Pentagram map

Informal definition On polygons pentagon. Initially, the pentagram map was defined for convex polygons (with at least five sides) on the euclidean plane. Given such a polygon P with n sides, one can draw the "shortest diagonals", meaning the segments whose endpoints are a vertex and one of its second neighbors (as in the picture). The intersections of the shortest diagonals are then taken as the vertices of a new n-gon T(P); this new polygon is the output of the pentagram map. The same construction can be done on non-convex polygons, but there are several complications. First, some consecutive short diagonals may not intersect, so one must extend the segments to lines. Second, the image T(P) can fail to be a new n-gon because some consecutive vertices could coincide. However, this generically doesn't happen. Finally, it is possible that two diagonals are parallel and do not intersect on the euclidean plane. This is resolved by extending the euclidean plane to the real projective plane by the addition of a line at infinity, where the intersection point lies. Hence, the pentagram map is defined for generic polygons on the real projective plane. More generally, the construction of the pentagram map is well defined whenever the concepts of lines and their intersections make sense. This is encompassed by the notion of a general projective plane, of which the real projective plane is one example; but the pentagram map can also be considered over other fields, for instance the complex numbers, which give the complex projective plane. On the moduli space of polygons Since the pentagram map is defined by taking lines and their intersections, it commutes with any map that sends lines to lines. Such maps are called projective transformations. This allows to identify polygons up to projectivity. This identification gives a quotient space (technically called a moduli space) of classes of polygons. The pentagram map on polygons induces another dynamical system on the moduli space, whose behavior differs quite a lot from the initial one. Historical elements The pentagram map for general polygons was introduced in , but the simplest case is the one of pentagons, hence the name "pentagram". Their study goes back to , and . The pentagram map is similar in spirit to the constructions underlying Desargues's theorem, Pappus's theorem and Poncelet's porism. ==Definitions and first properties==

Definition of the map (in particular, non-convex) pentagon. The vertex w_2 is on the line at infinity, because it is the intersection of two parallel lines. Let n\geq 5 be an integer. A polygon P with n sides, or n-gon, is a tuple of vertices (v_1,\dots,v_n) lying in some projective plane \mathbb P ^2, where the indices are understood modulo n. The dimension of the space of n-gons is 2n. Suppose that the vertices are in sufficiently general position, meaning that no consecutive triple of points are collinear. Taking the intersection of the two consecutive "shortest" diagonals{{Efn|Meaning the line between a vertex v_k and a "second neighbour" v_{k\pm 2}.}} defines a new point w_k := \overline{v_{k-1} v_{k+1}} \cap \overline{v_{k} v_{k+2}}. This procedure defines a new n-gon T(P)=(w_1,\dots,w_n). The labeling of the indices of T(P) is not canonical. In most papers, a choice is made at the beginning of the paper and the formulas are tuned accordingly. The pentagram map on polygons is a birational map T:(\mathbb P^2)^n(\mathbb P^2)^n. Indeed, each coordinate of w_k is given as a rational function of the coordinates of v_{k-1},\dots,v_{k+2}, since it is defined as the intersection of lines passing by them. Moreover, the inverse map is given by taking the intersections \overline{w_{k-2} w_{k-1}} \cap \overline{w_{k} w_{k+1}} , which is rational for the same reason. Moduli space The pentagram map is defined by taking lines and intersections of them. The biggest group which maps lines to lines is the one of projective transformations \mathbb P \mathrm{GL}_{3}. Such a transformation M acts on a polygon P by sending it to M \cdot P:=(Mv_1,\dots,Mv_n). The pentagram map commutes with this action, and thereby induces another dynamical system on the moduli space of projective equivalence classes of polygons. Its dimension is 2n-8. Twisted polygons on the real plane. The pentagram map naturally generalizes on the larger space of twisted polygons. For any integer n\geq5, a twisted n-gon P is the data of: • a bi-infinite sequence of points (v_k)_{k\in\mathbb Z} in the projective plane (called the vertices), • a projective transformation M \in \mathbb P \mathrm{GL}_3 (called the monodromy), such that for any k \in \mathbb Z, the property v_{k+n}=Mv_k is satisfied. The dimension of the space of twisted n-gon is 2n+8. When M=\mathrm{Id}, this gives back the initial definition of polygons (which are said to be closed). The space of closed n-gons is of codimension 8 in the space of twisted ones. The action of projective transformations over the space of closed polygons generalizes to the space of twisted ones (the monodromy is changed by conjugation). This provides again a moduli space, of dimension 2n. == Collapsing of convex polygons ==

Exponential shrinking , exhibiting the convergence. Let P be a closed strictly convex polygon lying on the real plane. One of the first results proved by Richard Schwartz it that its iterates under the pentagram map shrink exponentially fast to a point. This follows from two facts. • The image of a strictly convex polygon is contained in its interior, and is also strictly convex. • There exists a constant 0, depending on P, such that for any N \in \mathbb N, the diameters of the iterates verify the inequality \operatorname{diam}(T^N(P))\leq\eta_P^N \operatorname{diam}(P). Hence, by Cantor's intersection theorem, the sequence of polygons collapses toward a point. The behavior on the moduli space is very different, since the dynamic is recurrent. It is even a quasiperiodic motion, as discussed in the section about integrability. Coordinates of the limit point The limit point coordinates are found in . They satisfy some degree 3 polynomial equations, whose coefficients are rational functions in the coordinates of the vertices of the starting polygon. The proof relies on the fact that the limit point must be an eigenline of a certain linear operator of \mathbb R^3. This operator was reinterpreted in as the infinitesimal monodromy of the polygon. The scaling symmetry is used to deform a closed polygon P into a family of twisted ones (P_z)_{z\in \mathbb C^*} with monodromy M_z. The infinitesimal monodromy is defined to be: \left.\frac{dM_z}{dz}\right|_{z=1}. Generalization The collapsing of polygons may also happen in some generalization of the pentagram map, when considering some specific configurations of polygons in the real plane. The coordinates of the collapse point are given by a formula analogous to the one for the original pentagram map. == Periodic orbits on the moduli space ==

For some configurations of closed polygons, the iterate of the pentagram map will send P to a projectively equivalent polygon (up to some shift of the indices). This means that, on the moduli space, the orbit of the class of P is periodic. Pentagons and hexagons The two following facts are proved by checking cross-ratio equalities, so they are true for polygons in any projective plane (not just the real one). The pentagram map T is the identity on the moduli space of pentagons. The second iterate T^2 is the identity on the space of labeled hexagons, up to a shift of labeling. This phenomenon doesn't generalize to generic polygons with at least seven sides, for which the motion is quasi-periodic. The action of the pentagram map on pentagons and hexagons is similar in spirit to classical configuration theorems in projective geometry such as Pascal's theorem, Desargues's theorem and others. Generalization The result about pentagons and hexagons generalizes to some higher pentagram maps in \mathbb P ^k, for polygons with k+3 or 2k+2 sides. The proof uses a generalization of the Gale transform. Poncelet polygons A polygon is said to be Poncelet if it is inscribed in a conic and circumbscribed about another one. For a convex Poncelet n-gons P lying on the real projective plane, the polygon T^2(P) is projectively equivalent to P. In fact, when n is odd, the converse is also true. However, this converse statement is no longer true when the polygons are considered over the complex projective plane. ==Coordinates for the moduli space==

Corner coordinates Define the cross-ratio of four collinear points to be : [a,b,c,d]=\frac{(a-b)(c-d)}{(a-c)(b-d)}. The corner invariants are a system of coordinates on the space of twisted polygons, constructed by taking intersections as on the figure. The left and right invariants are respectively defined as the following cross-ratios: : x_k:=[v_{k-2},v_{k-1},\overline{v_{k-2}v_{k-1}}\cap\overline{v_{k}v_{k+1}},\overline{v_{k-2}v_{k-1}}\cap\overline{v_{k+1}v_{k+2}}], : y_k:=[\overline{v_{k+1}v_{k+2}}\cap\overline{v_{k-2}v_{k-1}}, \overline{v_{k+1}v_{k+2}}\cap\overline{v_{k-1}v_{k}},v_{k+1},v_{k+2}]. Since the cross-ratio is projective invariant, the sequences (x_k)_{k \in \mathbb Z} and (y_k)_{k \in \mathbb Z} associated to a twisted n-gon are n periodic. The corner invariants are elements of \mathbb{P}^1\smallsetminus\{0,1,\infty\}, and they realize an isomorphism of variety between the moduli space of twisted n-gons and (\mathbb{P}^1\smallsetminus\{0,1,\infty\})^{2n}. ab-coordinates There is a second set of coordinates for the moduli space of twisted n-gons defined over a field F satisfying \mathrm{SL}_3(F)\cong \mathbb P\mathrm{GL}_3(F), and such that n is not divisible by 3. The vertices v_k's in the projective plane \mathbb P^2(F) can be lifted to vectors V_k's in the affine space F^3 so that each consecutive triple of vectors spans a parallelepiped having determinant equal to 1. This leads to the relation : V_{k+3} = a_k V_{k+2} + b_k V_{k+1} + V_k. This bring out an analogy between twisted polygons and solutions of third order linear ordinary differential equations, normalized to have unit Wronskian. They are linked to the corner coordinates by: : x_k=\frac{a_{k-2}}{b_{k-2}b_{k-1}}, : y_k=-\frac{b_{k-1}}{a_{k-2}a_{k-1}}. ==Formulas on the moduli space==

As a birational map The pentagram pentagram map is a birational map on the moduli space, because it can be decomposed as the composition of two birational involutions. The corner invariants change in the following way: : x_k'=x_k\frac{1-x_{k-1} y_{k-1}}{1-x_{k+1}y_{k+1}}, : y_k'=y_{k+1}\frac{1-x_{k+2} y_{k+2}}{1-x_k y_k}. The scaling symmetry The multiplicative group F\smallsetminus\{0\} acts on the moduli space in the following way: : R_s\cdot(x_1,\dots,x_n,y_1,\dots,y_n)=(sx_1,\dots,sx_n,s^{-1}y_1,\dots,s^{-1}y_n), where R is called the scaling action an s is the scaling parameter. This action commutes with the pentagram map on the moduli space (as presented in the previous formulas). This property is called the scaling symmetry, and is instrumental in proving the complete integrability of the dynamic. An homogeneous polynomial Q is said to have weight k if : Q(R_s\cdot(x_1,\dots,y_n))=s^kQ(x_1,\dots,y_n). ==Invariant structures==

Monodromy invariants The monodromy invariants, introduced in , are a collection of functions on the moduli space that are invariant under the pentagram map. The simplest example of them are : O_n= x_1x_2\cdots x_{n}, \quad E_n = y_1y_2\cdots y_n. The other monodromy invariants can be retrieved through different points of view: through the scaling symmetry, as combinatorial objects, or as some determinants. The one involving scaling symmetry is presented here. Let M\in \mathrm{GL}_3 be a lift of the monodromy of a twisted n-gon. The quantities : \Omega_1=\frac{\operatorname{trace}^3(M)}{\det(M)}, \quad \Omega_2=\frac{\operatorname{trace}^3(M^{-1})}{\det(M^{-1})}, are independent of the choice of lift and are invariant under conjugation, so they are well defined for the projective class of the polygon. They are invariant under the pentagram map, since the monodromy matrix doesn't change. Now, the quantities : \tilde{\Omega}_1=O_n^2E_n\Omega_1, \quad \tilde{\Omega}_2=O_nE_n^2\Omega_2, have the same properties, but turn out to be polynomials in the corner invariants. They can be written as : \tilde{\Omega}_1=\biggl(\sum_{k=0}^{\lfloor n/2\rfloor}O_k\biggr)^3, \quad \tilde{\Omega}_2=\biggl(\sum_{k=0}^{\lfloor n/2\rfloor}E_k\biggr)^3, where each O_k,E_k are homogeneous polynomials respectively of weight k and -k (with respect to the rescaling action). The quantities O_1,\dots,O_{\lfloor n/2 \rfloor},O_n, E_1,\dots,E_{\lfloor n/2 \rfloor},E_n, are unchanged by the dynamic, and are called the monodromy invariants. Moreover, they are algebraically independent. Polygons on conics Whenever P is inscribed on a conic section, one has O_k(P)=E_k(P) for all k. Moreover, if P is circumscribed about another conic, then its monodromy invariants are characterized by the pair of conics. For such odd-gons, the translation on the Jacobian variety is restricted to the Prym variety (which is a half-dimensional torus in the Jacobian). Poisson bracket An invariant Poisson bracket on the space of twisted polygons was found in . The monodromy invariants commute with respect to it: \{O_i,O_j\}=\{O_i,E_j\}=\{E_i,E_j\}=0 for all i,j. The Poisson bracket is defined in terms of the corner coordinates by: \begin{align} \{x_i,x_{i\pm1}\} &= \mp x_i x_{i+1}, \\ \{y_i,y_{i\pm 1}\} &= \mp y_i y_{i+1}, \\ \{x_i,x_j\} &= \{y_i,y_j\} = \{x_i,y_j\} = 0 \end{align} for all other i,j. The spectral curve Let \zeta be an element of the multiplicative group and P_\zeta be the polygon obtained by applying the rescaling action R_\zeta on P. A Lax matrix \hat{T}(\zeta) \in \mathrm{GL}_3 is a lift of the monodromy of P_\zeta satisfying a zero-curvature equation. Then, the spectral function is the bivariate characteristic polynomial Q(\lambda,\zeta) := \det(\lambda\operatorname{Id}-\hat{T}(\zeta)), or some renormalization it. The spectral curve is the projective completion of the affine curve defined by the equation Q(\lambda,\zeta)=0. It is invariant under the pentagram map, and the monodromy invariants appear as the coefficients of Q. Its geometric genus is n-1 if n is odd, and n-2 if n is even. It was first introduced in for his proof of algebraic integrability. ==Complete integrability==

The pentagram map on the moduli space has been proved to be a completely integrable discrete dynamical system, both in the Arnold-Liouville and the algebro-geometric senses. In any case, this means that the moduli space is almost everywhere foliated by flat tori (or in the algebraic setting, Abelian varieties), where the motion is a translation. This generically makes a quasiperiodic motion. Arnold–Liouville integrability The proof of the integrability of the pentagram map on a real twisted polygon was achieved in . This is done by noticing that the monodromy invariants O_n and E_n are Casimir invariants for the bracket, meaning (in this context) that \{O_n,f\}=\{E_n,f\} = 0 for all functions f. When n is even, this is also true for the monodromy invariants O_{\lfloor n/2 \rfloor } and E_{\lfloor n/2 \rfloor }. This allows to consider the Casimir level set, where each Casimir has a specified value. Because of Sard's theorem, any generic level set is a smooth manifold. They form a foliation in symplectic leaves, on which the Poisson bracket gives rise to a symplectic form. Each of these symplectic leaves has an iso-monodromy foliation, namely, a decomposition into the common level sets of the remaining monodromy functions. By using again Sard's theorem, they are generically Lagrangian manifolds. Moreover, they are compact. Since the monodromy invariants Poisson-commute and there are enough of them, the discrete Liouville–Arnold theorem can be applied to prove that the level sets are flat tori over which the dynamic is a translation. The integrability for real closed polygons was proved in by restricting the Hamiltonian vector fields of monodromy functions to smaller dimensional tori, and showing that enough of them are still independent. Algebro-geometric integrability In , it was shown that the pentagram map admits a Lax representation with a spectral parameter, which allows to prove its algebraic-geometric integrability. This means that the space of polygons (either twisted or closed) is parametrized by its spectral data, consisting of its spectral curve, with marked points and a divisor given by a Floquet–Bloch equation. This gives an embedding to the Jacobian variety through the Abel–Jacobi map, where the motion is expressed in term of translation. The previously defined Poisson bracket is also retrieved. This integrability was generalized in from the field of complex numbers to any algebraically closed field of characteristic different from 2. The translation on a torus is replaced by a translation on an Abelian variety (in fact, a Jacobian variety again). Dimension of the invariant manifold For a twisted n-gons, the dimension of the invariant tori (or Jacobian varieties) is : \begin{cases} n-1 & \text{when }n \text{ is odd,}\\ n-2 & \text{when }n \text{ is even,} \end{cases} and drops by 3 for closed n-gons. Moreover, when n is even, there are two isomorphic Jacobians on which the iterates of the pentagram map alternate. But on each of them, the second iterate of the pentagram is a translation. ==Connections to other topics==

The Boussinesq equation The continuous limit of a convex polygon is a parametrized convex curve in the plane. When the time parameter is suitably chosen, the continuous limit of the pentagram map is the classical Boussinesq equation. This equation is a classical example of an integrable partial differential equation. Here is a description of the geometric action of the Boussinesq equation. Given a locally convex curve C:\mathbb R\to \mathbb R^2 and real numbers x and t, consider the chord connecting C(x-t) to C(x+t) . The envelope of all these chords is a new curve C_t(x) . When t is extremely small, the curve C_t(x) is a good model for the time t evolution of the original curve C_0(x) under the Boussinesq equation. This construction is also similar to the pentagram map. Moreover, the pentagram invariant bracket is a discretization of a well known invariant Poisson bracket associated to the Boussinesq equation. Cluster algebras The pentagram map and some of its generalizations are identified as special cases of cluster algebra. This provides a link with the Poisson–Lie groups, dimer models and other so-called cluster-integrable systems. These methods allow to retrieve the Poisson-bracket and Hamiltonians used to prove complete integrability and provide Lax representations. Singularity theory The pentagram map exhibit a property called singularity confinement, which is typical from integrable systems. It states that if a polygon P is singular for the pentagram map T, then there exists an integer m such that P not singular for the iterate map T^m. Moreover, the pentagram map (along with some of its generalizations and other discrete dynamical systems) exhibit the Devron property. This means that if a polygon P is singular for some iterate of the pentagram map T^m, then it will also be singular for some iterate of the inverse map T^{-m'}. == Generalizations ==

The definition of twisted polygons still makes sense in any projective space \mathbb P^d, under the action of the projective group \mathbb P \mathrm{GL}_{d+1}. The pentagram map can be generalized in many ways, and some of them are presented here. Not all of them are integrable. Some are discretizations of PDEs from the KdV hierarchy, seen as higher dimensional version of Boussinesq or KP equations. The description of all generalized pentagram maps as cluster algebras is still an open question. Polygons in general positions Let d \geq 2 and P be a twisted polygon of \mathbb P^d in general position. Short diagonal pentagram maps The k-th short diagonal hyperplane H_k^{sh} is uniquely defined by passing through the vertices v_k,v_{k+2},\dots,v_{k+2d-2}. Generically, the intersection of d consecutive hyperplanes uniquely defines a new point : T_{sh}v_k:=H_k^{sh}\cap H_{k+1}^{sh}\cap \dots \cap H_{k+d-1}^{sh}. Doing this for every vertex defines a new twisted polygon. This map, denoted by T_{sh}, is again projectively equivariant. Generalized pentagram maps The previous procedure can be generalized. Let I=(i_1,\dots,i_{d-1}),~J=(j_1,\dots,j_{d-1}) be two sets of integers, respectively called the jump tuple and the intersection tuple. Define the k-th hyperplane H_k^I to be passing through the vertices v_k,v_{k+i_1},\dots,v_{k+i_1+\dots+i_{d-1}}. A new point is given by the intersection : T_{I,J}v_k:=H_k^I \cap H_{k+j_1}^I \cap \dots \cap H_{k+j_1+\dots +j_{d-1}}^I. The map T_{I,J} is called a generalized pentagram map. It is conjectured that the maps T_{I,I} are integrable for any I, but that the general case is not (based on numerical experiments that seem to disprove the diophantine integrability test). Some of these maps are discretizations of higher dimensional counterpart of the Boussinesq equation in the KdV hierarchy. Dented pentagram maps Fix an integer m\in \{1,\dots ,d-1\}. Consider the jump tuple I_m:=(1,\dots,1,2,1,\dots,1), where the 2 is at the m-th place, and the intersection tuple J:=(1,\dots,1). The dented pentagram map is T_m :=T_{I_m,J}. They are proved to be integrable. For an integer p \geq 2, the deep dented pentagram map (of depth p) T_m^p is the same map as before, but the number 2 in the definition of I_m is replaced by p. This kind of pentagram maps are again integrable. Corrugated polygons A twisted polygon P lying in \mathbb P^d is said to be corrugated if for any k\in \mathbb Z, the vertices v_k,v_{k+1},v_{k+d},v_{k+d+1} span a projective two-dimensional plane. Such polygons are not in general position. A new point is defined by : T_\text{cor}v_k:=\overline{v_k v_{k+d}}\cap \overline{v_{k+1} v_{k+d+1}}. The map T_\text{cor} yields a new corrugated polygon. They are completely Liouville-integrable. In fact, they can retrieved as some dented pentagram map applied on corrugated polygons. Grassmannians polygons Let d \geq 3, m \geq 1 be integers. The pentagram map can also be generalized to the space of Grassmannians \mathrm{Gr}(m,md), which consists of m-dimensional linear subspaces of an md-dimensional vector space. When m=1, the linear subspaces are lines, which retrieves the definition of projective spaces \mathbb P^d. A point in v\in\operatorname{Gr}(m,md) is represented by an m \times md matrix X_v such that its columns form a basis of v. Consider the diagonal action of the general linear group \mathrm{Gl}_{md} on each column of X_v. This defines an action on the Grassmannian, even though it's not faithfull. Hence, the polygons of \mathrm{Gr}(m,md) and their moduli spaces are defined as before, after the change of underlying group. Depending on the parity of d, one can define linear subspaces spanned by some X_{v_k}'s such that taking their intersection generically define a new point of v\in\mathrm{Gr}(m,md). This generalization of the pentagram map is integrable in a noncommutative sense. Over rings The pentagram map admits a generalization by considering projective planes over stably finite rings, instead of fields. In particular, this retrieves the pentagram map over Grassmanians. Again, it admits a Lax representation. == References ==

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