Search

The pentagram map takes a planar polygon $P$ to a polygon $P'$ whose vertices are the intersection points of consecutive shortest diagonals of $P$. The orbit of a convex polygon under this map is a sequence of polygons which converges exponentially to a point. Furthermore, as recently proved by Glick, coordinates of that limit point can be computed as an eigenvector of a certain operator associated with the polygon. In the present paper we show that Glick's operator can be interpreted as the infinitesimal monodromy of the polygon. Namely, there exists a certain natural infinitesimal perturbation of a polygon, which is again a polygon but in general not closed; what Glick's operator measures is the extent to which this perturbed polygon does not close up., Comment: 10 pages, 4 figures; final version accepted to IMRN

In this paper we define a generalization of the pentagram map to a map on twisted polygons in the Grassmannian space Gr(n;mn). We define invariants of Grassmannian twisted polygons under the natural action of SL(nm), invariants that define coordinates in the moduli space of twisted polygons. We then prove that when written in terms of the moduli space coordinates, the pentagram map is preserved by a certain scaling. The scaling is then used to construct a Lax representation for the map that can be used for integration.

Consider the map $S$ which sends a planar polygon $P$ to a new polygon $S(P)$ whose vertices are the intersection points of second nearest sides of $P$. This map is the inverse of the famous pentagram map. In this paper we investigate the dynamics of the map $S$. Namely, we address the question of whether a convex polygon stays convex under iterations of $S$. Computer experiments suggest that this almost never happens. We prove that indeed the set of polygons which remain convex under iterations of $S$ has measure zero, and moreover it is an algebraic subvariety of codimension two. We also discuss the equations cutting out this subvariety, as well as their geometric meaning in the case of pentagons., 9 pages, 9 figures

The pentagram map is a discrete dynamical system defined on the space of polygons in the plane. In the first paper on the subject, R. Schwartz proved that the pentagram map produces from each convex polygon a sequence of successively smaller polygons that converges exponentially to a point. We investigate the limit point itself, giving an explicit description of its Cartesian coordinates as roots of certain degree three polynomials., Comment: 11 pages, 2 figures

We provide a rigorous treatment of continuous limits for various generalizations of the pentagram map on polygons in $\mathbb{RP}^d$ by means of quantum calculus. Describing this limit in detail for the case of the short-diagonal pentagram map, we verify that this construction yields the $(2,d+1)$-KdV equation, and moreover, the Lax form of the pentagram map in the limit is proved to become the Lax representation of the corresponding KdV system. More generally, we introduce the $\chi$-pentagram map, a geometric construction defining curve evolutions by directly taking intersections of subspaces through specified points. We show that its different configurations yield certain other KdV equations and provide an argument towards disproving the conjecture that any KdV-type equation can be discretized through pentagram-type maps., Comment: 21 pages, 3 figures. v2: minor changes for clarity; to appear in Journal of Geometry and Physics

We survey some recent results concerning projective configuration theorems in the spirit of the classical theorems of Pappus, Desargues, Pascal, …We hope that this modern take on the old theorems makes this evergreen topic fresh again. We connect configuration theorems to completely integrable systems, identities in Lie algebras of motion, modular group, and other subject of contemporary interest.

The pentagram map was introduced by R. Schwartz more than 20 years ago. In 2009, V. Ovsienko, R. Schwartz and S. Tabachnikov established Liouville complete integrability of this discrete dynamical system. In 2011, M. Glick interpreted the pentagram map as a sequence of cluster transformations associated with a special quiver. Using compatibility of Poisson and cluster structures and Poisson geometry of directed networks on surfaces, we generalize Glick's construction to include the pentagram map into a family of discrete integrable maps and we give these maps geometric interpretations. The appendix relates the simplest of these discrete maps to the Toda lattice and its tri-Hamiltonian structure.

The pentagram map is a discrete integrable system on the moduli space of planar polygons. The corresponding first integrals are so-called monodromy invariants $E_1, O_1, E_2, O_2,\dots$ By analyzing the combinatorics of these invariants, R.Schwartz and S.Tabachnikov have recently proved that for polygons inscribed in a conic section one has $E_k = O_k$ for all $k$. In this paper we give a simple conceptual proof of the Schwartz-Tabachnikov theorem. Our main observation is that for inscribed polygons the corresponding monodromy satisfies a certain self-duality relation. From this we also deduce that the space of inscribed polygons with fixed values of the monodromy invariants is an open dense subset in the Prym variety (i.e., a half-dimensional torus in the Jacobian) of the spectral curve. As a byproduct, we also prove another conjecture of Schwartz and Tabachnikov on positivity of monodromy invariants for convex polygons., 11 pages, 7 figures

The pentagram map is a discrete dynamical system first introduced by Schwartz in 1992. It was proved to be integrable by Schwartz, Ovsienko, and Tabachnikov in 2010. Gekhtman, Shapiro, and Vainshtein studied Poisson geometry associated to certain networks embedded in a disc or annulus, and its relation to cluster algebras. Later, Gekhtman et al. and Tabachnikov reinterpreted the pentagram map in terms of these networks, and used the associated Poisson structures to give a new proof of integrability. In 2011, Mari Beffa and Felipe introduced a generalization of the pentagram map to certain Grassmannians, and proved it had a Lax representation. We reinterpret this Grassmann pentagram map in terms of noncommutative algebra, in particular the double brackets of Van den Bergh, and generalize the approach of Gekhtman et al. to establish a noncommutative version of integrability.

We revisit recent results on integrable cases for higher-dimensional generalizations of the 2D pentagram map: short-diagonal, dented, deep-dented, and corrugated versions, and define a universal class of pentagram maps, which are proved to possess projective duality. We show that in many cases the pentagram map cannot be included into integrable flows as a time-one map, and discuss how the corresponding notion of discrete integrability can be extended to include jumps between invariant tori. We also present a numerical evidence that certain generalizations of the integrable 2D pentagram map are non-integrable and present a conjecture for a necessary condition of their discrete integrability., Comment: 16 pages, 12 figures

The pentagram map is now known to be a discrete integrable system. We show that the integrals for the pentagram map are constant along Poncelet families. That is, if P 1 P 1 and P 2 P 2 are two polygons in the same Poncelet family, and f f is a monodromy invariant for the pentagram map, then f(P 1 )=f(P 2 ) f ( P 1 ) = f ( P 2 ) . Our proof combines complex analysis with an analysis of the geometry of a degenerating sequence of Poncelet polygons.

These are notes for a series of lectures presented at the ASIDE conference 2016. The definition of a cluster algebra is motivated through several examples, namely Markov triples, the Grassmannians \(\mathit{Gr}_{2}(\mathbb{C}^{n})\), and the appearance of double Bruhat cells in the theory of total positivity. Once the definition of cluster algebras is introduced in several stages of increasing generality, proofs of fundamental results are sketched in the rank 2 case. From these foundations we build up the notion of Poisson structures compatible with a cluster algebra structure and indicate how this leads to a quantization of cluster algebras. Finally we give applications of these ideas to integrable systems in the form of Zamolodchikov periodicity and the pentagram map.

In this paper we investigate discretizations of AGD flows whose projective realizations are defined by intersecting different types of subspace in $\mathbb{RP}^{m}$ . These maps are natural candidates to generalize the pentagram map, itself defined as the intersection of consecutive shortest diagonals of a convex polygon, and a completely integrable discretization of the Boussinesq equation. We conjecture that the r-AGD flow in m dimensions can be discretized using one (r−1)-dimensional subspace and r−1 different (m−1)-dimensional subspaces of $\mathbb{RP}^{m}$ .

Agraïments/Ajudes: Fundación Séneca de la Región de Murcia grant number 19219/PI/14. The second author is AGAUR grant number 2014SGR-568, and the grants FP7-PEOPLE-2012-IRSES 318999 and 316338. The objective of this paper is to provide information on the set of periodic points of a continuous self--map defined in the following compact spaces: S^n (the n--dimensional sphere), S^n S^m (the product space of the n--dimensional with the m--dimensional spheres), CP^n (the n--dimensional complex projective space) and HP^n (the n--dimensional quaternion projective space). We use as main tool the action of the map on the homology groups of these compact spaces.

We introduce a rich family of generalizations of the pentagram map sharing the property that each generates an infinite configuration of points and lines with four points on each line. These systems all have a description as $Y$-mutations in a cluster algebra and hence establish new connections between cluster theory and projective geometry. Our framework incorporates many preexisting generalized pentagram maps due to M. Gekhtman, M. Shapiro, S. Tabachnikov, and A. Vainshtein and also B. Khesin and F. Soloviev. In several of these cases a reduction to cluster dynamics was not previously known., 48 pages, 22 figures, to appear in Proceedings of the London Mathematical Society

We study the stability map from the rigid analytic space of semistable points in P3 to convex sets in the building of Sp2 over a local field and construct a pure affinoid covering of the space of stable points.

In this paper we prove that the generalization to $\mathbb{RP}^n$ of the pentagram map defined in \cite{KS} is invariant under certain scalings for any $n$. This property allows the definition of a Lax representation for the map, to be used to establish its integrability.

These notes summarize two different connections between two discrete integrable systems, the $A_d$ $T$-system and its infinite-rank analog, the octahedron relation, and the pentagram map and its various generalizations., Comment: 21 pages, contribution to FDIS13, July 2013

We prove that, over any field, the dimension of the indeterminacy locus of a rational map \(f: \mathbf{P}^{n} --\rightarrow \mathbf{P}^{n}\) defined by monomials of the same degree d with no common factors is at least \((n - 2)/2\), provided that the degree of f as a map is not divisible by d. This implies upper bounds on the multidegree of f and in particular, when f is birational, on the degree of f −1.

The pentagram map is defined on the space of convex $n$-gons (considered up to projective equivalence) by drawing the diagonals that join second-nearest-neighbors in an $n$-gon and taking the new $n$-gon formed by the intersections. We prove that this map is recurrent; thus, for almost any starting polygon, repeated application of the pentagram map will show a near copy of the starting polygon appear infinitely often under various perspectives.

We introduce a criterion called the Devron property that a discrete dynamical system can possess. The Devron property is said to occur when a class of highly singular inputs of a mapping F are carried by some iterate of $F^{-1}$ to a class of highly singular inputs of $F^{-1}$. The inspiration for this definition is the discovery by R. Schwartz that the pentagram map exhibits this kind of behavior. We investigate occurrences of the Devron property in a number of different dynamical systems., 41 pages, 8 figures; minor revisions; to appear in Journal of Geometry and Physics

The pentagram map was introduced by R. Schwartz in 1992 for convex planar polygons. Recently, V. Ovsienko, R. Schwartz, and S. Tabachnikov proved Liouville integrability of the pentagram map for generic monodromies by providing a Poisson structure and the sufficient number of integrals in involution on the space of twisted polygons. In this paper we prove algebraic-geometric integrability for any monodromy, i.e., for both twisted and closed polygons. For that purpose we show that the pentagram map can be written as a discrete zero-curvature equation with a spectral parameter, study the corresponding spectral curve, and the dynamics on its Jacobian. We also prove that on the symplectic leaves Poisson brackets discovered for twisted polygons coincide with the symplectic structure obtained from Krichever-Phong's universal formula., 33 pages, 1 figure; v3: substantially revised

We give a minimal triangulation η: S 12 3 →S 4 2 of the Hopf map h: S 3→S 2 and use it to obtain a new construction of the 9-vertex complex projective plane.

Several recently introduced and studied planar curve evolution equations turn out to be iterative smoothing procedures that are invariant under the actions of the Euclidean and affine groups of continuous transformations. This paper discusses possible ways to extend these results to the projective group of transformations. Invariant polygon evolutions are also investigated.

The pentagram map is a discrete dynamical system defined on the moduli space of polygons in the projective plane. This map has recently attracted a considerable interest, mostly because its connection to a number of different domains, such as: classical projective geometry, algebraic combinatorics, moduli spaces, cluster algebras and integrable systems. Integrability of the pentagram map was conjectured by R. Schwartz and later proved by V. Ovsienko, R. Schwartz and S. Tabachnikov for a larger space of twisted polygons. In this paper, we prove the initial conjecture that the pentagram map is completely integrable on the moduli space of closed polygons. In the case of convex polygons in the real projective plane, this result implies the existence of a toric foliation on the moduli space. The leaves of the foliation carry affine structure and the dynamics of the pentagram map is quasi-periodic. Our proof is based on an invariant Poisson structure on the space of twisted polygons. We prove that the Hamiltonian vector fields corresponding to the monodoromy invariants preserve the space of closed polygons and define an invariant affine structure on the level surfaces of the monodromy invariants., a revised version. with minor changes to clarify some points

The paper deals with features of a general polygon which are invariant with respect to projective transform. First, some properties of the area of a triangle under projective transform are discussed. New projective triangular invariants of polygons are derived as the quotient of two different products of the areas of triangles formed by the vertices of the polygon. The features are proved to be invariant to numbering of the vertices of the polygon. The number of projective triangular invariants for polygons with a given number of vertices is derived. A method for reduction of the number of invariants in the case of polygons with a high number of vertices is presented. Numerical experiments dealing with three octagons and one polygon with 18 vertices deformed by projective transforms are described.

We extend the definition of the pentagram map from 2D to higher dimensions and describe its integrability properties for both closed and twisted polygons by presenting its Lax form. The corresponding continuous limit of the pentagram map in dimension $d$ is shown to be the $(2,d+1)$-flow of the KdV hierarchy, generalizing the Boussinesq equation in 2D., 11 pages, 3 figures; This note is a short announcement of arXiv:1204.0756

The pentagram map that associates to a projective polygon a new one formed by intersections of short diagonals was introduced by R. Schwartz and was shown to be integrable by V. Ovsienko, R. Schwartz and S. Tabachnikov. Recently, M. Glick demonstrated that the pentagram map can be put into the framework of the theory of cluster algebras. In this paper, we extend and generalize Glick's work by including the pentagram map into a family of discrete completely integrable systems. Our main tool is Poisson geometry of weighted directed networks on surfaces developed by M. Gekhtman, M. Shapiro, and A. Vainshtein. The ingredients necessary for complete integrability -- invariant Poisson brackets, integrals of motion in involution, Lax representation -- are recovered from combinatorics of the networks. Our integrable systems depend on one discrete parameter $k>1$. The case $k=3$ corresponds to the pentagram map. For $k>3$, we give our integrable systems a geometric interpretation as pentagram-like maps involving deeper diagonals. If $k=2$ and the ground field is $\C$, we give a geometric interpretation in terms of circle patterns., One figure and more references added

The pentagram map is a natural iteration on projective equivalence classes of (twisted) n-gons in the projective plane. It was recently proved ([OST]) that the pentagram map is completely integrable, with the complete set of Poisson commuting integrals given by the polynomials O1,...,O[n/2],On and E1,...,E[n/2],En, previously constructed in [S3]. These polynomials are somewhat reminiscent of the symmetric polynomials. It was observed in computer experiments that if a polygon is inscribed into a conic then Oi=Ei for all i. The goal of the paper is to prove this theorem. The proof is combinatorial, and it was also suggested by computer experimentation.