) (respectively, π − t ( d ) ) is the looser. We define a subinterval J of I by J = (cid:26) I \ T ( I π − b ( d ) ) if T is of type t ; I \ I π − t ( d ) if T is of type b.The image of T by the Rauzy-Veech induction R is defined as the first return map of T tothe subinterval J . This is again a labeled interval exchange transformation, defined on d letters. The combinatorial datum of this new interval exchange transformation is very easyto calculate in terms of the one of T . Indeed, let α ∈ A ( resp. β ∈ A ) be the winner ( resp. looser). Let λ ′ ∈ R A such that: λ ′ α = λ α − λ β λ ′ ν = λ ν for all ν ∈ A\{ α } Then, R ( T ) = ( R ε ( π ) , λ ′ ) , were ε is the type of T , and R t , R b are the following combina-torial maps:(1) R t : let k = π b ( π − t ( d )) with k ≤ d − . Then, R t ( π t , π b ) = ( π ′ t , π ′ b ) where π t = π ′ t and π ′− b ( j ) = π − b ( j ) if j ≤ kπ − b ( d ) if j = k + 1 π − b ( j − otherwise.(2) R b : let k = π t ( π − b ( d )) with k ≤ d − . Then, R b ( π t , π b ) = ( π ′ t , π ′ b ) where π b = π ′ b and π ′− t ( j ) = π − t ( j ) if j ≤ kπ − t ( d ) if j = k + 1 π − t ( j − otherwise. he maps R t , R b are called the combinatorial Rauzy moves. It is easy to define similarmaps for reduced permutations. We first identify π with the corresponding ( Id, π b ) andperform the Rauzy move. Then, if needed, we renumber the result so that it corresponds toa reduced permutation. We will still denote the corresponding maps by R t , R b , since it willalways be clear, when the distinction is needed, whether or not the objects are labeled orreduced.We define the Rauzy induction for reduced interval exchange maps by considering labeledinterval exchange maps up to renumbering. Definition 2.3. A Rauzy class , usually denoted by R is a minimal set of labeled or reducedpermutations invariant by the combinatorial Rauzy moves.A Rauzy diagram is a graph whose vertices are the elements of a Rauzy class and whosevertices are the combinatorial Rauzy maps.A Rauzy class or Rauzy diagram will be called labeled or reduced depending on thecorresponding permutations.
Example . • Let τ n = ( ... nn n − ... ) . Rauzy proved that the cardinality of R ( τ n ) is n − − for the reduced case, and one can prove “by hand” that it is the same forthe labeled case. • Let π n = (cid:0) ... n − nn n − ... (cid:1) . Contrary to the previous case, the labeled and reduceddiagrams are not isomorphic anymore. The structure of the labeled and reducedRauzy diagrams is precisely described in [BL10]. It is in particular shown that thecardinality of the reduced diagram is n − − n and the cardinality of the labeleddiagram is (2 n − − n )( n − . • Consider π = ( ) . The reduced Rauzy class is of size Translation surfaces and moduli space.
Translation surfaces. A translation surface is a (real, compact, connected) genus g surface X with a translation atlas i.e. a triple ( X, U , Σ) such that Σ is a finite subset of X (whose elements are called singularities ) and U = { ( U i , z i ) } is an atlas of X \ Σ whosetransition maps are translations. We will require that for each s ∈ Σ , there is a neighborhoodof s isometric to a Euclidean cone. One can show that the holomorphic structure on X \ Σ extends to X and that the holomorphic 1-form ω = dz i extends to a holomorphic − formon X were Σ corresponds to the zeroes of ω and maybe some marked points. We usuallycall ω an Abelian differential .For g ≥ , we define the moduli space of Abelian differentials H g as the moduli space ofpairs ( X, ω ) where X is a genus g (compact, connected) Riemann surface and ω non-zeroholomorphic − form defined on X . The term moduli space means that we identify thepoints ( X, ω ) and ( X ′ , ω ′ ) if there exists an analytic isomorphism f : X → X ′ such that f ∗ ω ′ = ω . The group SL ( R ) naturally acts on the moduli space of translation surfaces bypost composition on the charts defining the translation structures.One can also see a translation surface obtained as a polygon (or a finite union of polygons)whose sides come by pairs, and for each pairs, the corresponding segments are parallel and This can be computed for instance using Zorich’s MATHEMATICA software, or using the SAGE packagedeveloped by Delecroix f the same lengths. These parallel sides are glued together by translation and we assumethat this identification preserves the natural orientation of the polygons. In this context,two translation surfaces are identified in the moduli space of Abelian differentials if andonly if the corresponding polygons can be obtained from each other by cutting and gluingand preserving the identifications. Also, the SL ( R ) action in this representation is just thenatural linear action on the polygons.The moduli space of Abelian differentials is stratified by the combinatorics of the zeroes;we will denote by H ( k n , . . . , k n r r ) the stratum of H g consisting of (classes of) pairs ( X, ω ) such that ω possesses exactly n i zeroes on X with multiplicities k i for all i ∈ { , . . . , r } ,and no other zeroes. It is a well known part of the Teichmüller theory that these spaces are(Hausdorff) complex analytic, and in fact algebraic, spaces. These strata are non-connectedin general but each stratum has at most three connected components (see [KZ03] for acomplete classification, or see Section 4).Given a translation surface X , we will call separatrix an oriented half line (possibly finite)starting from a singularity of X . A horizontal separatrix l will be outgoing if it goes on theright in a translation chart, and incoming otherwise.2.4. Suspension data.
The next construction provides a link between interval exchangetransformations and translation surfaces. A suspension datum for T = ( π, λ ) is a collectionof vectors { τ α } α ∈A such that • ∀ ≤ k ≤ d − , P π t ( α ) ≤ k τ α > , • ∀ ≤ k ≤ d − , P π b ( α ) ≤ k τ α < .We will often use the notation ζ = ( λ, τ ) . To each suspension datum τ , we can associatea translation surface ( X, ω ) = X ( π, ζ ) in the following way.Consider the broken line L t on C = R defined by concatenation of the vectors ζ π − t ( j ) (in this order) for j = 1 , . . . , d with starting point at the origin. Similarly, we consider thebroken line L b defined by concatenation of the vectors ζ π − b ( j ) (in this order) for j = 1 , . . . , d with starting point at the origin. If the lines L t and L b have no intersections other than theendpoints, we can construct a translation surface X by identifying each side ζ j on L t withthe side ζ j on L b by a translation. The resulting surface is a translation surface endowed withthe form ω = dz . Note that the lines L t and L b might have some other intersection points.But in this case, one can still define a translation surface by using the zippered rectangleconstruction , due to Veech ([Vee82]). See for instance Figure 1.Let I ⊂ X be the horizontal interval defined by I = (0 , P α λ α ) × { } . The reducedinterval exchange transformation T is precisely the one defined by the first return map to I of the vertical flow on X .We can extend the Rauzy induction to suspension data in the following way: let τ be asuspension data over ( π, λ ) , we define R ( π, λ, τ ) = ( π ′ , λ ′ , τ ′ ) by: • R ( π, λ ) = ( π ′ , λ ′ ) • τ ′ α = τ α − τ β , where α (resp. β ) is the winner (resp. looser) for T = ( π, λ ) This extension is known as the
Rauzy–Veech induction , and is used as a discretization ofthe Teichmüller flow.
Remark . By construction the two translation surfaces X ( π, ζ ) and X ( π ′ , ζ ′ ) define thesame element in the moduli space. Figure 1.
The zippered rectangle construction, for two examples of suspen-sion data.
Remark . Note that λ, τ define natural local parameters for the stratum of the modulispace of Abelian differentials.
Remark . The left end of the two lines in the previous construction L t , L b is a singularity,and the horizontal half line starting from this point to the right corresponds to a choice ofa horizontal separatrix starting from this singularity, and it is easy to see that this combi-natorial data is preserved under the Rauzy–Veech induction. Let us denote by l ( π, ζ ) thisseparatrix. 3. Rauzy diagrams and framed translation surfaces
Moduli space of framed translation surface. A frame on a translation surface X a map F X from a discrete alphabet A , to a set DC X of discrete combinatorial data on thesurface X .Let C be a connected component of a stratum of moduli space of translation surfaces, anda collection F of frames for translation surfaces in C , with a fixed alphabet. One can definethe corresponding moduli space of framed translation surfaces: two elements ( X, F X ) and ( X ′ , F ′ X ′ ) are identified if there is a translation mapping X → X ′ which is consistent withthe frames. Then, we will denote by C ( F ) the corresponding moduli space.The sets DC X can be many things: incoming or outgoing horizontal separatrices, H ( X, Z ) ,etc. . . Here we will not study the precise conditions on the collections of frames so that C ( F ) is a “nice” space. We will just introduce three cases, for which C ( F ) are coverings of C . Wefirst ask that DC X = S X is the set of horizontal outgoing separatrices , then we consider thethree families of all frames that satisfy the following conditions respectively:(1) F k : the set A is a singleton and the image of F X is any separatrix adjacent to adegree k singularity. F sat : F X is a one-to-one mapping from A to S X .(3) F comp : A = ⊔ k A k , the map F X is injective, and we require that for each k anysingularity of S of degree k has a unique separatrix in F ( A k ) .We will denote respectively by C ( F k ) , C ( F sat ) , and C ( F comp ) , the corresponding modulispaces, which are finite coverings of C .(1) The space C ( F k ) is the moduli space of pairs ( X, l ) , were X ∈ C and l is a separatrixadjacent to a singularity of degree k in X .(2) The space C ( F sat ) corresponds to translation surfaces were we label each horizontaloutgoing separatrix by an element in A . It corresponds to a “saturated case”.(3) The space C ( F comp ) corresponds to translation surfaces were we label exactly oneseparatrix for each singularity, accordingly to the degree of the singularity.As we will see, the spaces C ( F k ) (resp. C ( F sat ) ) will appear naturally in the study ofreduced (resp. labeled) Rauzy classes. We will then reduce the problem to the study of thespace C ( F comp ) .Note that the space C ( F comp ) was also introduced recently by Marchese in [Ma10] (Sec-tion 3.2).3.2. Moduli space of reduced suspension data.
The set of suspension data associatedto a labeled or reduced permutation is connected (in fact, convex). Hence, for a Rauzyclass R , all flat surfaces obtained from the Veech construction are in the same connectedcomponent of a stratum of the moduli space of Abelian differentials. In fact, according toRemark 2.7 there is a natural map: Φ : H R = (cid:26) ( π, ζ ) , π ∈ R,ζ susp. dat. for π (cid:27) / R → C ( F k )[( π, ζ )] (cid:0) X ( π, ζ ) , l ( π, ζ ) (cid:1) Example . Let us consider the permutations given in Example 2.4. We have: • For τ n = ( ... nn n − ... ) , the corresponding connected component is H hyp ( n − or H hyp ( n − − , n − − depending on the parity of n . • For π n = (cid:0) ... n − nn n − ... (cid:1) , the corresponding connected component is H hyp (0 , n − or H hyp (0 , n − − , n − − , the singularity which is marked by the Veech con-struction being of degree 0. • For π = ( ) , the corresponding stratum is H (1 , , ,
1) = H (1 ) and isconnected.When R is a Rauzy class of reduced permutations, then the following theorem was provenin [Boi09]. Theorem 3.2.
The map Φ is a homeomorphism on its image. The complement of the imageof Φ is contained in a codimension 2 subset of C ( F k ) , which is connected. Moduli space of labeled suspension data.
The previous section describes whatrepresents “geometrically” a reduced Rauzy diagram (or more precisely, the correspondingmoduli space of suspension data): a suspension data for a reduced permutation, modulo theRauzy–Veech induction corresponds to a translation surface with a marked separatrix, i.e. an element of F k ( C ) . n this section, we give an analogous description of a labeled Rauzy diagram. In this case, asuspension data for a labeled permutation, modulo the Rauzy–Veech induction correspondsto a translation surface where all the separatrices are labeled, i.e an element of C ( F sat ) .Then, we will show next that we can reduce the problem to studying the moduli space oftranslation surfaces with a single marked separatrix for each singularities. l l l l Figure 2.
A framing of a surface issued from the Veech construction. Herewe have l i = F ( i ) for i = 1 , , , and l = l .Let π = ( π t , π b ) be a labeled permutation and let ζ be a suspension data for π . The zip-pered rectangle construction naturally defines a framed translation surface (see Figure 2) inthe following way: for each rectangle in the Veech construction, the left vertical side containsa unique singularity. Hence, we can label the corresponding outgoing horizontal separatrixwith the letter of the rectangle. Furthermore, two of these rectangles (corresponding to π − t (1) and π − b (1) ) intersect the corresponding singularity at a left corner (the bottom forone, the top for the other), and the corresponding horizontal outgoing separatrix is the same,so is labeled twice: once by the symbol π − t (1) , and once by the symbol π − b (1) . For all theother rectangles, the singularity on the left is in the interior of the left vertical side, hence,each corresponding separatrix is uniquely labeled. Therefore one gets a one-to-one map: F : A ′ → S X were A ′ is the quotient of A by the equivalence relation π − t (1) ∼ π − b (1) .The element F ( { π − t (1) , π − b (1) } ) will be refered as the doubly labeled separatrix. Lemma 3.3.
Let ( π, ζ ) as previously. The framed translation surface constructed as beforefrom ( π, ζ ) and the one defined by R ( π, ζ ) are the same element in C ( F sat ) .Proof. This is an elementary check. (cid:3)
The following proposition transforms the initial combinatorial question into a topologicalone on the moduli space of Abelian differentials.
Proposition 3.4.
There is a natural one-to-one correspondence between labeled Rauzy classesand connected components of the moduli space of framed translation surfaces C ( F sat ) . Thedegree of the mapping from a labeled Rauzy diagram to the reduced one is then precisely thedegree of the natural mapping from a connected component of C ( F sat ) to C ( F k ) . roof. Let R all be the set of labeled permutations that corresponds to C ( F k ) , and let: H R all = { ( π, ζ ) , π ∈ R all , ζ suspension data for π } / R . By Lemma 3.3 there is a map Φ sat : H R all → C ( F sat ) such that the following diagramcommutes. H R all Φ sat −−→ C ( F sat ) ↓ p ↓ p H R Φ −−→ C ( F k ) Where p is the canonical map that replace a labeled permutation by a reduced one, and p is the map that “forget” all labels except for the doubly labeled separatrix. The maps Φ and Φ sat are homeomorphisms on their images, and onto up to codimension 2 subsets (see[Boi09], Section 2 for details.). Hence, H R all and C ( F sat ) have the same number of connectedcomponents, and the degree of the maps p and p , when restricted to a connected componentare the same. But the degree of the map p restricted to a connected component is preciselythe degree of natural map from the labeled Rauzy diagram to the reduced one. (cid:3) Moduli space of translation surfaces with frame.
There are obvious invariants for the connected components of the moduli space C ( F sat ) .Indeed, two elements of C ( F sat ) that are in the same connected component must satisfy thefollowing property: • The labels that correspond to a given singularity on one surface must correspond toa same singularity on the other surface. • The canonical cyclic order on the set of labels obtained by rotating clockwise arounda singularity must be the same.Hence, a connected component of C ( F sat ) is clearly isomorphic to a connected componentof C ( F comp ) Let α ∈ A k be a label associated to a degree k singularity. There is a natural covering p α from C ( F comp ) to C ( F k ) obtained by “forgetting” all the markings, except the one thatcorresponds to α . The following proposition summarizes the discussion of this section, andis equivalent to Proposition 1.2. Proposition 3.5.
Let R lab be a labeled Rauzy class and R be the corresponding reduced one.Let k be the degree of the marked singularity associated to R . The ratio | R lab || R | equals the degreeof the canonical projection p α : C ( F comp ) → C ( F k ) , restricted to a connected component of C ( F comp ) , where α is a label associated to a degree k singularity.Proof. It was proven in [Boi09] that C ( F k ) is connected. A connected component of C ( F comp ) is naturally isomorphic to a connected component of C ( F sat ) . Then, we just apply Proposi-tion 3.4. (cid:3) Topological invariants for framed translation surfaces
From now, a framed translation surface will be an element in C ( F comp ) .As seen in Proposition 3.5, the formula given in Theorem 1.1 is related to the number ofconnected components of C ( F comp ) . Also, the degree of the covering C ( F comp ) → C , restrictedto a connected component of C ( F comp ) , is clearly Π ri =1 n i !( k i +1) ni c , were c is the number of onnected component of C ( F comp ) , since Π ri =1 n i !( k i + 1) n i is the number of possible frame F ∈ F comp on a surface.In this section, we give lower bounds on the number of connected components of C ( F comp ) .There are two cases. • The “hyperelliptic case”. If the corresponding surfaces are all hyperelliptic and havetwo singularities of degree g − , with possibly some added regular marked points.Then, C ( F comp ) cannot be connected due to the extra symmetries of the underlyingtranslation surfaces. • The ”odd singularity case”. When there are odd degree singularities, we can defineon C ( F comp ) a topological invariant which generalizes the well known spin structureinvariant for the moduli space of Abelian differentials, found by Kontsevich andZorich.Recall that a Riemann S surface is hyperelliptic if there exists an involution τ such that S/τ = CP . Since CP does not have any nontrivial Abelian differential, then for anytranslation surface ( S, ω ) such that S is hyperelliptic the corresponding involution τ satisfies τ ∗ ω = − ω . In particular, this means that the translation surface ( S, ω ) have an isometricinvolution which reverse the vertical direction. Kontsevich and Zorich have shown thatfor each genus g ≥ , there are exactly two strata that contain a connected componentwhich consists only of hyperelliptic translation surfaces. These are the strata H (2 g − and H ( g − , g − . Of course, for each strata, one can also define new ones by adding regularmarked points on the surfaces. Proposition 4.1.
Assume that C consists only of hyperelliptic translation surfaces with twosingularities of degree g − and n regular marked points. Then, C ( F comp ) has at least g connected components.Proof. Let X ∈ C ( F comp ) . We denote by l and l the marked separatrices associated to thedegree g − singularities, and we denote by P i the singularity corresponding to l i .The hyperelliptic involution interchanges P and P . Hence, there is a well defined (incom-ing) separatrix l ′ adjacent to P which is the image of l . The angle θ between l ′ and l isan odd multiple of π and is constant under continuous deformations of X inside the ambientstratum. Note that the value of θ does not depend on any choice. Hence the value of θ is aninvariant of the connected component of C ( F comp ) . Since all the values π, π, . . . , (2 g − π are possible, we see that the number of connected components of C ( F comp ) is at least g . (cid:3) Proposition 4.2.
Assume that C consists of translation surfaces with at least one odd degreesingularity. Then, C ( F comp ) has at least connected components. We postpone the proof of this proposition to the end of this section. We first define the“spin structure” invariant for C ( F comp ) .Let X be a completely framed surface with at least one odd degree singularity. Note thatthe number of odd degree singularities of X is necessarily even, since the sum of the degree ofthe singularities must be equal to g − by the Riemann-Roch formula. For each singularity,we have given a name α ∈ A to a horizontal outgoing separatrix. Now let us fix a total orderon the finite alphabet A , so that the marked separatrices are naturally ordered. This orderinduces an oriented pairing of the separatrices corresponding to odd degree singularities.Now let ( l , l ) be such a pair. We rotate the first separatrix clockwise by an angle π/ , andthe second one counterclockwise by an angle π/ . We obtain pairs of vertical separatrices, +1 l − a b c d ee d c b ae d c b aa b c d e e d c b aa b c d e Figure 3.
Building a surface with only even degree singularities.the first one being on the positive direction, the second one on the negative direction. Wedenote by ( l +1 , l − ) this pair of positive/negative vertical separatrices, let us also denote by k , k the degree of the corresponding zeroes. According to Hubbard–Masur [HM78], thereexists a (smooth) path ν transverse to the horizontal foliation which starts being tangent to l +1 and ends being tangent to l − . Now we consider the following surgery: we cut the surface X along the path and paste in a “curvilinear parallelogram” with two small horizontal sidesand two opposite sides that are isomorphic to ν (see Figure 3). Then, gluing together thehorizontal sides of the parallelogram, one obtains a translation surface where the pair ofsingularities corresponding to l +1 , l − have become a singularity of degree k + k + 2 , whichis even. We will refer to this construction as the parallelogram construction with parameters ( l , l ) .Then, we apply this procedure on all the pairs of vertical separatrices that were definedpreviously. The resulting translation surface only has even singularities and is of genus atleast 3, since the minimal genus case corresponds to starting from H (1 , and ending in H (1 + 1 + 2) .Recall that the strata of the moduli space of Abelian differentials corresponding to onlyeven degree singularities are not connected as soon as the genus is greater than or equal to3, and are distinguished by the parity of spin structure. We will prove the following lemma: Lemma 4.3.
The connected component of the resulting surface in the previous constructiondoesn’t depend on the chosen paths. roof. Up to a small deformation of the surface X , one can assume that it is obtained bythe Veech construction starting from a data ( π, λ, τ ) . Then, for a pair l , l of separatrices aspreviously, the surface obtained after the parallelogram construction with parameters l , l also arises from the Veech construction, where the corresponding permutation is obtainedfrom π by adding a new label on the top before the symbol corresponding to l and the samelabel on the bottom before the label corresponding to l . For instance, in Figure 3, the labeledpermutation ( a b c d ee d c b a ) becomes ( a b c d ee d c b a ) , since l corresponds to c and l corresponds to b . In particular, the permutation after removing all the odd degree singularities doesn’tdepend on the choices of the paths, but only on the order that we have chosen on A . (cid:3) Proof of Proposition 4.2.
We just need to show that the spin structure invariant definedbefore reaches all the possible values. First we recall Kontsevich–Zorich formula for the parityof spin structure for a translation surface X ′ of genus g ′ with only even degree singularities.Let a , b , . . . , a g ′ , b g ′ be a collection of closed paths that represent a symplectic basis of thehomology H ( X ′ ; Z ) , and such that each path does not pass trough any singularity. We canassume that the a i , b i are parametrized by the arc length. For each a i (resp. b i ), we defineind ( a i ) (resp. ind ( b i ) ) to be the index of the map S → S , t a ′ i ( t ) (resp. t b ′ i ( t )) ).Then, the parity of spin structure of X ′ is defined by the following formula: g ′ X i =1 ( ind ( a i ) + 1)( ind ( b i ) + 1) mod 2 The result does not depend on the choice of the symplectic basis and is therefore an invariantof connected components of the strata of the moduli space of Abelian differentials (see[KZ03]).In the definition of the invariant for C ( F comp ) , we successively glue together some pairsof odd degree singularities. We can also glue all pairs except one and therefore we canassume that there is only one pair ( P , P ) of odd degree singularities, of degree k and k respectively, on the surface.We present such surface X as coming from the Veech construction with parameters ( π, λ, τ ) . Let g be the genus of this surface. As in Figure 3, we have a pair l +1 , l − ofpositive/negative vertical separatrix, we choose a path γ transverse to the horizontal folia-tion. There exists a collection of closed paths a , b , . . . , a g , b g that do not intersect γ andthat represent a symplectic basis of the homology H ( X, Z ) . Let also be a a small circlearound the singularity P .When doing the parallelogram construction with parameters l , l using γ , the closed paths a i , b i persists and also the path a . Considering a path isometric to γ inside the parallelogram,one obtains a closed path b , that intersect a only once, and that does not intersect a i , b i for all i ∈ { , . . . , g } . Hence, one gets symplectic basis on the homology of the newly builtsurface, that can be used to compute the corresponding parity of spin structure. Here, aswe will see later, the only relevant data are the indices of a and b . We clearly have: • ind ( a ) = k + 1 mod 2 = 0 mod 2 . • ind ( b ) = 0 . +1 l − l − γ a b a b ′ a Figure 4.
Building two surfaces with different spin structureNow we start again from the surface X and replace the separatrix l by the separatrix l ,obtained by rotating l by the angle π . Then, we do the parallelogram construction withparameters ( l , l ) . We consider the following symplectic basis on the resulting flat surface: • The path a i , b i , for i ∈ { . . . g } which persist under this construction. • The path a , which also persists under this construction. • A path b ′ obtained as in Figure 4The indices of a , a , . . . , a g and of b , . . . , b g are the same as previously, but ind ( b ′ ) = 1 .Since, ind ( a ) + 1 = 1 mod 2 , the surface obtained from ( l , l ) has a different parity ofspin structure as the one obtained from ( l , l ) . Hence, the two corresponding flat surfacesare in different connected components of the moduli space of Abelian differentials. Thisproves that C ( F comp ) has at least connected components. (cid:3) Number of connected components of C ( F comp ) In the previous section, we have used topological invariants to find lower bounds on thenumber of connected components of C ( F comp ) . Here, we show that they are the exact values.5.1. Three elementary surgeries.
Here we describe some elementary closed paths in C that lift to unclosed paths in C ( F comp ) .Recall that a saddle connection γ joining two distinct singularities is simple if there existsno other saddle connection homologous to γ . In particular, it means that up to a smalldeformation of the surface in the ambient stratum, there is no other saddle connection inthe surface parallel to γ . Then, deforming suitably the surface with the Teichmüller geodesicflow (see [Boi08, Boi09] for instance), one gets a surface for which the saddle connectioncorresponding to γ is very short compared to the other ones. Then, one can show that uch surface is obtained by the breaking up a zero surgery (see [EMZ03]). We give a shortdescription of this surgery. εε εεεε ε − δε − δ ε − δε − δε + δε + δ π π + 4 π∂V ε Figure 5.
Local surgery that break a zero of degree k + k into two zeroesof degree k and k respectively. Breaking up a singularity.
Let k , k be the degree of the zeroes that are the endpoints of γ . We start from a zero P of degree k + k . The neighborhood V ε = { x ∈ X, d ( x, P ) ≤ ε } of this conical singularity is obtained by considering k + k ) + 2 Euclidean half disks ofradii r and gluing each half side of them to another one in a cyclic order. We can break thezero into two smaller one by changing continuously the way they are glued to each other asin Figure 5. Note that in this surgery, the metric is not modified outside V ε . In particular,the boundary ∂V ε is isometric to (a connected covering) of an Euclidean circle. Note that isthis construction, we can “rotate” the two singularities by an angle θ by cutting the surfacealong ∂V ε , rotating V ε by an angle θ and regluing it. Move 1.
Let X be a framed surface and let P , P be two distinct singularities of degree k , joined by a simple saddle connection γ . We deform slightly the surface so that no saddleconnection is parallel to γ . Then, using the Teichmüller geodesic flow, we contract the saddleconnection γ until it is very small compared to any other saddle connection. So the newsurface X ′ is obtained by breaking a zero of degree k into two zeroes P ′ and P ′ of degree k . Now we continuously rotate these two zeroes by the angle θ = (2 k + 1) π . The resultingunframed surface is the same as X ′ , but this procedure interchanges P ′ and P ′ . Then, wecome back to the initial surface X , but the labeled zeroes P and P have been interchanged.The labels on the separatrices adjacent to the other singularities have not changed.The projection of this move in C is a closed path. This move in C ( F comp ) interchanges P and P , and fixes the separatrices adjacent to the other singularities.The idea of this previous move is, as we will see, to authorize us to do any degree preservingpermutation on the set of simply marked singularities. This explains the terms n i ! in theformula of Theorem 1.1. Now the next two moves will fix the labeled singularities and changemarked the outgoing separatrices. Move 2.
Let X be a framed surface and let P , P be two distinct singularities of degree k and k respectively, joined by a simple saddle connection γ . Here we do not assume that k = k . We perform the same as in Move 1, but we turn P ′ and P ′ by ( k + k + 1)2 π instead.This move clearly corresponds to a closed path in C . It also preserves P , P pointwise.Let us look how changes the marked separatrices. For this, we can fix once for all a marked eparatrix for all the singularities. Then, for a singularity of degree k , we can identify the setof corresponding horizontal separatrices to Z / ( k + 1) Z by ordering them counterclockwise.We have the following lemma. Lemma 5.1.
Let l ∈ Z / ( k + 1) Z and let l ∈ Z / ( k + 1) Z be the separatrices associated to P and P . Then, Move 2 acts on the set of separatrices in the following way: • l becomes l − k mod k + 1 • l becomes l − k mod k + 1 • All the other labeled separatrices remain unchangedProof.
Note that it is enough to prove this lemma in the case when P , P is obtained afterbreaking up a singularity. The last statement of the lemma is obvious by construction: wedo not change the metric outside a small neighborhood of γ .Now we look at the surgery, keeping track of the labeled separatrices. When turningcontinuously the set V ε by an angle θ , one must simultaneously change the separatrices byan angle − θ , so that they stay horizontal. So at the end, they have moved by the angle − ( k + k + 1)2 π each, so for i = 1 , , l i is replaced by l i − ( k + k + 1) modulo k i + 1 , whichgives the result. (cid:3) Note that this move is especially useful when k = k . Then, the corresponding transfor-mation is ( l , l ) ( l + 1 , l + 1) .Before describing the last move, we first describe a surgery which is analogous to the onepresented in [KZ03]. Bubbling r handles: We start from a singularity of degree p ∈ { , } . Let us consider asmall polygonal line L with no self intersection starting from the singularity. Let r be thenumber of segments s , . . . , s r of L . We consider r parallelograms, each one having a pairon sides parallel to one of the s i . Then, we cut the surface along each s i and paste in thecorresponding parallelogram, and we glue by translation each remaining opposite sides ofeach parallelogram. We obtain a translation surface of genus g ( X ) + r , and the degree psingularity have been replaced by a degree p + 2 r singularity. Note that this surgery can beperformed without changing the metric outside a small neighborhood of the singularity ofdegree p . Note that we can “rotate” the construction in the following way: the surgery isperformed inside a ε neighborhood V ε of the initial singularity of degree p . The boundary ∂V ε remains a metric covering of a euclidean circle after bubbling the handles. Now we cancut the surface X along this circle and reglue it after a rotation by θ .Now we can describe the last move. Move 3.
Assume that the translation surface X was obtained after bubbling r handles andlet P be the corresponding singularity. We continuously rotate the construction as explainedpreviously, by a angle of ( p + 2 r + 1)2 π .As before, the underlying surface in C is the same after Move 3 and any separatrix thatdoes not correspond to the singularity P remains unchanged. Lemma 5.2.
Let l ∈ Z / ( p + 2 r + 1) Z be the separatrix corresponding to P . Then, Move 3changes l in the following way: • If p = 0 , l is replaced by l − . If p = 1 , l is replaced by l − .Proof. It is easy to see that, as in the case of Move 2, a marked separatrix attached tothe singularity is changed by the transformation l l − ( p + 1) mod p + 2 r + 1 , and theseparatrices associated to the other singularities remain unchanged. (cid:3) In particular if p = 0 , we reach all possible separatrices adjacent P in this way. If p = 1 ,then p + 2 r + 1 is even, and we reach only half of the separatrices adjacent to P in this way.5.2. Generating the monodromy group.Proposition 5.3.
Assume that C contains nonhyperelliptic surfaces, then the following holds: • The set C ( F comp ) is connected if all the singularities have even degree • The set C ( F comp ) has two connected components otherwise. We start with the following lemma.
Lemma 5.4.
Let C be a connected component of H ( k n , . . . , k n r r ) . Choose an ordering onthe set with multiplicities { k , . . . , k , . . . , k r , . . . , k r } . There exists X ∈ C and a polygonalline in X that consists of simple saddle connections and that joins all the singularities of X in that order.Proof. The proof is the same as the proof of Proposition 3.5 in [Boi09]. (cid:3)
Proof of Proposition 5.3.
We first assume that there exists odd degree singularities in theunderlying stratum. Since we assume that it is not a hyperelliptic stratum, it is connected(see [KZ03]). We write this stratum as H ( k n , . . . , k n s s , (2 k ′ ) β , . . . , (2 k ′ s ′ ) β s ′ ) , with s + s ′ = r ,and k , . . . , k s are odd.Now we start from a surface in H ( k n − , , . . . , k n s − s , (i.e. we don’t take the even degreesingularities and we replace one singularity of each odd degree by a singularity of degree one).From the previous lemma, we can assume that there is a polygonal path of simple saddleconnections which has no self intersection and that joins successively all the singularities inthe following order: • first the singularities of degree k , • then, a singularity of degree 1, • then, the singularities of degree k , • then, a singularity of degree 1, • and so on . . .Now for each singularity of degree that ends a group of singularities of degree k i , webubble ( k i − / handles as in section 5.1. This replace the singularity of degree by asingularity of degree k i . Note that the polygonal path of simple saddle connections persistsunder this surgery. We will denote by γ this polygonal line.Now for each i ∈ { , . . . , s ′ } , we consider a polygonal line γ i joining β i regular points,such that the paths γ, ( γ i ) i have no intersection points. Then, for each vertex, we bubble n i handles. We obtain s ′ chains of simple saddle connections that join each collection of singular-ities of degree k ′ i . The resulting surface is therefore in H ( k n , . . . , k n s s , (2 k ′ ) β , . . . , (2 k ′ s ′ ) β s ′ ) ,which is the stratum that we study.Now using Move 1, we see that for any polygonal line of simple saddle connections joiningsingularities with the same degree, we can perform any transposition of two consecutivesingularities. Hence we can arbitrarily permute the singularities sharing the same degree. sing Move 3, we see that we can reach any choice of separatrices for the even degreesingularities.Now we consider the chain γ of simple saddle connections joining all the separatrices of odddegree, that was constructed before. The first n vertices of γ makes a chain of singularitiesof degree k . Let us name the singularities P , , . . . , P ,n according to the order given bythe polygonal path γ . If n > , we reach any choice of separatrices for P , , . . . , P ,n − byapplying successively Move 2 on the pairs ( P ,i , P ,i +1 ) for i ∈ { , . . . , n − } , in order tochoose arbitrarily a labeled separatrix of P ,i . Note that once this is done for some i , the nextmoves don’t change the marked separatrix corresponding to P ,i . Then, for the singularity P ,n , we use Move 3 to rotate the corresponding separatrix by any even number (recallthat the set of outgoing separatrices corresponding to a singularity of degree k is naturallyidentified with Z / ( k + 1) Z ) ). If P ,n is not the end of γ , i.e. the polygonal line γ continuesto some other (odd) degree singularity P , , then Move 2 on the pair P ,n , P , will act onthe marked separatrice of P ,n as l l − k . Hence, it will be changed by a odd number,so in combination with Move 3, we obtain all possible choices.If we iterate this procedure until the last singularity of degree k s , we see that we can reachany choice of separatrices for the singularities of odd degree, except the last one of the chainwhere we obtain only half of the possibilities. This proves the proposition in the case whenthere exists odd singularities.If there does not exists any singularity of odd degree, the procedure described above works(with γ = ∅ ) as soon as we can find a surface like above in the connected component that westudy. But in this case, the corresponding stratum of translation surfaces is not connected.Consider a translation surface obtained from a torus with the "bubbling r handles" con-struction. We can easily show that in this case, each singularity contribute to zero to thespin structure. See Figure 6. Hence the resulting parity of spin structure is the same as forthe flat torus, which is odd. a b a b a b a a a Figure 6.
The “bubbling r -handles standard construction. We have ind ( a i ) =1 , so the collection ( a i , b i ) i ∈{ ...β } contributes to for the spin structure. f there exists a singularity of degree k ≥ . It is easy to see that one can slightly changethe construction to make this singularity contribute to 1 to the spin structure, and obtain asurface with even spin structure (see Figure 7) a b a b a b a a a a a Figure 7.
A slight change in the r -handle construction, for r ≥ changesthe spin structure, since in this case, Ind ( a ) = 0 mod 2 .The last remaining case to see is when all the singularities are of degree , and the parityof spin structure is even. If there are exactly singularities, then the connected component is H even (2 ,
2) = H hyp (2 , , which is a hyperelliptic case. If there are at least singularities. Wecan find a surface with a chain of simple saddle connections joining all the singularities, andsuch that the last element of the chain is obtained by the "bubbling a handle" construction.Then, combining Move 2 along the chain, and Move 3 at the end of the chain, we obtainthat C ( F comp ) is connected. This concludes the proof of the proposition. (cid:3) Proof of Theorem 1.3.
The nonhyperelliptic case is given by Proposition 5.3. For the hyper-elliptic case, the lower bound on the number of connected components is given by Proposi-tion 4.1, and the upper bound is easy and left to the reader. (cid:3)
Proof of Theorem 1.1.
Recall that we denote by H ( k n , . . . , k n r r ) the ambient stratum of themoduli space of Abelian differentials. The degree of the covering C ( F comp ) → C , restrictedto a connected component of C ( F comp ) , is clearly Π ri =1 n i !( k i +1) αi c , were c is the number ofconnected component of C ( F comp ) and is given by Theorem 1.3.Let k be the degree of the marked singularity associated the Rauzy class R , and let n be the number of singularities of degree k . The set C k is connected and the degree of theprojection C k → C is n ( k + 1) . Hence we have: | R lab || R | = Π ri =1 n i !( k i + 1) α i c. ( k + 1) .n Which gives Theorem 1.1. (cid:3) ppendix A. Rauzy classes for quadratic differentials
Half-translation surfaces are a natural generalization of translation surfaces. They aresurfaces with an atlas such that the changes of coordinates are not only translations, butcan also be half-turns. They corresponds to Riemann surfaces with quadratic differentials .Danthony and Noguiera have generalized interval exchange transformations and Rauzyinduction to describe first return maps of nonoriented measured foliations on transversesegment. One gets linear involutions , see [DN90].The relation between quadratic differentials and linear involutions was described by theauthor and Lanneau in [BL09].One can wonder if there is a analogous result for Rauzy classes appearing in this context.There doesn’t seem to be a natural relation between "labeled generalized permutations"and framed half-translation surfaces. In particular, there is no “quadratic” equivalent ofLemma 3.3.Numerical experiments on SAGE, suggest that the ratio between a labeled Rauzy classand its corresponding reduced one is always n ! or n !2 , were n is the number of underlyingintervals, which is generally much more than the Abelian case. In particular, using Rauzyinduction and labelling the intervals, these numerical experiments suggests that one obtaineither any renumbering of the intervals, or any even renumbering of the intervals dependingon the stratum.One can also look at extended Rauzy classes , where we add Rauzy moves that correspond tocutting on the left of the interval (see [KZ03]). In this case, numerical experiments suggeststhat the ratio is also n ! or n ! / depending on the stratum. References [AGY06]
A. Avila, S. Gouëzel and
J.-C. Yoccoz – “Exponential mixing for the Teichmüller flow ”,
Publ. Math. IHES (2006), pp. 143–211.[AV07]
A. Avila, and
M. Viana – “Simplicity of Lyapunov spectra: proof of the Zorich-Kontsevichconjecture ”,
Acta Math. (2007), no. 1, pp. 1–56.[Boi08]
C. Boissy – “Degenerations of quadratic differentials on CP ”, Geometry and Topology (2008)pp. 1345-1386[BL09] C. Boissy , and
E. Lanneau – “Dynamics and geometry of the Rauzy-Veech induction for qua-dratic differentials ”,
Ergodic Theory Dynam. Systems (2009), no. 3, pp. 767–816.[Boi09] C. Boissy – “Classification of Rauzy classes in the moduli space of quadratic differentials. ”, preprint arXiv:0904.3826 (2009).[BL10]
C. Boissy , and
E. Lanneau – “Pseudo-Anosov homeomorphisms on translation surfaces inhyperelliptic components have large entropy ”, preprint arXiv:1005.4148 (2010).[Bu06]
A. I. Bufetov – “Decay of correlations for the Rauzy-Veech-Zorich induction map on the spaceof interval exchange transformations and the central limit theorem for the Teichmüller flow on themoduli space of abelian differentials”,
J. Amer. Math. Soc. (2006), no. 3, pp.579–623[DN90] C. Danthony, and
A. Nogueira –“Measured foliations on nonorientable surfaces”,
Ann. Sci.École Norm. Sup. (4) (1990), pp. 469–494.[Del10] V. Delecroix – “Cardinality of Rauzy classes ”, preprint (2010).[EMZ03]
A. Eskin , H. Masur, and
A. Zorich – “Moduli spaces of Abelian differentials: the principalboundary, counting problems, and the Siegel–Veech constants ”.
Publ. Math. IHES (2003),pp. 61–179.[HM78] J. Hubbard, H. Masur – “Quadratic differentials and foliations”,
Acta Math. , (1979), pp. 221–274. Ke85]
S. P. Kerckhoff – “Simplicial systems for interval exchange maps and measured foliations”,
Er-godic Theory Dynam. Systems (1985), no. 2, pp.257–271.[KZ03] M. Kontsevich, and
A. Zorich – “Connected components of the moduli spaces of Abeliandifferentials with prescribed singularities”,
Invent. Math. (2003), no. 3, pp. 631–678.[Ma10]
L. Marchese – “Khinchin type condition for translation surfaces and asymptotic laws for theTeichmüller flow ”, preprint (2010).[MMY05]
S. Marmi, P. Moussa and
J.-C. Yoccoz – “The cohomological equation for Roth type intervalexchange transformations”,
Journal of the Amer. Math. Soc. (2005), pp. 823–872.[Ma82] H. Masur – “Interval exchange transformations and measured foliations”,
Ann of Math. (1982) pp. 169–200.[Ste09]
W. A. Stein et al – “
Sage Mathematics Software (Version 4.2.1) ”, The Sage DevelopmentTeam, 2009, .[Rau79]
G. Rauzy – “Échanges d’intervalles et transformations induites”,
Acta Arith. (1979), pp. 315–328.[Vee82] W. Veech – “Gauss measures for transformations on the space of interval exchange maps”,
Ann.of Math. (2) (1982), no. 1, pp. 201–242.
Université Aix-Marseille 3 LATP, case cour A, Faculté de Saint Jérôme Avenue Es-cadrille Normandie-Niemen, 13397 Marseille cedex 20
E-mail address : [email protected]@latp.univ-mrs.fr