LENGTHS SPECTRUM OF HYPERELLIPTIC COMPONENTS

We propose a general framework for studying pseudo-Anosov homeomorphisms on translation surfaces. This new approach, among other consequences, allows us to compute the systole of the Teichmüller geodesic flow restricted to the hyperelliptic connected components of the strata of Abelian differentials, settling a question of [Far06]; This is the first description of the systole in any component of any stratum outside of genus two and three. We stress that all proofs and computations can be performed without the help of a computer. As a byproduct, our methods give a way to describe the bottom of the lengths spectrum of the hyperelliptic components and we provide a picture of that for small genera.


Introduction
Every affine pseudo-Anosov map φ on a half-translation surface S has an expansion factor λ(φ) ∈ R recording the exponential growth rate of the lengths of the curves under iteration of φ. The set of the logarithms of all expansion factors (when fixing the genus g of the surfaces) is a discrete subset of R: this is the lengths spectrum of the moduli space M g (for the Teichmüller metric).
There are other lengths spectra spec(H) or spec(C) ⊂ spec(Mod g ) for various subgroups H < Mod(g) of the mapping class group, or for various connected components C of the strata of the moduli spaces of quadratic differentials.
These objects have a long history and have been the subject of many recent investigations (see e.g. the work of McMullen [McM00], Farb-Leininger-Margalit [FLM08], Agol-Leininger-Margalit [ALM16]). Different approches have be used in order to tackle these questions, thanks to the work by Thurston, McMullen (via train-tracks and hyperbolic 3-manifolds) or by the work of Veech and Yoccoz (via the Rauzy-Veech induction).
In this paper we present a general framework for studying pseudo-Anosov homeomorphisms. This applies to many different connected components of the strata of Abelian and quadratic differentials. Among them the hyperelliptic components are simplest to understand (we refer to the work of Kontsevich and Zorich [KZ03] for a description of these components): they consist entirely of hyperelliptic curves. These components are importants since they are related to the braid groups (or the mapping class groups Mod(0, n)). Recently a classification of affine invariant submanifolds in hyperelliptic connected components was obtained in [Api17]. Apisa proved that there is no primitive affine invariant submanifolds, except perhaps Teichmüller curves. In the present paper, our framework applies to this setting, and completes the geometric description of these connected components. More precisely we will give a very precise description of L(spec(C)) when C = C hyp is a hyperelliptic connected component of the stratum H(2g − 2) or H(g − 1, g − 1). We will establish: Theorem A. The minimum value of the expansion factor λ(φ) over all affine pseudo-Anosov maps φ on a translation surface S ∈ C hyp is given by the largest root of the polynomial Moreover the conjugacy mapping class realizing the minimum is unique.
Thus L(spec(C hyp )) is the logarithm of this largest root. This theorem settles a question of Farb [Far06,Problem 7.5] for hyperelliptic component and gives the very first instance of an explicit computation of the systole for an infinite family of strata of quadratic differentials.
With more efforts, our method also gives a way to compute the other elements of the spectrum Spec(C hyp ) where C hyp ranges over all hyperelliptic connected components, for any genus. As an instance we will also prove (see Theorem 2.3): Theorem B. For any even g, g ≡ 2 mod 3, g ≥ 9, the second least dilatation of an affine pseudo-Anosov maps φ on a translation surface S ∈ H hyp (2g − 2) is given by the largest root of the polynomial Moreover the conjugacy mapping class realizing the minimum is unique.
For small values of g, as an illustration of our construction, we are able to produce a complete description of the bottom of the spectrum of C hyp . For g ≤ 10, the lengths l of the closed Teichmüller geodesics on H hyp (2g − 2) satisfying, say, l < 2 are recorded in Table 1 below. For genus between 4 and 10, we only indicate the number of expansion factors. Obviously our techniques also provide a way to investigate the bottom of the spectrum for the stratum H hyp (g − 1, g − 1) for various g and various bound (not necessarily 2). We also emphasise that our techniques can be applied to a general stratum of quadratic differentials, not necessarily a hyperelliptic connected component. This will be done in a forthcoming work.
Organization of the paper and remarks on the proofs. Our strategy is to convert the computation of mapping classes and their expanding factors into a finite combinatorial problem. This is classical in pseudo-Anosov theory: the Rauzy-Veech theory and the train track theory are now well established. However a major difficulty comes from the very complicated underlying combinatorics. We conclude with quick description of the different steps of the proof. It involves three tools of different nature.
(1) Geometrical part (Sections 3 and 4). We introduce a new construction of pseudo-Anosov homeomorphism producing all such maps: to any non closed path γ (satisfying some conditions) in a Rauzy diagram is associated a pseudo-Anosov homeomorphism φ γ , whose expansion factor θ γ is the leading eigenvalue of a matrix V γ . (2) Dynamical part (Section 5). In view of computing the systole (or, more generally, the bottom of the spectrum) we use the concept of renormalization in order to reduce the range of paths to analyse to a simpler set of paths. (3) Combinatorial part (Section 6). By using the geometry of the Rauzy class, it is possible to further reduce the above set to a finite set Γ g of paths, for every g ≥ 2.
The technical computations was isolated to avoid overloading the main body of the paper. This is the content of Appendix A and Appendix B.
(4) Computation 1 (Appendix A). We compute the matrices V γ and characteristic polynomials P γ for any γ ∈ Γ g by the help of the "Rome" method. (5) Computation 2 (Appendix B). We compare the top roots of each polynomial.
As explained in the abstract, the computations are done without the help of the computer, except perhaps in the last step (Appendix B) for the concrete evaluation of roots for a few polynomials that appear in low genera.
For a first reading, the reader can skip Appendix A and Appendix B and go directly to Appendix D: it provides an elementary proof of Theorem A when n is even (systole of Spec(H hyp (2g − 2))) without any technical computations. However all the main steps of the strategy already reveals all the difficulties.
Appendix A and B are more technical and are mostly used for computing the systole of Spec(H hyp (g−1, g−1)) and the second least element of Spec(H hyp (2g−2)) (these two problems are of the same order of difficulty). In this situation, we also have to deal with the existence of imprimitive matrices V γ .
Acknowledgments. The authors thank Artur Avila and Jean-Christophe Yoccoz for helpful conversations and for asking the question on the systoles. This article would not be possible without the seminal works of Jean-Christophe Yoccoz and Bill Veech, whose mathematics is still a source of inspiration for both authors.
The second author would like to thank Vincent Delecroix and Alex Eskin for stimulating discussions. This collaboration began during the program sage days at CIRM in March 2011, and continued during the program "Flat Surfaces and Dynamics on Moduli Space" at Oberwolfach in March 2014. Both authors attended these programs and are grateful to the organizers, the CIRM, and MFO. This work was partially supported by the ANR Project GeoDyM and the Labex Persyval and CIMI.

Overall strategy
In this section, we state more precisely the main theorems. For a definition of the notions of interval exchange transformations, suspension data, Rauzy-Veech induction, see Section 3 or [MMY05].
2.1. Hyperelliptic connected components. In the sequel, for any integer n ≥ 4, we will consider the hyperelliptic Rauzy diagram D n of size 2 n−1 − 1 containing the permutation π n = 1 2 . . . n n n − 1 . . . 1 and by C hyp n the associated connected component. If n = 2g is even then C hyp n = H hyp (2g − 2) and if n = 2g + 1 is odd then C hyp n = H hyp (g − 1, g − 1). The precise description of these diagrams was given by Rauzy [Rau79]. Equipped with this and the Kontsevich-Zorich [KZ03] classification, one can convert our Main Theorem into Theorem 2.1 below: Theorem 2.1. For any n ≥ 4, the minimum value of the expansion factor λ(φ) over all affine pseudo-Anosov map φ on a translation surface S ∈ C hyp n is given by the largest root of the polynomial Moreover the conjugacy mapping class realizing the minimum is unique.
The same applies to the theorem for the second least expanding factor : Theorem 2.2. For any n ≥ 18, if n ≡ 4 mod 6 is even then the second least element of Spec(C hyp n ) is given by the largest root of the polynomial Moreover the conjugacy mapping class realizing the minimum is unique.
As mentioned above, one possible way to tackle Theorem 2.1 is by the use of the Rauzy-Veech induction. It is now well established that closed loops in the Rauzy diagram D n furnishes pseudo-Anosov maps fixing a separatrix (see Section 3). Hence the "usual" construction is not sufficient to capture all relevant maps. In [BL12], it is shown that the square of any pseudo-Anosov maps fixes a separatrix, up to adding a regular point. The cost to pay is that this produces very complicated Rauzy diagrams and the computation of the precise systole seems out of reach with this method.
We will propose a new construction in order to solve these two difficulties at the same time. The proof of our Main Theorem is divided into three parts of different nature: geometric, dynamical, combinatoric.
2.2. Geometrical part. In this paper we propose a new construction of pseudo-Anosov map, denoted the Symmetric Rauzy-Veech construction. One key of this construction is the following definition.
Definition 2.1. A pseudo Anosov homeomorphism is positive (respectively, negative) if it fixes (respectively, reverses) the orientation of the invariant measured foliations.
The classical Rauzy-Veech construction associates to any closed path γ in the Rauzy diagram a non negative matrix V (γ). If this matrix is primitive (we will also say Perron-Frobenius), i.e. a power of V (γ) has all its coefficients greater than zero, we obtain a positive pseudo-Anosov map φ(γ) with expansion factor ρ(V (γ)).
In Section 4 we will associate to any non-closed path γ in a given Rauzy diagram D connecting a permutation π to its symmetric s(π) (see Section 4), a matrix V (γ). If the matrix is primitive we obtain a negative pseudo-Anosov map φ(γ) with expansion factor ρ(V (γ)).
The converse holds in the following sense (see Theorem 4.4 and Proposition 4.5).
(1) Any affine negative pseudo-Anosov map φ on a translation surface S that fixes a point is obtained by the Symmetric Rauzy-Veech construction.
(2) Any affine pseudo-Anosov φ on a translation surface S ∈ C hyp n is obtained as above, up to replacing φ by τ • φ, where τ is the hyperelliptic involution.
2.3. Dynamical part. From now on, we assume that the underlying connected component is hyperelliptic. The theorem above reduces our problem to a combinatorial problem on graphs. However the new problem is still very complicated since the complexity of paths is too large. To bypass this difficulty we introduce a renormalization process (ZRL for Zorich Right Left induction), analogously to the Veech-Zorich and Marmi-Moussa-Yoccoz acceleration of the Rauzy induction, see Section 5. This allows us to reduce considerably the range of paths to analyze in the Rauzy diagram D n .
The particular shape of D n will be strongly used. We will usually refer to the permutation π n as the central permutation. For any permutation π ∈ D n , there is a unique shortest path from π n to π, that can be expressed as a word w with letters 't' and 'b'. We will often write this permutation as π n .w. The loop in D n composed by the vertices {π n .t k } k=0,...,n−2 will be referred to as the central loop.
In the next, an admissible path is a path in D n from a permutation π to s(π) whose corresponding matrix V (γ) is primitive. The path is pure if the corresponding pseudo-Anosov map is not obtained by the usual Rauzy-Veech construction. This definition is motivated by Proposition 3.3 and the main result of [BL12]: pseudo-Anosov homeomorphism in hyperelliptic connected components with small expansion factors are not obtained by the usual Rauzy-Veech construction.
We show (see Theorem 5.1): Theorem (Dynamical Statement). There exists a map (denoted by ZRL) defined on the set of pure admissible paths on D n satisfying the following property: The paths γ and ZRL(γ) define the same pseudo-Anosov homeomorphism up to conjugacy, and there exists an iterate m ≥ 0 such that γ = ZRL (m) (γ) is an admissible path starting from a permutation π that belongs to the central loop. In addition the first step of γ is of type 'b'.
This results allows us to restrict the computation to paths γ starting from the central loop. As a byproduct one can actually compute all the expansion factors less than two i.e. one can compute explicitly the finite set for small values of g up to 20 (see Table 1).
2.4. Combinatorial part. The last part of the proof deals with the estimates of the spectral radius of all the paths γ starting from the central loop. We first show that this problem reduces to a problem on a finite number of paths (for a given n ≥ 4). Then we show how to compare spectral radius ρ(V (γ)) for various γ starting from the permutation π := π n .t k for some k. At this aim, for a given n ≥ 4, we introduce several notations.
Notation. For n ∈ N we set K n = n 2 − 1 and L n = n − 2 − K n . For any k ∈ {1, . . . , K n } we define the path For any k ∈ {1, . . . , K n } and l ∈ {1, . . . , 2n − 2 − 3k} we define the path Loosely speaking γ n,k is the shortest path from π to s(π) starting with a label 'b', i.e. it is the concatenation of the small loop attached to π labelled by 'b' followed by the shortest path from π to s(π). Similarly, γ n,k,l is a path joining π to s(π) obtained from γ n,k by adding a loop: if l ≤ n − 2 − k then it is a 't' loop based at π n .t k b l , if l ≥ n − 1 − k then it is a 'b' loop based at π n .t l−(n−1−2k) (see Figure 2).
For any path γ in the Rauzy diagram, we will denote by θ(γ) the maximal real eigenvalue of the matrix V (γ). We will also use the notations θ n,k (respectively, θ n,k,l ) for θ(γ n,k ) (respectively, θ(γ n,k,l )). Observe that the matrix V (γ) is not necessarily primitive, but the spectral radius is still an eigenvalue by the Perron-Frobenius theorem. We show (see Proposition 6.1, Lemma 4.2 and Proposition A.3): Theorem (Combinatorial Statement). Let n ≥ 4 be any integer. The followings hold.
If V (γ n,k ) is not primitive, we can also assume that l = n − 1 − k in the previous statement.
. . . Figure 2. Half of the Rauzy diagram. In bold the path γ n,k .

2.5.
Proof of the Main Theorem. Theorems 2.1 and 2.2 will follow from Theorem 2.3, Theorem 2.5 and Theorem 2.4 below.
(2) If n ≡ 4 mod 6 and n ≥ 18 is even then the second least element of Spec(C hyp n ) is log(θ n,Kn−1 ).
Moreover the conjugacy mapping class realizing the minimum is unique.
Letting P n,k the characteristic polynomial of V (γ n,k ) multiplied by X + 1, by definition θ n,k is the maximal real root of P n,k . By Lemma A.1 we have, when n is even: and when n ≡ 4 mod 6 and gcd(n − 1, K n − 1) = 1: that are the desired formulas.
Proof of Theorem 2.3. We will show that L(Spec(C hyp n )) = log(θ n,Kn ). Observe that K n = n/2 − 1 thus gcd(n − 1, K n ) = 1 and the matrix V n,Kn is primitive by the Combinatorial Statement. This shows L(Spec(C hyp n )) ≤ log(θ n,Kn ). A simple computation shows θ n,Kn < 2 (see Lemma A.1). Now by the Dynamical Statement, L(Spec(C hyp n )) = log(θ(γ)) for some path γ starting from π = π n .t k with first step of type 'b'. By the Combinatorial Statement, θ(γ) ≥ θ n,k for some k ∈ {1, . . . , K n }. Thus θ n,k ≤ θ n,Kn . We need to show that k = K n . Let us assume that k < K n .
(2) If gcd(k, n − 1) = d > 1 then by the second point of the Combinatorial Statement, θ n,k = θ n ,k where n = n−1 d + 1 and k = k d . Since n is even, k = K n and gcd(k , n − 1) = 1, the previous step applies and θ n ,k > θ n ,K n . By Lemma B.2 the sequence (θ 2n,K 2n ) n is a decreasing sequence, hence θ n ,K n > θ n,Kn . Again we run into a contradiction. In conclusion k = K n and L(Spec(C hyp n )) = log(θ n,Kn ). By construction and since all inequality above are strict, the conjugacy mapping class realizing this minimum is unique.
We now finish the proof of the theorem with the second least dilatation. The assumption n ≡ 4 mod 6 implies gcd(K n − 1, n − 1) = 1. Hence V n,Kn−1 is primitive and the second least dilatation is less than θ n,Kn−1 . As before, a simple computation shows that θ n,Kn−1 < 2 (see Lemma A.1).
The theorem will follow from the following two assertions, that are proven in Proposition B.8.
Theorem 2.4. If n ≥ 5 and n ≡ 1 mod 4 then L(C hyp n ) = log(θ n,Kn ). By Lemma A.1, when n = 1 mod 4, we have that θ n,Kn is the maximal real root of Proof of Theorem 2.4. We follow the same strategy than that of the proof of the previous theorem. Namely L(C hyp n ) ≤ log(θ n,Kn ) since V n,Kn is irreducible (by the Combinatorial statement).
This finishes the proof of Theorem 2.4.
Proof of Theorem 2.5. We follow the strategy used in the two previous proofs. Namely L(C hyp n ) ≤ log(θ n,Kn,Ln ) since V n,Kn,Ln is irreducible (see the Combinatorial statement where n ≡ 3 mod 4 and L n is odd).
Case 1. l is even. By the Combinatorial statement, one has θ n,Kn,l = θ n ,K n ,l with n = (n + 1)/2, l = l/2 and K n = K n /2. If l < L n then l < L n and Lemma B.7 applies (since n ≥ 4 is even): θ n ,K n ,l > θ n ,K n ,L n .
If l > L n then l = L n +3. Again the Combinatorial Statement gives θ n,Kn,Ln+3 = θ n ,K n ,L n +2 . In all cases, Proposition B.6 applies and: θ n ,K n ,L n > θ n,Kn,Ln running into a contradiction.
Case 2. l is odd. If l < L n then by Proposition B.5 we have θ n,Kn,l > θ n,Kn,Ln . This is again a contradiction. The case l > L n , namely l = L n + 2, is ruled out by Lemma B.4: θ n,Kn,Ln+2 > 2 > θ n,Kn,Ln .
In conclusion l = L n and γ = γ n,Kn,Ln . The Main Theorem is proved.

Rauzy-Veech induction and pseudo-Anosov homeomorphism
In this section, we briefly recall the notions of interval exchange transformations, suspension data, Rauzy-Veech induction, and the associated construction of pseudo-Anosov homeomorphisms. We also provide a slight generalization of these standard notions. See [MMY05,BL12] for references.
3.1. Interval exchange transformation. Let I ⊂ R be an open interval and let us choose a finite partition of I into d ≥ 2 open subintervals {I j , j = 1, . . . , d}. An interval exchange transformation is a one-to-one map T from I to itself that permutes, by translation, the subintervals I j . It is easy to see that T is precisely determined by the following data: a permutation that encodes how the intervals are exchanged, and a vector with positive entries that encodes the lengths of the intervals.
We will attribute a name to each interval I j . In this case, we will speak of labeled interval exchange maps. One gets a pair of one-to-one maps (π t , π b ) (t for "top" and b for "bottom") from a finite alphabet A to {1, . . . , d} in the following way. In the partition of I into intervals, we denote the k th interval, when counted from the left to the right, by I π −1 t (k) . Once the intervals are exchanged, the interval number k is I π −1 b (k) . Then with this convention, the permutation encoding the map T is π b • π −1 t . We will denote the length of the intervals by a vector λ = (λ α ) α∈A .
We will call the pair (π t , π b ) a labeled permutation, and π b • π −1 t a permutation (or reduced permutation). One usually represents labeled permutations π = (π t , π b ) by a table: .
3.2. Suspension data and weak suspension data. The next construction provides a link between interval exchange transformations and translation surfaces. A suspension datum for T = (π, λ) is a vector (τ α ) α∈A such that π b (α)≤k τ α < 0. We will often use the notation ζ = (λ, τ ). To each suspension datum τ , we can associate a translation surface (X, ω) = X(π, ζ) in the following way.
Consider the broken line L t on C = R 2 defined by concatenation of the vectors ζ π −1 t (j) (in this order) for j = 1, . . . , d with starting point at the origin. Similarly, we consider the broken line L b defined by concatenation of the vectors ζ π −1 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 the endpoints, we can construct a translation surface X by identifying each side ζ j on L t with the side ζ j on L b by a translation. The resulting surface is a translation surface endowed with the form dz 2 . 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 rectangle construction, due to Veech [Vee82].
We will need to extend a little the definition of a suspension datum.
Definition 3.1. Let T = (π, λ) an interval exchange map. A weak suspension data for T is an element τ ∈ R A , such that there exists h ∈ R such that: The parameter h above will be called height of the weak suspension datum τ . Note that a usual suspension datum corresponds to the case h = 0: in this case, the first two conditions imply the two others. Figure 3. Zippered rectangle construction for a weak suspension datum Observe that, in a very similar way as for the case of suspension data, one can associate to a pair (π, λ, τ, h) a translation surface X = X(π, λ, τ ) and a horizontal interval I h ⊂ X. One use a small variation of the zippered rectangle construction: let I h be a horizontal interval of length α λ α . For α ∈ A, we consider the rectangle R α of width λ α and of height By definition of weak suspension datum, h α > 0 for each α. Note that, for instance, Condition iii-is needed to insure that h a > 0 for a permutation π of the form ( a * * * * * * a )). Then, we glue the rectangles (R α ) α in a similar way as for the usual zippered rectangle construction. Two values h 1 , h 2 give canonically isometric surfaces X(π, λ, τ ), where the intervals I h 1 and I h 2 differs by a vertical translation (of length h 2 − h 1 ). In other words, there is a immersed Euclidean rectangle of height |h 2 − h 1 | whose horizontal sides are I h 1 and I h 2 . Also, vertical leaves starting from the endpoints of I h satisfy the "classical conditions", i.e. each leaf will hit a singularity before intersecting I h (in the positive or negative direction depending on h and the suspension data). Conversely, let X be a translation surface and I ⊂ X be a horizontal interval with the same classical conditions as above. We assume that there are no vertical saddle connections. In a similar way to the classical case, there exists a (unique) weak suspension datum (π, λ, τ, h) such that (X, I) = (X(π, λ, τ ), I h ). The datum (π, λ) is given by considering the interval exchange T defined by the first return map of the vertical flow on I, h is the time (positive or negative) for which the vertical geodesic starting from the left end hits a singularity. The parameters τ α are obtained by considering vertical geodesics starting from the discontinuities of T and the time where they hit singularities: the corresponding time t k for the k-th discontinuity is then Condition iii-corresponds to the fact that the vertical geodesics starting from the right end of I hits a singularity before intersecting I again (and similarly for Condition iv-).

Rauzy-Veech induction and other Rauzy operations. The Rauzy-Veech induction
R(T ) of T is defined as the first return map of T to a certain subinterval J of I (see [Rau79,MMY05] for details).
We recall very briefly the construction. Following [MMY05] we define the type of When T is of type t (respectively, b) we will say that the label π −1 t (d) (respectively, π −1 b (d)) is the winner and that π −1 b (d) (respectively, π −1 t (d)) is the loser. We define a subinterval J of I by The image of T by the Rauzy-Veech induction R is defined as the first return map of T to the subinterval J. This is again an interval exchange transformation, defined on d letters (see e.g. [Rau79]). The data of R(T ) are very easy to express in term of those of T .
There are two cases to distinguish depending whether T is of type t or b; the labeled permutations of R(T ) only depends on π and on the type of T . If ε ∈ {t, b} is the type of T , this defines two maps R t and R b by R(T ) = (R ε (π), λ ). We will often make use of the following notation: if ε ∈ {t, b} we denote by 1 − ε the other element of {t, b}.
If τ is a suspension data over (π, λ) then we define R(π, λ, τ ) by where ε is the type of T = (π, λ) and V is the corresponding transition matrix. In other terms Remark 3.1. By construction the two translation surfaces X(π, ζ) and X(π , ζ ) are naturally isometric (as translation surfaces).
Remark 3.2. We can extend the Rauzy-Veech operation on the space of weak suspension data by using the same formulas. We easily see that V −1 ζ is a weak suspension data for R ε (π). Also, X(R(π, ζ)) and X(π, ζ) are also naturally isometric as translation surfaces. Now if we iterate the Rauzy induction, we get a sequence (α k , β k ) of winners/losers. Denoting R (n) (π, λ) = (π (n) , λ (n) ), the transition matrix that relates λ (n) to λ is the product of the transition matrices: (1) Now, we define other Rauzy moves that will be used later. Let π be a labeled permutation. .
We define the symmetric of π, denoted by s(π), the following labeled permutation. .
Observe that if τ is a weak suspension datum for (π, λ), then τ is also a weak suspension datum for (s(π), λ) (it is not necessarily true for usual suspension data). For simplicity, we define s(π, λ, τ ) = (s(π), λ, τ ). Also, the translation surfaces X(π, λ, τ ) and X(s(π, λ, τ )) are related by the element −I ∈ SL(2, R). Left Rauzy induction can be defined analogously as the Rauzy induction, by "cutting" the interval on the left. It can also be defined by R L = s • R • s. From the above study, we see that R L preserves weak suspension data, and that X((π, λ, τ )) and X(R L (π, λ, τ )) are naturally isometric as translation surfaces.
3.4. Labeled Rauzy diagrams. For a labeled permutation π, we call the labeled Rauzy diagram, denoted by D(π), the graph whose vertices are all labeled permutations that we can obtained from π by the combinatorial Rauzy moves. From each vertices, there are two edges labeled t and b (the type) corresponding to the two combinatorial Rauzy moves. We will denote by π α,β − − → π for the edge corresponding to R ε (π) = π where ε ∈ {t, b} and α/β is the winner/loser. To each path γ of this diagram, there is thus a sequence of winners/losers. We will denote by V (γ) the product of the transition matrices in Equation (1).
Similarly, one can define the reduced Rauzy diagram D red (π) by considering (reduced) permutations as vertices. There is clearly a canonical map D(π) → D red (π).
3.5. Coordinates on the hyperelliptic Rauzy diagrams. The hyperelliptic Rauzy diagram D n of size 2 n−1 − 1 is the one that contain the permutation When n is even, it corresponds to the hyperelliptic connected component H hyp (n − 2) and when n is odd, it corresponds to the hyperelliptic component H hyp ( n−1 2 , n−1 2 ). The precise description of these diagrams was given by Rauzy [Rau79]. See also [BL12, Section 3.7.1]. An easy corollary is the following proposition.
Proposition 3.1. Let n ≥ 2. For any π ∈ D n there exists a unique shortest path joining π n to π.
For a permutation π in D n as above. We can write the path of the proposition as a unique sequence of n 1 > 0 Rauzy moves of type ε ∈ {t, b}, then n 2 > 0 Rauzy moves of type 1−ε, etc. The sequence of non negative integers n 1 , . . . , n k−1 defines the permutation π once ε is chosen. Observe that replacing the starting Rauzy type ε by 1 − ε changes π = (π t , π b ) by (π b , π t ) up to renaming. For our purpose, we wont need to distinguish these two cases. Observe also that we necessarily have i=1,...,k−1 n i < n − 1.
Definition 3.2. Let π be a permutation in D n and let n 1 , . . . , n k−1 be as above, the coordinates of π are (n 1 , . . . , This definition is motivated by the following easy proposition.
• If k is even then π and s(π) belong to the same component of D n \{π n }.
• If k is odd then π and s(π) belong to a different component of D n \{π n }.
Proof of Proposition 3.2. The proof is straightforward. To see that the last two assertions hold, we can remark that π and s(π) belong to the same connected component of D n \{π n } if and only if the minimal paths (from π n ) to π and s(π) have the same starting Rauzy type.
We end this section by recalling the construction of Veech on pseudo-Anosov homeomorphisms.
3.6. Pseudo-Anosov homeomorphism and Rauzy-Veech induction. We follow Veech's work [Vee82]. To any path γ in the labeled Rauzy diagram, whose image in the reduced Rauzy diagram is closed, one can associate a matrix V as follows. We denote by (π t , π b ) and (π t , π b ) the labeled permutations corresponding to the endpoints of γ. By definition of γ, they both define the same underlying permutation. We associate to it a matrix V as before. Let P be the permutation matrix defined by permuting the columns of the d × d identity matrix according to the maps π t , π t , i.e. P = [p αβ ] α,β∈A 2 , with p αβ = 1 if β = π −1 t (π t (α)) and 0 otherwise. The transition matrix associated to the path γ is then: Observe that V is obtained from C by replacing, for each k ∈ {1, . . . , d}, the column π −1 t (k) by the column π −1 t (k) or for each letter α, the column labelled α by the column labelled Assume now that the matrix V is a primitive (i.e. it as a power with only positive entries). Let θ > 1 be its Perron-Frobenius eigenvalue. We choose a positive eigenvector λ for θ. It can be shown that V is symplectic [Vee82], thus let us choose an eigenvector τ for the eigenvalue θ −1 with τ π −1 t (d) > 0. It turns out that τ defines a suspension data over T = (π, λ). Indeed, the set of suspension data is an open cone, that is preserved by V −1 . Since the matrix V −1 has a dominant eigenvalue θ (for the eigenvector τ ), the vector τ must belong to this cone. If ζ = (λ, τ ), one has, for some integer k Hence the two surfaces X(π, ζ) and g t X(π, ζ) differ by some element of the mapping class group (see Remark 3.1). In other words there exists a pseudo-Anosov homeomorphism φ affine with respect to the translation surface X(π, ζ) and such that Dφ = g t . The action of φ on the relative homology of (X, ω) is V (γ) thus the expanding factor of φ is θ. By construction φ fixes the zero on the left of the interval I and thus it fixes a horizontal separatrix adjacent to this zero (namely, the oriented half line corresponding to the interval I). Conversely: Theorem (Veech). Any pseudo-Anosov homeomorphism (preserving a orientable foliation, and) fixing a horizontal separatrix is obtained by the above construction.
Recall that the main result of [BL12] was based on the following key proposition: . For any n ≥ 2 and any pseudo-Anosov homeomorphism φ that is affine with respect to (X, ω) ∈ C hyp n , if φ fixes a horizontal separatrix attached to a zero of ω, or a marked point, then its expanding factor is bounded from below by 2.

Obtaining all "small" pseudo-Anosov homeomorphisms
The proof of Theorem 2.1 uses a generalization of the Rauzy-Veech construction of pseudo-Anosov homeomorphisms. We explain in Appendix C why a naive generalization of the Rauzy induction does not work. 4.1. Construction of negative pseudo-Anosov maps. The usual Rauzy-Veech construction naturally produces pseudo-Anosov maps that preserve the orientation of the stable and unstable foliation. The proposed generalization produces maps that reverse the unstable foliation.
We consider a non closed path γ in the labeled Rauzy diagram such that its starting point π and its ending point π satisfy s(π) = π up to a relabelling. By a slight abuse of language we say that γ is a path from π to s(π).
As above, we associate to such path a matrix V by multiplying the corresponding product of the transition matrices by a suitable permutation matrix that corresponds to the relabelling between s(π ) and π. As before, V is symplectic, thus let us choose an eigenvector λ for the eigenvalue θ and τ for the eigenvalue θ −1 . It turns out that τ is not necessarily a suspension datum, but it is a weak suspension datum, as we show in the next Proposition.
Proposition 4.1. Let γ by a path in a Rauzy diagram from π to s(π), and let V := V (γ) be the corresponding matrix. We assume that V is primitive. If λ, τ are the eigenvectors as above then • up to multiplying τ by −1, (λ, τ ) is a weak suspension data for π.
• the constructed surface S(π, λ, τ ) admits an affine map φ whose derivative is If τ ∈ ∂C then it still defines a nice translation surface since there is no vertical saddle connection (see [BL09, Figure 14] for a detailed argument). In this case the corresponding horizontal segment contains some singularities and thus [τ ] cannot be fixed by V −1 that is a contraction. Hence V −1 τ = θτ ∈C. Furthermore, θ is the greatest eigenvalue of V −1 , hence of V . We construct the surface (X, ω) = X(π, λ, τ ). If ζ = (λ, τ ), one has, for some integer k 0 θ and the composition φ = f 2 • f 1 gives a pseudo-Anosov map affine on S 1 with θ as expansion factor and that reverse vertical and horizontal foliations. Now we prove that φ has a regular fixed point. Let h be a height of (π, λ, τ ), and I h the corresponding interval. Then, Rauzy-Veech induction gives a subinterval I h of I h , and the image of I h by φ gives a interval I h 2 defining the same weak suspension datum, with a different height (see Figure 4). Hence, there is a map from φ( 4.2. An example. We use here the following convention (see Notation 2.4 and Figure 2): For any k = 1, . . . , K n and any l = 1, . . . , n − 2 − k we define the path Lemma 4.2. Let n ≥ 4 and 1 ≤ k ≤ K n . Set d = gcd(n − 1, k). We denote by n = n−1 d + 1 and k = k d . Then the matrix V n ,k is primitive and θ n,k = θ n ,k . Proof of Lemma 4.2. We first assume that k and n − 1 are relatively prime. We compute the matrix V n,k associated to the path γ n,k . For the sequel, in order to be able to compare the top eigenvalues of the matrices V n,k we will compute them with a labelling depending on k, with alphabet A = {1, . . . , n}. To do this, we start from the central permutation, with the following labelling This is well defined since k and n − 1 are relatively prime. In particular 1 = α k+1 , 2 = α 2k+1 , α 1 = n − 1, and more generally, α i+k = α i + 1 mod n − 1. The starting point of γ k,n is The path γ n,k consists of the Rauzy moves b n−1−k t n−1−2k , hence we have the following sequence of winners/losers: • 1 = α k+1 is successively winner against α k+2 , . . . , α n−1 , n • then, n is successively winner against α k+1 , . . . , α n−1−k . Note that k + 1 < n − 1 − k. also, the first line of the labeled permutation s(π ), where π is the endpoint of γ n,k is In any case, we obtain, α i = α i − 1. Hence the matrix V n,k is obtained from the product of elementary Rauzy-Veech matrices by translating cyclically the first n − 1 columns by 1 on the right. Finally, we have Where for i ∈ {2, . . . , n − 2}, we have: and b i = 1 if and only if a i = 2, and b i = 0 otherwise. Note that α 1 = n − 1, hence a n−1 = 0 = b n−1 . Also, α n−k = n − 2 hence a n−2 = 1 and b n−2 = 0. The matrix V n,k is clearly irreducible (this can be easily seen by inspecting its adjacency graph) and thus primitive since there is a non zero diagonal element.
Now if d = gcd(n − 1, k) > 2, we define α in the following way: • The other α i are chosen in any way. The matrix V n,k , with this labelling, is where the bottom right block is a permutation matrix. This ends the proof of the lemma.

4.3.
Converse of the Symmetric Rauzy-Veech construction. As for the usual Rauzy-Veech construction, we have a converse.
Definition 4.1. Let φ be an affine pseudo-Anosov on S, that that reverse the orientation of the foliations, and with a regular fixed point (with positive index). A curve L is suitable for φ if (1) it is made by a horizontal segment, starting from p, and then followed by a vertical segment, ending at a singular point. We do not allow L to have self-intersections. Proof. The fact that the base segment I =]a, b[ determines an interval exchange transformation T = (π, λ) and a weak suspension datum τ, h for T is similar to the case for classical suspension data, except that the segment is not attached on the left to a singularity. This is left to the reader.
We only need to check that T defines a Rauzy path from π to s(π) such that λ, τ are the corresponding eigenvectors.
The key remark is the following: the horizontal part of φ(L) ∪ φ 2 (L) is a segment I ⊂ I that has the same left end as I. Observe that φ(L) and φ 2 (L) do not have intersection on their interior hence, by a classical argument, the interval exchange transformation T associated to I is obtained from T by applying a finite number of time the Rauzy induction on the right to T . Similarly, for the weak suspension datum, we get (π , λ , τ ) = (π (n) , λ (n) , τ (n) ).
Rotating by 180 • the picture we have, up to relabelling, s(π (n) ) = π, λ (n) = 1/θλ and τ (n) = θτ . This ends the proof. Now we prove the first part of the Geometrical Statement.
Theorem 4.4. Let φ be an affine pseudo-Anosov on S having a regular fixed point P and that reverse horizontal and vertical foliations. Then S is obtained by the above construction.
Proof of Theorem 4.4. By Proposition 4.3, all we need to show is to produce a suitable curve L for φ.
We start from any oriented curve L in X made by a finite horizontal segment, starting from P , and then followed by a finite vertical segment, ending at a singular point. We do not allow L to have self-intersections. Such a curve always exists (e.g. by using the Veech's polygonal representation of translation surfaces). We denote by L x and L y the (oriented) components of L: they bound a rectangle R whose opposite corners are P and a zero σ of ω.
We will perform several operations on L, in order to obtain the required condition. The strategy is the following: • 1st Step: arrange that L bounds an immersed euclidean rectangle i(R).
• 2nd Step: Step: change the fixed point in order to get suitable curve. 1st Step. We assume that h is minimal in the following sense: for each x ∈ L x , the unit speed vertical geodesic starting from x (in the same direction as L y ), does not hit a singularity at a time less than h. Now, let R ⊂ R 2 be the open rectangle of width |L x | and of height h.There is a natural translation map i : R → S that sends the bottom side of R to L x . By the above hypothesis, i(R) does not contain any singularity. Note that i might not be an embedding but just an immersion. However, P is not in i(R), otherwise one easily see that the interior of L x intersects the interior of L y (see also Figure 5).
Assume that h is not minimal, then there is a x 0 ∈ L x whose corresponding vertical geodesic hits a singularity for a time h 0 minimal. We then consider the new oriented curve L 0 , starting from P such that L 0x is the segment joining P to x 0 and L 0y is the vertical segment of length h 0 . Note that L 0 still satisfies the no self intersection hypothesis otherwise we would find an element x 1 ∈ L x with h 1 < h 0 , contradicting the minimality hypothesis.
Note also that if φ(L x ) ∩ • L y = ∅, then it is still also the case for L 0 . Indeed, the rectangle R 0 of width |L x | and of height h 0 immerses in S (in a similar way as R). If φ(L 0x ) ∩ • L 0y = ∅, then we find P in i(R 0 ), which is not possible for the same reason as above (see Figure 5). 2d Step. Now we assume φ(L x ) ∩ Step since there is no singularity inside i(R).
Step 1: If φ(L x ) ∩ L y = ∅, and h is not minimal Now we replace L by L as follows: We choose Q in φ(L y ) ∩ • L x such that the horizontal distance from P to Q is minimal. Then, we define L by considering the horizontal segment, starting from P and ending at Q , and the vertical segment from Q and ending at φ(σ). Since • L y = ∅ as required. Now, up to shortening L as in the first step, we can assume that L x and L y bound an immerse rectangle Step. Let S be the universal covering of S. Choose P a preimage of P , L a preimage of L attached to P . Now the rectangle R, as defined in the 1st Step, embeds as a rectangle R in S with L as bottom and right sides. For any lift φ of φ, such that φ( L y ) intersects R, φ( L y ) intersects the interior of L x (in a unique point Q, since we are working on the universal cover). Now, we choose a lift φ of φ that minimize the length d of the vertical segment joining Q to the singular point that is the end of φ( L y ). Now, we easily see that φ( R) intersects R as in Figure 6. As in the proof of Proposition 4.1, we find x ∈ L x such that the corresponding vertical leaf is fixed by φ. Then, we find in R a fixed point P for φ.
Now we consider L obtained as follows: take the horizontal segment with left end P and whose right end is in L x , then consider the vertical segment that ends in the same singularity as L (see Figure 6). We claim that L projects in S into a suitable curve L . Indeed, otherwise L x ∪ φ(L x ) either intersects the interior of L y or the interior of φ(L y ). In both cases, we find another intersection point between L x and φ(L y ) which contradicts the minimality of d. Therefore, we have found a suitable curve, the theorem is proven.
The next proposition implies the second part of the Geometrical Statement. • φ − is obtained by the symmetric Rauzy-Veech construction.
• If φ + is not obtained by the usual Rauzy-Veech construction, then there are exactly two regular fixed points for φ − , that are interchanged by the hyperelliptic involution.
Proof. We prove the first part. From the previous theorem, all we need to show is that φ − has a regular fixed point. For a homeomorphism φ, we denote by φ * the linear action of φ on the homology H 1 (S, R). We recall the Lefschetz formula: When φ is of type pseudo-Anosov, and x a fixed point, we can show that: • Ind(φ, x) < 0 if there is a fixed separatrix.
• Ind(φ, x) = 1 otherwise. Assume that φ = φ + and the underlying translation surface is in H(2g − 2). Then the unique singularity P is necessarily fixed and has index ≤ 1. All the other fixed point (possibly zero) are regular point so have negative index. Hence 2 − T r(φ * ) ≤ 1. Since T r(τ • φ) = −T r(φ), we conclude that 2 − T r(φ − * ) ≥ 3. Therefore, there must be at least 2 regular fixed points for φ − . Now assume that the underlying translation surface is in H(g−1, g−1). The two singularities P 1 , P 2 are either fixed or interchanged by φ + .
• If P 1 , P 2 are interchanged by φ + , then 2 − T r(φ + * ) ≤ 0, hence 2 − T r(φ − * ) ≥ 4, there are at least four fixed points so, at least 2 regular fixed points. Now, we see that in the above proof, the case where φ + is not obtained by the usual Rauzy-Veech construction (i.e. when φ + does not have negative index fixed point) corresponds to the equality case. There is exactly one pair of regular fixed points {Q 1 , Q 2 }. Since φ − and τ commute, we see that τ (Q 1 ) is a fixed point, hence Q 1 or Q 2 . It cannot be Q 1 otherwise τ • φ − = φ + has a regular fixed point, contradicting the hypothesis. Hence, τ (Q 1 ) = Q 2 , this concludes the proof.
We end this section with the following useful proposition.
Proposition 4.6. If an admissible path from π to s(π) passes through the central permutation then τ • φ is obtained by the usual Rauzy-Veech construction, where τ is the hyperelliptic involution.
Proof of Proposition 4.6. The map ψ = τ • φ preserves the orientation of the vertical and horizontal leafs. Hence, all we need to show is that there is a fixed separatrix.
Let (π, λ, τ ) be the weak suspension data associated to the admissible path. Let h be a height, and I = I h ⊂ X(π, λ, τ ).
We claim that there is an immersed Euclidean rectangle with ψ(I) as one horizontal side, and the other one is a subinterval of I. Assuming the claim, there is an isometry f from ψ(I) to I obtained by following a vertical leaf. The map f • ψ is therefore a contracting map from I to itself (its derivative is θ −1 ), hence has a fixed point. It means that there is an element x in I whose image by ψ is in the vertical leaf l passing through x. Thus, this vertical leaf l is preserved by ψ. Since ψ, restricted to l as derivative θ = 1, there is a fixed point of ψ on l. This fixed point is either a conical singularity or a regular point. In any case, ψ fixes a vertical separatrix. Hence ψ fixes also a horizontal separatrix. It is therefore obtained by the usual Rauzy-Veech construction. Now we prove the claim. By construction of φ, there is base segment I such that I = φ(I) ⊂ I. This subinterval I is obtained from I after doing Rauzy-Veech induction until the end of the defining path in the Rauzy diagram. By hypothesis, there is a step of the form (π n , λ , τ ), where π n is the central permutation, and that corresponds to a interval I , with I ⊂ I ⊂ I. Now, we easily see that τ (I ) and I are the horizontal sides of a immersed rectangle. We now conclude as in the proof of Proposition 4.1 that there is a fixed separatrix.

Renormalization and changing the base permutation
The aim of this section is to reduce our analysis to the set of paths of D n starting from the central loop. First, recall that, from [BL12], pseudo-Anosov homeomorphisms in hyperelliptic connected components obtained by the Rauzy-Veech construction have expansion factors at least 2, and are not the minimal ones. Recall that an admissible path is pure if the corresponding map is not obtained by the usual Rauzy-Veech construction.
Theorem 5.1. Let γ be a pure admissible path in D n joining a permutation π to its symmetric s(π), and let φ be the corresponding pseudo-Anosov map. Then, φ can also be obtained by a path γ starting from the central loop. In addition the first step of γ is on a secondary loop (i.e. of type 'b').
The construction of γ in terms of γ is not obvious. To prove the theorem, we appeal to the dynamics of the induction. 5.1. Renormalization: The ZRL acceleration. In this section we will consider only pure admissible paths from π to s(π). If we denote by (n 1 , . . . , n k ) the coordinates of π then k is even (otherwise Proposition 3.2 implies that γ passes through the central permutation and Proposition 4.6 is a contradiction with the path being pure).
Convention 1. We will always assume that the first Rauzy move is 't', i.e. π is obtained by applying the sequence t n 1 b n 2 . . . t n k−1 to π n .
Proposition 5.2. Let α = π −1 b (n) and β = π −1 t (1). Then φ(ζ α ) = ζ β . In particular there exists a canonical base segment I obtained from the previous one as follows : We apply a right induction until α is is loser ( i.e. Rauzy path of type b l t, for some l ≥ 0) followed by a left induction until β is loser ( i.e. left Rauzy path of typet mb , for some m ≥ 0). This segment I defines a new Rauzy path γ starting point π of γ satisfying Convention 1 (up to permuting top and bottom). Definition 5.1. The map ZRL from the space of pure admissible paths satisfying Convention 1 to itself is defined by ZRL(γ) = γ , where γ, γ are as in Proposition 5.2.
Proof of Proposition 5.2. By hypothesis, π is in the connected component of D n \{π n } where β = π −1 t (1) is never winner. Since γ does not pass through the central permutation, α is never winner, hence the parameter ζ β is unchanged during the whole Rauzy path. In particular, by definition of φ (see Figure 4), the segment corresponding to ζ α is sent by φ to the segment corresponding to ζ α . Hence, the curve L as in Figure 7 is admissible and defines a new admissible path γ from a permutation π to s(π ). The horizontal part of L ∪ φ(L ) defines a new base segment I and parameters (π , ζ ) as described in Section 3.2. We obtain (π , ζ ) by the sequence of right and left Rauzy move described in the statement of the proposition. If π does not satisfy Convention 1, we interchange the two lines, that is equivalent to interchanging "up" and "down" on the surface, and therefore conjugating φ with the surface orientation changing homeomorphism.
Remark 5.1. ZRL stands for Zorich acceleration of Right-Left induction. By construction the two surfaces S(γ) and S(γ ) are in the same Teichmüller orbit (perhaps up to an affine isometry with derivative map 1 0 0 −1 ). It is difficult to express the path γ in a simple way in terms of the path γ. However the coordinates for the permutations (introduced in Section 3.5) allow us to express the new starting point π in an easier way, only in terms of the old starting point (see Proposition 5.4 below).
In view of considering iterates of the map ZRL, we will use the following lemma.
Lemma 5.3. Assume that γ is a pure admissible path. Then the ZRL orbit of γ is infinite and all the letters are winner and loser infinitely often.
Proof. By Proposition 5.2, the new path γ is a pure admissible path. Hence, the ZRL orbit is infinite. In this proof, we do not interchange top and bottom line in order to follow Convention 1, but consider a sequence of base segment (I n ) n∈N on the same underlying surface S, with I n+1 ⊂ I n . Note that the vertical flow on S is minimal (and uniquely ergodic) since S carries an affine pseudo-Anosov homeomorphism. We identify those segments as subsets of the real line, I n =]b n , a n [, with (b n ) increasing and (a n ) decreasing. As for the usual Rauzy induction, all letter are winner infinitely often if and only if |I n |= a n − b n tends to zero. Let us assume that it is not the case. Then a n → a, b n → b with φ(a) = b. We consider the sequence (λ n , τ n ) of corresponding suspension data. Define t n to be the time when the vertical unit speed geodesic starting from a n reach a singularity, note that t n can be positive or negative, observe that |t n |→ ∞ otherwise the set of singularities in the surface S would not be discrete. Without loss of generalities, we can assume that there is a subsequence (n k ) such that t n k > 0. There are two cases.
(1) If the geodesic in the positive direction starting from a, denoted as g a , is infinite, then it follows the (finite) one starting from a n k for a longer and longer time. By density there is a time t such that g b (t) intersects ]b, a[, hence for k large enough, the geodesics starting from b n k intersects ]b, a[, hence I n,k before reaching a singularity. Contradiction.
(2) If the geodesic g a is finite in the positive direction, then it is necessarily infinite in the negative direction. If there is a subsequence t m k < 0, the same argument as above gives a contradiction. Hence, for n large enough t n > 0. The same argument also works for an non accelerated Right-Left induction. Hence for n large enough, the 'right' part of the ZRL move is necessarily b. Similarly, the 'left' part of the ZRL move is necessarilȳ t. But such move cannot hold for an infinite number of step. Contradiction.
The lemma is proved.

5.2.
Action of ZLR on the starting point. As promised we now explain how ZRL acts on the starting point, in terms of the coding introduced in Section 3.5.
Proposition 5.4. Let γ be an admissible path, starting from a permutation π. Let (n 1 , . . . , n k ), k ≥ 4 be the coding of π. The coding of the starting point of ZRL(γ) is obtained by the following rules.
where l, m are positive integers with l < n k − 1, m < n 1 − 1, andl, respectivelym, is the remainder of the Euclidean division of l, respectively m, by n k , respectively n 1 .
Proof of Proposition 5.4. By convention, the starting permutation π is of the form t n 1 b n 2 . . . t n k−1 . The map ZRL acts on π by a sequence of Rauzy moves of the form b l tt mb and then followed (perhaps) by a permutations of the lines (which does not change the coding).
The remaining part is obtained similarly: we must act on the left by the movest mb , which is equivalent to the moves s.b m .t.s. This proves the proposition. 5.3. Proof Theorem 5.1. We are now ready to prove the theorem announced at the beginning of this section.
Proof Theorem 5.1. Let γ be an admissible path, with corresponding expansion factor λ < 2. We need to show that there exists an iterate of ZRL which starts from a permutation π starting from the central loop, i.e. satisfying π −1 t (n) = π −1 b (1). We first observe the following: let (l, . . . , l ) be a coordinate of π, then l ≤ l . Indeed, if l > l , then a path joining π to s(π) must pass through the central permutation. Now, let (n 1 , , n 2 , . . . , n k−1 , n k ) = (l, x, . . . , x , l ) be the coding of a permutation, with k ≥ 4 even. We prove by induction the following property P(l): (1) l = l (2) After applying a finite sequence of ZRL, we reach the permutation coded by (l + x, . . . , x + l ), and during this sequence, the letters "between the blocks corresponding to x and x ", i.e. π −1 ε (j) for j ∈ {l + x + 2, . . . , n − 1 − l − x } are constant along the ZRL-orbit and non-winner. Note that if k ≥ 4, this set of letters is nonempty. Initialisation corresponds to the case (1, x, . . . , x , l ). Assume that l > 1, then after one step of ZRL, the left part must be (1 + x, . . . ), but in this case, in order to preserve the parity of the numbers of blocks, we must have: (x + 1, . . . , x , l − 1, 1) with x + 1 > 1, which contradicts the initial observation. So, we have l = 1, and ZRL maps (1, x, . . . , x , 1) to (x + 1, . . . , x + 1), and we see directly that the second condition is fulfilled. Now, let 1 < l ≤ l and let the initial permutation be coded by (l, x, . . . , x , l ). Assume P(l ) for any l < l. By Proposition 5.4 and observing again that the parity of the number of blocks is constant, after one step of ZRL the coding is one of the following: (1) (1, l − 1, x, . . . , x , l − 1, 1).
Contradiction. (5) By induction hypothesis, we have l 1 = l 1 and after some steps, we have (l 1 +1, l 2 , x, . . . , x , l 2 , l 1 + 1), and again after some steps, we have (l, x, . . . , x , l ). During these two sequences of ZRL, all the letters between the blocks corresponding to l 2 and l 2 are unchanged and non-winner, and this set is nonempty. Note that it is impossible to repeat infinitely the steps (1) or (5) by Lemma 5.3, hence we will eventually get Step (2), proving P(l).
Hence, after a finite number of ZRL steps, we obtain a permutation with a coding with two blocks (k=2), which corresponds to a starting point in the central loop. This proves the first part of Theorem 5.1.
We now turn into the proof of the second part: the first step of the corresponding admissible Rauzy path leaves the central loop, i.e. is of type b. In this case, ZRL acts on the starting point π by tb. Write π as: a * * * b b * * * c We easily see that tb preserves π, and λ b becomes λ b − λ a − λ c . Iterating ZRL, after a finite number of steps, b is not winner any more. Theorem 5.1 is proved.

Reducing to a finite number of paths
In view of Section 4, for a given n, one needs to control the spectral radii of matrices V (γ) for all paths γ in D n . Section 5 shows that it is enough to consider paths γ starting from the central loop of D n . However this still produces an infinite number of paths to control. The proposition below shows how one can restrict our problem to a finite number of paths.
Proposition 6.1. Let γ be an admissible path starting from a permutation π = π n .t k in the central loop. We assume that the first step goes in the secondary loop.
For two paths γ, γ we will write γ ≤ γ if the path γ is a subset of the graph γ (viewed as a ordered collection of edges). For a real matrix A ∈ M n (R), we will write A ≥ 0 (respectively, A 0) to mean that A ij ≥ 0 (respectively, A ij > 0) for all indices 1 ≤ i, j ≤ n, and similarly for vectors v ∈ R n . The notation A ≥ B means A − B ≥ 0. Before proving Proposition 6.1, we will use the following: Proposition 6.2. Let γ ≤ γ i.e. the path γ is obtained from the path γ by adding (possibly zero) closed loops. Then V (γ ) ≤ V (γ) and θ(γ ) ≤ θ(γ). Moreover if V (γ) is primitive and γ = γ then θ(γ ) < θ(γ).
Proof of Proposition 6.2. If V (γ ) = V 1 · V 2 · . . . · V l · P is the matrix associated to the path γ , where V i are the elementary Rauzy-Veech matrices and P is a permutation matrix, then the matrix associated to γ has the form where: • N 1 , . . . , N l are products of (possibly empty) elementary Rauzy-Veech matrices, hence of the type I + N i , where I is the identity matrix and N i is a matrix with nonnegative coefficients. • P is the permutation matrix corresponding to the end point of γ.
Since the labeled Rauzy diagram and the reduced Rauzy diagram coincide, the endpoints of γ and γ also coincide in the labeled Rauzy diagram, hence P = P . From these facts we deduce that V (γ ) ≤ V (γ). Let us show that θ(γ ) ≤ θ(γ).
Recall that V (γ ) is not necessarily primitive. However there is a permutation matrix P σ , where the matrices A i are primitive matrices. Up to a change of basis one can assume that the spectral radius of A s is achieved by θ(γ ).
Proof of Proposition 6.1. We prove the first assertion. Let k be any integer satisfying k > (n − 1)/2. Then n − 1 − k < k, hence s(π) = t n−1−k π n is "before" π in the central loop. Since γ is admissible, γ is passes thought the central permutation. Now if k = (n − 1)/2 then, up to relabeling, π = s(π). By using the alphabet A = {1, . . . , n}, we have: Hence the relabeling does not changes the letter n. In particular it must be winner at least once. This imply that γ contains the step π → t.π, hence passes through the central permutation. In both situation θ > 2.
We now turn to the proof of the second part of the proposition. The Rauzy diagram that we consider have the particular property that removing any vertex disconnects it (except the the particular ones that satisfies t.π = π or b.π = π). This implies that the path γ is obtained from the path γ n,k by adding (possibly zero) closed loops, namely γ n,k ≤ γ. Hence we are in a position to apply Proposition 6.2: this finishes the proof of this part.
The proof of the last statement is similar, once we remark that if V n,k is not primitive, then the path γ is obtained from γ n,k by adding at least one closed loop. Assuming that θ(γ) < 2 this gives by definition a path of the form γ n,k,l for some l ∈ {1, . . . , 2n − 2 − 3k}. Thus γ n,k,l ≤ γ and Proposition 6.2 again applies. If l = n − 1 − k then γ n,k,l is obtained from γ n,k by adding twice the same loop. Hence V n,k,l is not primitive and γ = γ n,k,l . The same argument above applies and γ must contain γ n,k,l for some l = n − 1 − k. We conclude with Proposition 6.2. In the sequel we will denote by P n,k (respectively, P n,k,l ) the characteristic polynomial of the matrices V (γ n,k ) (respectively, V (γ n,k,l )) multiplied by (X + 1). Their maximal real root are θ n,k and θ n,k,l , respectively.
A.1. The rome technique (after [BGMY1980]). To compute the characteristic polynomial of matrices we will use the rome method, developed in [BGMY1980]. To this end it is helpful to represent a matrix V into the form of a combinatorial graph which amounts to draw all paths of length 1 associated to V .
A subset R ⊂ {1, 2, . . . , k} is called a "rome" if there is no loop outside R. Given r i , r j ∈ R, a path from r i to r j is a "first return path" is it does not intersect R, except at its starting and ending points. This allows us to define an r × r matrix-valued real function V R (X), where r is the size of R, by setting V R (X) = (a ij (X)), where a ij (X) = η w(η) · X −l(η) , where the summation is over all first return paths beginning at r i and ending at r j .
Theorem (Theorem 1.7 of [BGMY1980]). If R is a rome of cardinality r of a n × n matrix V then the characteristic polynomial χ V (X) of V is equal to Remark A.1. The matrices V = V n,k or V = V n,k,l can be seen as the action of homeomorphisms on absolute homology if n is even and relative homology otherwise. Thus their characteristic polynomials are reciprocal polynomials when n is even, and either reciprocal or anti-reciprocal polynomials when n is odd, depending whether φ fixes or interchanges the two singularities), i.e. χ V (X) = X n χ V (X −1 ) or χ V (X) = −X n χ V (X −1 ). Thus A.2. The paths γ n,k . We briefly recall Notation 2.4 (see also Figure 2). For any k = 1, . . . , K n and any l = 1, . . . , n − 2 − k we define the path γ n,k : π := π n .t k −→ s(π) : b n−1−k t n−1−2k Lemma A.1. Let n ≥ 4 and 1 ≤ k ≤ K n . If gcd(n − 1, k) = 1 then . . , k − 1 and ε = 0, 1 . In particular: (1) For even n P n,Kn = X n+1 − 2X n−1 − 2X 2 + 1.
(2) For n = 1 mod 4, we have gcd(n − 1, K n ) = 1 and (3) For n = 3 mod 4, we have gcd(n − 1, K n − 1) = 1 and (4) More generally, for n even, and k ≤ K n − 1, the highest nonzero monomial (except X n+1 and −2X n−2 ) has degree at least 2n 3 . Proof of Lemma A.1. We will use the rome method as explained in Section A.1. We use the notation of the proof of Lemma 4.2. We will write V n,k as V n,k = A n − B n,k where For any i = 1, . . . , k − 1, set l := i(n − 1) k + 1. Then In particular, all integers of the form i(n − 1)/k + ε for i ∈ {1, . . . , k − 1} and ε ∈ {0, 1} are mutually disjoint and in {3, . . . , n − 2}.  Figure 8. The graph associated to V n,k . In dashed line we have represented arrow that need to be removed from the graph in blue. Multiplicity are also indicated. To be more more precise, there is one arrow from vertex labelled 1 to the vertex labelled l and no arrow from vertex labelled 1 to the vertex labelled l+1. In the graph l = i(n−1) k + 1 for any i = 1, . . . , k − 1. Obviously the graph associated to A n is drawn in blue colour Clearly the set R = {1, n} is a rome for A n . Thus it is also a rome for V n,k (since we pass from A n to V n,k by removing some paths). The 2 × 2 matrix (A n ) R is easily obtained as To obtain the matrix (V n,k ) R one has to subtract the polynomial corresponding to the paths passing through arrows in dashed line (passing through vertices l and l+1 where l = i(n−1) k +1 for some i = 1, . . . , k − 1). At this aim we define The polynomial T n,k (respectively, 2Q n,k ) takes into account all simple paths from the vertex 1 to the vertex 1 passing through an arrows in dashed line connecting 1 to l (respectively, to l + 1). Hence Note that T n,k = XQ n,k . By using [BGMY1980], a straightforward computation gives: Hence, which implies the first statement of the lemma.
We give a little more information on the set J n,k . We write J n,k = {j 1 , . . . , j r } with j 1 < · · · < j r (possibly, r = 0). We have i(n−1) k = 2i + i(n−1−2k) k . In particular, we see that j s = 2i s +s, where i s ≥ 0 is the smallest integer such that is(n−1−2k) k > s, i.e i s = sk n−1−2k +1. Note also that P n,k must be reciprocal, hence j ∈ J n,k if and only if n + 1 − j ∈ J n,k . Now we compute particular cases.
Proposition A.2. Let n ≥ 4 be an even integer.
Proof. Again we will use the rome method in order to compute the characteristic polynomial. First we compute the matrix V n,K,l . Remark that n is even so n − 1 and K n = n/2 − 1 are relatively prime. Following the proof of Lemma 4.2 we consider the central permutation π n with the following labelling: Let l ∈ {1, . . . , L n }. By construction, γ n,Kn,l is a path obtained from γ n,Kn by adding a closed loop (of type 't' and of length n − 1 − K n − l) at b l t Kn π n . Hence the matrices V n,Kn and V n,Kn,l differ by a non negative matrix, namely V n,Kn,l − V n, On the way, the same argument applied to the matrix V n,Kn,Ln+2 applies, proving the last statement.
We are now in a position to compute P n,Kn,l . The graph associated to the matrix V n,Kn is already presented in the proof of Lemma A.1. For readability purposes, we reproduce this graph below, where we have added the edges corresponding to the graph associated to C n,l .
Clearly the set R = {1, 2l, n} is a rome for V n,Kn,l , and thus for V n,Kn (since we pass from V n,Kn,l to V n,Kn by removing some paths). The 3 × 3 matrix (V n,Kn ) R is easily obtained as Adding the matrix C n,l consists of adding two arrows form the vertex labelled by 2l to the vertices 2, 2l + 1 (with multiplicity 1) and K n − l arrows to the vertices 2l + 3, . . . , n − 3, n − 1 (with multiplicity 2). In this situation R is still a rome. To compute the matrix (V n,Kn,l ) R we need to consider all paths passing through a dashed edge in the graph in Figure 9. Thus (V n,Kn,l ) R (X) = (V n,Kn ) R (X) + C(X) where Hence by [BGMY1980] we draw χ V n,Kn,l = − det((V n,Kn ) R (X) + C(X) − Id 3 ). By multilinearity, one has: P n,Kn,l = P n,Kn − (1 + X) · det(W n,l ) where that is the desired result. The second assertion comes for free since 2L n = n − 2 and by Lemma A.1 one has P n,Kn = X n+1 − 2X n−1 − 2X 2 + 1. Proposition A.2 is proved. (1) If l is even then V n,Kn,l is reducible and θ n,Kn,l = θ n ,K n ,l with n = (n + 1)/2, l = l/2 and K n = K n /2. (2) If l ≤ L n is odd then V n,Kn,l is primitive and In particular, for l = L n , we have P n,Kn,Ln = X n+1 − 2X n−1 − 4X (n+3)/2 + 4X (n−1)/2 + 2X 2 − 1.
Proof of Proposition A.3. In the sequel, let m = K n + 1 = (n − 1)/2 (m is odd). We follow the strategy of the proof of the previous proposition. For first point we do the same as in proof of Lemma 4.2 in the case when d > 1. The labelling of the central permutation is given by: Then, computation of V n,Kn,l gives a upper triangular 2 × 2 block matrix, with V n ,K n ,l for the top left block and a permutation matrix for the bottom right bloc.
In the next, we compute V n,Kn,l by using the "standard" labelling" of the central permutation, i.e. π = 1 2 . . . n n . . . 1 . We have V n,Kn,l = A n + B n,l where We will again use the rome method in order to compute the characteristic polynomial. It is helpful to represent the matrices in form of a combinatorial graph which amounts to draw all paths. From this, we see that V n,Kn,l is irreducible, and therefore primitive since it has a nonzero diagonal element. The graph associated to A n is rather simple. Clearly a rome is made of the subsets labelled R = {n, n − 1, m}. The matrix A R (X) in this label is Adding the matrix B n,l consists of adding arrows form the vertex labelled by n − l to the vertices 1, . . . , m − l, n − 2 with multiplicity 1 and to the vertex n − 1 with multiplicity 2. In this situation R is still a rome. To compute the matrix (V n,Kn,l ) R we need to consider all paths passing through a dashed edge on the graph in Figure 9. Thus (V n,Kn,l ) R (X) = A R (X)+C(X) where Hence by [BGMY1980] we draw χ V n,Kn,l = − det(A R (X) + C(X) − Id 3 ). By using the fact and Q = X m (X l−1 − X) + XP we easily obtain the desired equality. The last assertion is also easy to derive from the fact that γ n,Kn,Ln+2 is obtained from γ n,Kn by adding a 'b' loop at t Kn+1 π n and from the shape of A n . This ends the proof of Proposition A.3.

Appendix B. Comparing roots of polynomials
This section is devoted to comparing θ n,k and θ n,k,l (the maximal real roots of the polynomials P n,k and P n,k,l introduced in Appendix A) for various n, k, l. Observe that by construction θ n,k > √ 2 and θ n,k,l > √ 2. One key ingredient for comparing maximal real roots of these polynomials is the easy Lemma B.1. Let P 1 , P 2 be two unitary polynomials of degree at least one such that for x > √ 2, P 1 (x) − P 2 (x) > 0. We assume that P 1 has a root θ 1 > √ 2. Then P 2 has a root θ 2 > θ 1 . Proof.
The assumption implies P 2 (θ 1 ) < 0. Since P 2 is unitary, the result follows by the mean value theorem.
Proof of Lemma B.3. From the proof of Lemma A.1, we have P n,k −P n,k = 2(x+1) Q n,k − Q n,k , where Q n,k = k−1 i=1 x i(n−1)/k . From Lemma B.1, we need to show that P n,k (x)−P n,k (x) > 0 for x > √ 2. First we observe that So, for any x > 1, P n,k (x) − P n,k (x) > x (k −k)(n−1) k > 0, proving the lemma.
Proof of Lemma B.5. We follow the notation of the proof of Proposition A.3. A simple computation shows: P n,Kn,l − P n,Kn,l = 2(X l − X l )(X m − 1)(X l+l + X 4+m ) X l+l (X − 1)(X + 1) .
(1) If n is odd (thus n ≡ 3 mod 4) then by the above case θ n ,k > θ n ,K n ,L n . We conclude with Lemma B.2.
(2) If n is even then we need to show directly that θ n ,k > θ n,Kn,Ln . Note n even implies that d is an odd multiple of 2. There are two cases: • d ≥ 6. In this case, θ n ,k ≥ θ n ,K n (k = K n is possible). We have We necessarily have n > 15 hence n − n ≥ n+3 2 + 2 hence the above polynomial is positive for x > √ 2. • d = 2. In this case θ n ,K n = θ n,Kn < θ n,Kn,L , hence the previous strategy does not work. We remark that necessarily, k < K n . However, n and K n − 1 are not necessarily relatively prime, so we need to compare directly θ n ,k with θ n,Kn,L by using Statement 4 of Lemma A.1. We have P n,Kn,L − x n−n P n ,k ≥ 2x n−n + 2n We necessarily have n ≥ 11 hence n − n + 2n 3 ≥ n+3 2 + 2, implying the desired inequality. This ends the proof of the proposition.
(1) If n ≡ 1 mod 4 then the previous argument shows that θ n ,k > θ n ,K n . By Lemma B.2, the sequence (θ n,Kn ) n is decreasing for n ≡ 1 mod 4, so we have θ n ,K n > θ n,Kn as desired.
Proof of Proposition B.9 is complete.
Appendix C. A naive attempt to generalize the Rauzy-Veech construction The classical construction of pseudo-Anosov homeomorphism by Rauzy induction necessarily produces maps that preserves a singularity, and a horizontal separatrix. This clearly comes from the fact that only right Rauzy induction is used. So, it is natural to expect to produce pseudo-Anosov homeomorphism that do not fix a separatrix by combining right and left induction.
For instance, we consider a path γ in the labeled (extended) Rauzy diagram such that.
• the image of γ in the reduced extended Rauzy diagram is closed.
• γ is the concatenation of a path γ 1 that consists only of right Rauzy moves, and a path γ 2 that consists only of left Rauzy moves.
As in Section 3.6 we associate to such path a matrix V by multiplying the corresponding product of the transition matrices by a suitable permutation matrix. Assume now that the matrix V is a primitive. Let θ > 1 be its Perron-Frobenius eigenvalue. We choose a positive eigenvector λ for θ. As before, V is symplectic, thus let us choose an eigenvector τ for the eigenvalue θ −1 . It turns out that τ is not necessarily a suspension datum, but it is a weak suspension datum (up to replacing it by its opposite). Indeed, the set of weak suspension data is a open cone W , which is by construction invariant by V −1 , and we conclude as previously (the proof is the same as in Proposition 4.1).
Proposition C.1. A pseudo-Anosov homeomorphism affine on a translation surface, and constructed as above fixes a vertical separatrix. In particular it is obtained by the usual Rauzy-Veech construction.
Proof. Let (π, λ, τ ) be the weak suspension datum defined as above and h be a height. We will denote the associated surface by X = X(π, λ, τ ) and by I = I h the corresponding horizontal interval.
After the prescribed the prescribed sequence of right and left Rauzy induction, we obtain the suspension datum (π, λ , τ ) = (π, 1 θ λ, θτ ) defining the same surface X, with corresponding interval I h = I ⊂ I (recall the Rauzy-Veech induction corresponds to cutting the interval on the right, or on the left). Also, θh is an obvious height for (π, 1 θ λ, θτ ), and the corresponding interval I θh is the image by φ of the interval I h .
Hence there is an isometry f from I θh to I h obtained by following a vertical leaf (see Section 3.2). The map f • φ is therefore a contracting map from I h to itself (its derivative is θ −1 ), hence has a fixed point. It means that there is an element x in I h whose image by φ is in the vertical leaf l passing through x. Thus, this vertical leaf l is preserved by φ. Since φ, restricted to l as derivative θ = 1, there is a fixed point of φ on l. This fixed point is either a conical singularity or a regular point. In any case, φ fixes a vertical separatrix. Hence φ fixes also a horizontal separatrix. It is therefore obtained by the usual Rauzy-Veech construction.
Appendix D. A direct proof of Theorem 2.1 when n is even In this appendix we will prove Theorem D.1. If n ≥ 4 is even then L(Spec(C hyp n )) = log(θ n,Kn ). Moreover the conjugacy mapping class realizing the minimum is unique.
Proof of Theorem D.1. By Remark 4.1 we have that log(θ n,Kn ) belongs to Spec(C hyp n ) when n is even. The Dynamical Statement implies L(Spec(C hyp n )) = log(θ(γ)) for some path γ starting from π = π n .t k with first step of type 'b'. By definition θ(γ) ≤ θ n,Kn . The Combinatorial Statement implies θ n,k ≤ θ(γ) for some k ∈ {1, . . . , K n }. Thus (3) θ n,k ≤ θ n,Kn where K n := n/2 − 1. Thus we only have to show that k = K n (this will also prove that the conjugacy mapping class realizing this minimum is unique). Let us assume by contradiction that k < K n . Let d = gcd(k, n − 1). We will distinguish two cases: d = 1 and d > 1.
We claim that V n,k v ≥ V n,Kn v (meaning that the inequality holds for each terms, and is strict for at least one term). Indeed all rows, except the first one, are obviously nonnegative. So we have to prove the claim for the first row only. Since k = K n , we can define l to be the first index so that a l = (−1) l +1 2 . From the above properties of the sequence (a i ), we see that l is necessarily even and that a l = 2. Setting θ = θ n,Kn to improve readability, the first row is equal to Since a i ∈ {0, 1, 2}, we only have to show that when a i − 1 < 0 (i is even) then (a i − 1)θ i−l + a i+1 θ i+1−l > 0. This is clear from the definition of a i s: if a i = 0 then a i+1 = 1 or a i+2 = 2. Thus (a i − 1)θ i−l + a i+1 θ i+1−l ≥ −θ i−l + θ i+1−l = θ i−l (θ − 1) > 0.
There is perhaps a problem with the last term, but we easily see that a n−2 = 1. The claim is proved. Let w T be a positive left eigenvector of V n,k : w T V n,k = θ n,k w T . From the inequality V n,k v ≥ θ n,Kn v draw: Hence θ n,k w T v > θ n,Kn w T v. In conclusion θ n,k > θ n,Kn that contradicts (3).
Second Case: k and n − 1 are not relatively prime. We have d = gcd(n − 1, k) > 1. Let n = n−1 d + 1 and k = k d . Lemma 4.2 gives θ n,k ≥ θ n ,k . By the previous discussion, the inequality θ n ,k > θ n ,K n holds (observe that k and n − 1 are relatively prime and n is an even integer).
By Remark 4.1, θ n,Kn is the largest root of the polynomial X n+1 − 2X n−1 − 2X 2 + 1. In particular the sequence m → θ m,Km , defined on even numbers, is a decreasing sequence. Since n < n we have θ n ,K n > θ n,Kn . In conclusion θ n,k ≥ θ n,Kn : this again contradicts (3). The proof of Theorem 2.1 when n is even is now complete.