The limit point of the pentagram map and infinitesimal monodromy
TThe limit point of the pentagram map and infinitesimal monodromy
Quinton Aboud ∗ and Anton Izosimov † Abstract
The pentagram map takes a planar polygon P to a polygon P ′ whose vertices are theintersection points of consecutive shortest diagonals of P . The orbit of a convex polygon underthis map is a sequence of polygons which converges exponentially to a point. Furthermore, asrecently proved by Glick, coordinates of that limit point can be computed as an eigenvectorof a certain operator associated with the polygon. In the present paper we show that Glick’soperator can be interpreted as the infinitesimal monodromy of the polygon. Namely, there existsa certain natural infinitesimal perturbation of a polygon, which is again a polygon but in generalnot closed; what Glick’s operator measures is the extent to which this perturbed polygon doesnot close up. The pentagram map, introduced by R. Schwartz in [10], is a discrete dynamical system on thespace of planar polygons. The definition of this map is illustrated in Figure 1: the image ofthe polygon P under the pentagram map is the polygon P ′ whose vertices are the intersectionpoints of consecutive shortest diagonals of P (i.e., diagonals connecting second-nearest vertices). P P ′ Figure 1: The pentagram map.
The pentagram map has been an especially popular topic in the last decade, mainly due to itsconnections with integrability [8, 12] and the theory of cluster algebras [2–4]. Most works on thethe pentagram map regard it as a dynamical system on the space of polygons modulo projectiveequivalence. And indeed that is the setting where most remarkable features of that map such asintegrability reveal themselves. That said, the pentagram map on actual polygons (as opposedto projective equivalence classes) also has interesting geometry. One of the early results in thisdirection was Schwartz’s proof of the exponential convergence of successive images of a convexpolygon under the pentagram map to a point (see Figure 2). That limit point is a natural ∗ Department of Mathematics, University of Arizona, e-mail: [email protected] † Department of Mathematics, University of Arizona, e-mail: [email protected] a r X i v : . [ n li n . S I] A ug igure 2: The orbit of a convex polygon under the pentagram map converges to a point. invariant of a polygon and can be thought of as a projectively natural version of the center ofmass. However, it is not clear a priori whether this limit point can be expressed in terms ofcoordinates of the vertices by any kind of an explicit formula. A remarkable recent result byM. Glick [5] is that this dependence is in fact algebraic. Moreover, there exists an operator in R whose matrix entries are rational in terms of polygon’s vertices, while the coordinates of thelimit point are given by an eigenvector of that operator. Therefore, coordinates of the limitpoint can be found by solving a cubic equation.Specifically, suppose we are given an n -gon P in the projectivization PV of a 3-dimensionalvector space V . Lift the vertices of the polygon to vectors V i ∈ V , i = , . . . , n . Define an operator G P ∶ V → V by the formula G p ( V ) ∶= nV − n ∑ i = V i − ∧ V ∧ V i + V i − ∧ V i ∧ V i + V i , (1)where all indices are understood modulo n . Note that this operator does not change under arescaling of V i ’s and hence depends only on the polygon P . What Glick proved is that thelimit point of successive images of P under the pentagram map is one of the eigenvectors of G P (equivalently, a fixed point of the associated projective mapping PV → PV ).We believe that the significance of Glick’s operator actually goes beyond the limit point. Inparticular, as was observed by Glick himself, the operator G P has a natural geometric meaningfor both pentagons and hexagons. Namely, by Clebsch’s theorem every pentagon is projectivelyequivalent to its pentagram map image, and it turns out that the corresponding projectivetransformation is given by G P − I , where I is the identity matrix. Indeed, consider e.g. thefirst vertex of the pentagon and its lift V . Then the above formula gives ( G P − I )( V ) = V − V ∧ V ∧ V V ∧ V ∧ V V − V ∧ V ∧ V V ∧ V ∧ V V . Taking the wedge product of this expression with V ∧ V or V ∧ V we get zero. This meansthat ( G P − I )( V ) ∈ span ( V , V ) ∩ span ( V , V ) , so the corresponding point in the projective plane is the intersection of diagonals of the pentagon.Furthermore, since Glick’s operator is invariant under cyclic permutations, the same holds forall vertices, meaning that the operator G P − I indeed takes a pentagon to its pentagram mapimage.Likewise, the second iterate of the pentagram map on hexagons also leads to an equivalenthexagon, and the equivalence is again realized by G P − I . Finally, notice that for quadrilaterals G P − I is a constant map onto the intersection of diagonals. These observations make usbelieve that the operator G P is per se an important object in projective geometry, whose fullsignificance is yet to be understood. n the present paper we show that Glick’s operator G P can be interpreted as infinitesimalmonodromy . To define the latter, consider the space of twisted polygons , that are polygonsclosed up to a projective transformation, known as the monodromy . Any closed polygon canbe viewed as a twisted one, with trivial monodromy. To define the infinitesimal monodromywe deform a closed polygon into a genuine twisted one. To construct such a deformation, weuse what is known as the scaling symmetry . The scaling symmetry is a 1-parametric group oftransformations of twisted polygons which commutes with the pentagram map. That symmetrywas instrumental for the proof of complete integrability of the pentagram map [8].Applying the scaling symmetry to a given closed polygon P we get a family P z of polygonsdepending on a real parameter z and such that P = P . Thus, the monodromy M z of P z isa projective transformation depending on z which is the identity for z =
1. By definition, theinfinitesimal monodromy of P is the derivative dM z / dz at z =
1. This makes the infinitesimalmonodromy an element of the Lie algebra of the projective group P GL ( P ) , i.e. a linear operatoron R defined up to adding a scalar matrix. The following is our main result. Theorem 1.1.
The infinitesimal monodromy of a closed polygon P coincides with Glick’s op-erator G P , up to addition of a scalar matrix. This result provides another perspective on the limit point. Namely, observe that for z ≈ M z of the deformed polygon is given by M z ≈ I + ( z − )( G P + λI ) , up to higher order terms. Thus, the eigenvectors of G P , and in particular the limit point,coincide with limiting positions of eigenvectors of M z as z →
1. At least one of the eigenvectorsof M z has a geometric meaning. Namely, the deformed polygon P ( z ) can be thought of as aspiral, and the center of that spiral must be an eigenvector of the monodromy. We believe thatas z → z goes to 1, the spiralapproaches the initial polygon, while its center approaches the limit point of the pentagrammap, see Figure 3. Figure 3: The image of a closed polygon under a scaling transformation is a spiral. As the scalingparameter goes to 1, the center of the spiral approaches the limit point of the pentagram map.3 e note that the scaling symmetry is actually only defined on projective equivalence classesof polygons as opposed to actual polygons. This makes the family of polygons P z we used todefine the infinitesimal monodromy non-unique. After reviewing basic notions in Section 2, weshow in Section 3 that the infinitesimal monodromy does not depend on the family used todefine it. The proof of Theorem 1.1 is given in Section 4.We end the introduction by mentioning a possible future direction. The notion of infinites-imal monodromy is well-defined for polygons in any dimension and any scaling operation. Formultidimensional polygons, there are different possible scalings, corresponding to different inte-grable generalizations of the pentagram map [6, 7]. It would be interesting to investigate theinfinitesimal monodromy in those cases, along with its possible relation to the limit point of thecorresponding pentagram maps. As for now, it is not even known if such a limit point exists forany class of multidimensional polygons satisfying a convexity-type condition.It also seems that the infinitesimal monodromy in P is related to so-called cross-ratio dy-namics, see [1, Section 6.2.1]. Acknowledgments.
The authors are grateful to Boris Khesin, Valentin Ovsienko, RichardSchwartz, and Sergei Tabachnikov for comments and discussions, as well to anonymous refereesfor their suggestions. A.I. was supported by NSF grant DMS-2008021.
In this section we briefly recall standard notions related to the pentagram map, concentratingon what will be used in the sequel.A twisted n -gon is a bi-infinite sequence of points v i ∈ P such that v i + n = M ( v i ) for all i ∈ Z and a certain projective transformation M ∈ P GL ( P ) called the monodromy . A twisted n -gongeneralizes the notion of a closed n -gon as we recover a closed n -gon when the monodromy isequal to the identity. We denote the space of twisted n -gons by P n .The pentagram map takes a twisted n -gon to a twisted n -gon (preserving the monodromy)so it can be regarded as a densely defined map from the space P n of twisted n -gons to itself.From now on, we will assume that polygons are in sufficiently general position so as to allow forall constructions to go through unhindered.We say that two twisted n -gons { v i } and { v ′ i } are projectively equivalent when there is aprojective transformation Φ such that Φ ( v i ) = v ′ i . Notice, if two twisted n -gons are projectivelyequivalent, then their monodromies M, M ′ are related by M ′ = Φ ○ M ○ Φ − .The pentagram map on twisted n -gons commutes with projective transformations and assuch descends to a map on the space P n / P GL ( P ) of projective equivalence classes of twisted n -gons.We now recall a construction of coordinates on the space P n / P GL ( P ) of projective equiv-alence classes of twisted n -gons. These coordinates are known as corner invariants and wereintroduced in [11].Let { v i ∈ P } be a twisted polygon. Then the corner invariants x i , y i of the vertex v i aredefined as follows. x i ∶= [ v i − , v i − , (( v i − , v i − ) ∩ ( v i , v i + )) , (( v i − , v i − ) ∩ ( v i + , v i + ))] ,y i ∶= [(( v i − , v i − ) ∩ ( v i + , v i + )) , (( v i − , v i ) ∩ ( v i + , v i + )) , v i + , v i + ] , where we define the cross-ratio [ a, b, c, d ] of 4 points a, b, c, d on a projective line as [ a, b, c, d ] ∶= ( a − b )( c − d )( a − c )( b − d ) . i − v i − v i v i + v i + Figure 4: Definition of corner invariants.
Consider Figure 4. The value of x i is the cross ratio of the four points drawn on the line ( v i − , v i − ) (i.e. the line on the left) and y i is the cross ratio of the four points drawn on theline ( v i + , v i + ) (i.e. the line on the right).These corner invariants are defined on almost the entire space P n of twisted n -gons. Fur-thermore, these numbers are invariant under projective transformations and hence descend tothe space P n / P GL ( P ) of projective equivalence classes of twisted polygons. As shown in[11], the functions x , . . . , x n , y , . . . , y n constitute a coordinate system on an open dense subsetof P n / P GL ( P ) . This in particular allows one to express the pentagram map, viewed as atransformation of P n / P GL ( P ) , in terms of the corner invariants.If we are given a twisted n -gon with corner invariants ( x i , y i ) , then the corner invariants ( x i , y i ) of its image under the pentagram are given by x ′ i = x i − x i − y i − − x i + y i + y ′ i = y i + − x i + y i + − x i y i . These formulas assume a specific labeling of vertices of the pentagram map image. For a differentlabeling the resulting formulas differ by a shift in indices. The choice of labeling, and moregenerally, the specific form of the above formulas will be of no importance to us. We will onlyuse the following corollary. Consider a 1-parametric group of densely defined transformations P n / P GL ( P ) → P n / P GL ( P ) given by R z ∶ ( x i , y i ) ↦ ( x i z, y i z − ) (2)These transformations are known as scaling symmetries . Proposition 2.1.
The scaling symmetry R z ∶ P n / P GL ( P ) → P n / P GL ( P ) on projective equiv-alence classes of twisted polygons commutes with the pentagram map for any z ≠ .Proof. The above formulas for the pentagram map in x, y coordinates remain unchanged if all x variables are multiplied by z and all y variables by are multiplied by z − .This proposition was a key tool in the proof of integrability of the pentagram map. Namely,consider a (twisted or closed) polygon P defined up to a projective transformation, and let P z beits image under the scaling symmetry. Then, since the pentagram map commutes with scalingand preserves the monodromy, it follows that the monodromy M z of P z (which does not have tobe the identity even if the initial polygon is closed!) is invariant under the map. Since P z is onlydefined as a projective equivalence class, this means that M z is only defined up to conjugation.Nevertheless, taking conjugation invariant functions (e.g. appropriately normalized eigenvalues)of M z , we obtain, for every z , functions that are invariant under the pentagram map. It is shownin [8] that the so-obtained functions commute under an appropriately defined Poisson bracketand turn the pentagram map into a discrete completely integrable system. See also [9] for amode detailed proof. In our paper we utilize pretty much the same idea, but instead of lookingat the eigenvalues of M z we will consider M z itself. It is not quite well-defined, but we will showthat its z derivative at z = Infinitesimal monodromy
In this section we define the infinitesimal monodromy and show that it does not depend onthe choices we need to make to formulate the definition, namely on the way we lift the scalingsymmetry (2) from projective equivalence classes of polygons to actual polygons.We start with a closed n -gon, P , in P . Let [ P ] ∈ P n / P GL ( P ) be its projective equivalenceclass. Then, applying the scaling transformation R z given by (2) to [ P ] , we get a path R z [ P ] in P n / P GL ( P ) such that R [ P ] = [ P ] . Now, choose a smooth in z lift P z of the path R z [ P ] tothe space P n of actual twisted polygons such that P = P (we will construct an explicit exampleof such a lift later on). Denote by M z ∈ P GL ( P ) the monodromy of P z . It is a family ofprojective transformations such that M is the identity, M = I . This family does depend on thechoice of the lift P z of the path R z [ P ] . However, as we show below, the tangent vector dM z / dz at z = infinitesimal monodromy . Definition 3.1.
The infinitesimal monodromy of a closed polygon P is the derivative dM z / dz at z =
1, where M z is the monodromy of any path P z of polygons such that P = [ P z ] = R z [ P ] .The infinitesimal monodromy is therefore a tangent vector to the projective group P GL ( P ) at the identity, and, upon a choice of basis, can be viewed as a 3 × Theorem 3.2 (=Theorem 1.1) . The tangent vector to P GL ( P ) represented by Glick’s operator G P coincides with the infinitesimal monodromy of P . The proof will be given in Section 4. But first we need to check that Definition 3.1 makessense, i.e. that the infinitesimal monodromy does not depend on the choice of the path P z . Thisis established by the following: Proposition 3.3.
Let P z and ˜ P z be two families of polygons such that P = ˜ P is a closedpolygon and ˜ P z is projectively equivalent to P z for every z . Then, for the monodromies M z and ˜ M z of these families, at z = we have dM z / dz = d ˜ M z / dz .Proof. Let Φ z be a projective transformation taking P z to ˜ P z . Since P = ˜ P , we have that Φ = I (a generic n -gon in P does not admit any non-trivial projective automorphisms, provided that n ≥ M z = Φ z M z Φ − z . Differentiating thisand using that Φ = I , we get ddz ∣ z = ˜ M z = ddz ∣ z = M z + [ ddz ∣ z = Φ z , M ] . This identity in particular shows that the infinitesimal monodromy of a twisted polygon is ingeneral not well-defined, due to the extra commutator term in the right-hand side. But for aclosed polygon we have M = I , so the extra term vanishes and we get the desired identity.Before we proceed to the proof of the main theorem, let us mention one property of theinfinitesimal monodromy: Proposition 3.4.
The infinitesimal monodromy of a closed polygon is preserved by the penta-gram map.Proof.
The pentagram map preserves the monodromy and commutes with the scaling. Theinfinitesimal monodromy is defined using monodromy and scaling and is thus preserved aswell.This result in fact follows from our main theorem, because Glick shows in [5, Theorem3.1] that his operator has this property. However, the proof based on Glick’s definition isquite non-trivial, while in our approach it is immediate. The observation that the infinitesimalmonodromy is preserved by the pentagram map was in fact our motivation to conjecture thatit should coincide with Glick’s operator. And, as we show below, this is indeed true. The infinitesimal monodromy and Glick’s operator
In this section we prove our main result, Theorem 1.1 (=Theorem 3.2). To that end, we explicitlyconstruct a deformation P z of a polygon P as in Definition 3.1. Such a deformation is not unique,but we know that the infinitesimal monodromy does not depend on the deformation. We will infact use this ambiguity to our advantage by choosing a deformation for which the infinitesimalmonodromy can be computed explicitly. We will then compute it and see that it coincides withGlick’s operator.Consider a closed n -gon P . Lift the n -periodic sequence { v i ∈ P } of its vertices to an n -periodic sequence of non-zero vectors V i ∈ R . Then, for every i ∈ Z , there exist a i , b i , c i ∈ R suchthat V i + = a i V i + + b i V i + + c i V i . (3)Furthermore, for a generic polygon the numbers a i , b i , c i are uniquely determined because thepoints v i , v i + , v i + are not collinear so the vectors V i , V i + , V i + are linearly independent. Also,we have c i ≠ i because the points v i + , v i + , v i + are not collinear. In addition to that,since V i + n = V i we have that the sequences a i , b i , c i are n -periodic. Finally, notice that for fixed a i , b i , c i the sequence V i is uniquely determined by equation (3) and initial condition V , V , V .Indeed, given V , V , V and using that c i ≠
0, we can successively find all V i ’s from (3). Thisgives us a way to deform the polygon P : keeping V , V , V unchanged, we deform the coefficientsin (3). Namely, consider the following equation V i + = a i V i + + z − ( b i V i + + c i V i ) , (4)We assume that the vectors V , V , V do not depend on z and coincide with the above-constructedlifts of vertices of P . For any z ≠
0, equation (4) has a unique solution with such initial condi-tion. For z = z we get its deformation.Note that for i ≠ , , V i of (4) are actually functions of the parameter z , i.e. V i = V i ( z ) . Proposition 4.1.
Taking the solution of (4) such that V , V , V are fixed lifts of vertices v , v , v of P and projecting the vectors V i ∈ R to P , we get a family P z of twisted poly-gons as in Definition 3.1. Namely, we have that P = P , and also [ P z ] = R z [ P ] , where R z isthe scaling symmetry (2) .Proof. First note that if a sequence V i is a solution of (4) with given initial condition, then V i ( z ) ≠ i and every z sufficiently close to 1, so we can indeed project those vectorsto get a sequence of points in P . Indeed, for z = z by continuity (in fact, one can show that V i ( z ) ≠ z ≠
0, notnecessarily close to 1).Further, observe that since the coefficients of equation (4) are periodic, its solution is quasi-periodic: V i + n ( z ) = M z V i ( z ) for a certain invertible matrix M z depending on z . Therefore, theprojections v i ( z ) ∈ P of the vectors V i ( z ) ∈ R form a twisted polygon whose monodromy isthe projective transformation defined by M z . Furthermore since equations (3) and (4) agree for z =
1, and the initial conditions are the same too, it follows that for the so-obtained family P z of twisted polygons we have P = P . Finally, we need to show that the projective equivalenceclasses of P and P z are related by scaling [ P z ] = R z [ P ] . To that end, we use formulas expressingcorner invariants in terms of coefficients of a recurrence relation satisfied by the lifts of vertices.Arguing as in the proof of [8, Lemma 4.5] one gets the following expressions for the cornerinvariants of P : x i + = a i c i b i b i + y i + = − b i + a i a i + . ccordingly, since equations (3) and (4) encoding P and P z are connected by the transformation b i ↦ z − b i , c i ↦ z − c i , the corner invariants of P z are given by x i + ( z ) = a i ( z − c i )( z − b i )( z − b i + ) = zx i + y i + ( z ) = − z − b i + a i a i + = z − y i + . Thus, the projective equivalence classes of the polygons P and P z are indeed related by scaling,as desired.We are now in a position to prove our main result. To that end, we will compute themonodromy of the polygon defined by (4), take its derivative at z =
1, and hence find theinfinitesimal monodromy.We put the vectors V i ( z ) into columns of matrices as follows: define W i ( z ) ∶= [ V i + ( z ) V i + ( z ) V i ( z )] . Then the relation (4) gives us the matrix equation W i + ( z ) = W i ( z ) U i ( z ) , where U i ( z ) ∶= ⎡⎢⎢⎢⎢⎢⎣ a i z − b i z − c i ⎤⎥⎥⎥⎥⎥⎦ . (5)We stop explicitly recording the dependence on z as it is notationally cumbersome. Inductively,we have that W i = W U U . . . U i − . In particular, W n = W U where U ∶= U U . . . U n − . At the same time, we have that V i + n = M z V i , where M z is a matrixrepresenting the monodromy of the polygon defined by the vectors V i . This means that W n = M z W . Relating these two expressions for W n we get W U = M z W ⇐⇒ M z = W U W − . Notice that because V , V , V are fixed we have that W = [ V V V ] is constant while z varies. This means that all the dependence of M z on z is contained in the expression for U .This gives dM z dz = ddz ( W U . . . U n − W − )= n − ∑ i = W U . . . U i − dU i dz U i + . . . U n − W − = n − ∑ i = W i dU i dz U i + . . . U n − W − , where the last equality uses that W i = W U . . . U i − . Further, observe that U i + . . . U n − = ( U . . . U i ) − ( U . . . U n − ) = ( W − W i + ) − ( W − W n ) = W − i + W n . Also using that W n W − = M z , we get dM z dz = n − ∑ i = W i dU i dz W − i + W n W − = ( n − ∑ i = W i dU i dz W − i + ) M z . urther, using that the monodromy satisfies M = I because we started with a closed n -gon, wearrive at dM z dz ∣ z = = n − ∑ i = S i , where S i ∶= ( W i dU i dz W − i + ) ∣ z = . Now, we will show that summing these S i with i = , , . . . , n − dU i dz ∣ z = = ⎡⎢⎢⎢⎢⎢⎣ − b i − c i ⎤⎥⎥⎥⎥⎥⎦ . Further, observe that for z = W i sends the standard basis to the lifts V i + , V i + , V i of the vertices of P . Therefore W − i + takes the vectors V i + , V i + , V i + to the standard basis, fromwhich we find that the matrix S i acts on these vectors as V i + ↦ − b i V i + − c i V i , V i + ↦ V i + ↦ . Using also (3), we find that S i ( V i ) = c i S i ( V i + ) = − b i c i V i + − V i , which means that S i ( V ) = ∣ V, V i + , V i + ∣∣ V i , V i + , V i + ∣ ( − V i − b i c i V i + ) ∀ V ∈ R , where ∣ A, B, C ∣ is the determinant of the matrix with columns A, B, C . Further, rewriting (3)as − V i − b i c i V i + = a i c i V i + − c i V i + we get S i ( V ) = ∣ V i + , V i + , V ∣∣ V i + , V i + , V i ∣ ( a i c i V i + − c i V i + ) = ∣ V i + , V i + , V ∣∣ V i + , V i + , c − i V i + ∣ ( a i c i V i + − c i V i + ) , where in the last equality we used (3) to express V i in terms of V i + , V i + , V i + . This can berewritten as S i ( V ) = ∣ V i + , V i + , V ∣∣ V i + , V i + , V i + ∣ a i V i + − ∣ V i + , V i + , V ∣∣ V i + , V i + , V i + ∣ V i + , (6)and the first term can be further rewritten as ∣ V i + , V i + , V ∣∣ V i + , V i + , V i + ∣ a i V i + = ∣ V i + , a i V i + , V ∣∣ V i + , V i + , V i + ∣ V i + = ∣ V i + , V i + − c i V i , V ∣∣ V i + , V i + , V i + ∣ V i + = − ∣ V i + , V, V i + ∣∣ V i + , V i + , V i + ∣ V i + + ∣ V i , V i + , V ∣∣ V i + , V i + , V i + ∣ c i V i + (7)where in the second equality we used (3) to express a i V i + in terms of V i , V i + , V i + . Furthermore,using (3) to express V i + in terms of V i , V i + , V i + , the last term in the latter expression can berewritten as ∣ V i , V i + , V ∣∣ V i + , V i + , V i + ∣ c i V i + = ∣ V i , V i + , V ∣∣ V i , V i + , V i + ∣ V i + . (8) ombining (6), (7), and (8), we arrive at the following expression S i ( V ) = − ∣ V i + , V, V i + ∣∣ V i + , V i + , V i + ∣ V i + + ∣ V i , V i + , V ∣∣ V i , V i + , V i + ∣ V i + − ∣ V i + , V i + , V ∣∣ V i + , V i + , V i + ∣ V i + . Since the last two terms only differ by a shift in index, and the sequence of V i ’s in n -periodic,we get dM z dz ∣ z = ( V ) = − n − ∑ i = S i ( V ) = n − ∑ i = ∣ V i + , V, V i + ∣∣ V i + , V i + , V i + ∣ V i + = − n − ∑ i = ∣ V i − , V, V i + ∣∣ V i − , V i , V i + ∣ V i , which coincides with Glick’s operator (1) up to a scalar matrix. Thus, Theorem 1.1 (=Theo-rem 3.2) is proved. References [1] M. Arnold, D. Fuchs, I. Izmestiev, and S. Tabachnikov. Cross-ratio dynamics on idealpolygons. arXiv:1812.05337 , 2018.[2] V.V. Fock and A. Marshakov. Loop groups, clusters, dimers and integrable systems. In
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