Pentagram map


In mathematics, the pentagram map is a discrete dynamical system acting on polygons in the projective plane. It defines a new polygon by taking the intersections of the "shortest" diagonals, and constructs a new polygon from these intersections. This is a projectively equivariant procedure, hence it descends to the moduli space of polygons and defines another dynamical system. It was first introduced by Richard Schwartz in 1992.
The pentagram map on the moduli space is famous for its complete integrability and its interpretation as a cluster algebra.
It admits many generalizations in projective spaces and other settings.

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 [hexagon theorem|Pappus's theorem] and Poncelet's porism.

Definitions and first properties

Definition of the map

Let be an integer. A polygon with sides, or -gon, is a tuple of vertices lying in some projective plane, where the indices are understood modulo. The dimension of the space of -gons is.
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 defines a new point
This procedure defines a new -gon.
The labeling of the indices of 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. Indeed, each coordinate of is given as a rational function of the coordinates of, since it is defined as the intersection of lines passing by them. Moreover, the inverse map is given by taking the intersections, 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. Such a transformation acts on a polygon by sending it to. 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.

Twisted polygons

The pentagram map naturally generalizes on the larger space of twisted polygons. For any integer, a twisted -gon is the data of:
such that for any, the property is satisfied. The dimension of the space of twisted -gon is.
When, this gives back the initial definition of polygons. The space of closed -gons is of codimension in the space of twisted ones.
The action of projective transformations over the space of closed polygons generalizes to the space of twisted ones. This provides again a moduli space, of dimension.

Collapsing of convex polygons

Exponential shrinking

Let 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.
  1. The image of a strictly convex polygon is contained in its interior, and is also strictly convex.
  2. There exists a constant, depending on, such that for any, the diameters of the iterates verify the inequality
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 formula for the limit point is found in. It is a degree 3 polynomial equation that the coordinates of the limit point must satisfy. The coefficients of the polynomial 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 the eigenline of a certain linear operator of.
This operator was reinterpreted in as the infinitesimal monodromy of the polygon. The scalling symmetry is used to deform a closed polygon into a family of twisted ones with monodromy. The infinitesimal monodromy is defined to be:

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 will map to a projectively equivalent polygon. This means that, on the moduli space, the orbit of the class of 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.
The pentagram map is the identity on the moduli space of pentagons.
The map is the identity on the space of labeled hexagons, up to a shift of labeling.
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.

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 -gons lying on the real projective plane, the polygon is projectively equivalent to. In fact, when 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
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:
Since the cross-ratio is projective invariant, the sequences and associated to a twisted -gon are periodic.
When working with -gon in the projective plane over a field, the corner invariants are elements of. The corner invariants realize an isomorphism of variety between the moduli space of twisted -gons and.

ab-coordinates

There is a second set of coordinates for the moduli space of twisted -gons defined over a field satisfying, and such that is not divisible by.
The vertices 's in the projective plane can be lifted to vectors 's in the affine space so that each consecutive triple of vectors spans a parallelepiped having determinant equal to. This leads to the relation
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:

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:

The scaling symmetry

The multiplicative group acts on the moduli space in the following way:
where is called the scaling action an is the scaling parameter. This action commutes with the pentagram map on the moduli space. This property is called the scaling symmetry, and is instrumental in proving the complete integrability of the dynamic.
An homogeneous polynomial is said to have weight if

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
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 be a lift of the monodromy of a twisted -gon. The quantities
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
have the same properties, but turn out to be polynomials in the corner invariants. They can be written as
where each are homogeneous polynomials respectively of weight and . The quantities are unchanged by the dynamic, and are called the monodromy invariants. Moreover, they are algebraically independent.

Polygons on conics

Whenever is inscribed on a conic section, one has for all. Moreover, if 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.

Poisson bracket

An invariant Poisson bracket on the space of twisted polygons was found in. The monodromy invariants commute with respect to it:
for all.
The Poisson bracket is defined in terms of the corner coordinates by:
for all other

The spectral curve

Let be an element of the multiplicative group and be the polygon obtained by applying the rescaling action on. A Lax matrix is a lift of the monodromy of satisfying a zero-curvature equation. Then, the spectral function is the bivariate characteristic polynomial
or some renormalization it. The spectral curve is the projective completion of the affine curve defined by the equation. It is invariant under the pentagram map, and the monodromy invariants appear as the coefficients of. Its geometric genus is if is odd, and if 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, where the motion is conjugated to 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 and are Casimir invariants for the bracket, meaning that
for all functions. When is even, this is also true for the monodromy invariants and.
This allows to consider the Casimir level set, where each Casimir has a specified value. 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. Since the monodromy invariants Poisson-commute and there are enough of them, the discrete Liouville–Arnold theorem can be applied to prove the result.
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 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.

Dimension of the invariant manifold

For a twisted -gons, the dimension of the invariant tori is
and drops by for closed -gons.
Moreover, when 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 and real numbers and, consider the chord connecting to. The envelope of all these chords is a new curve. When is extremely small, the curve is a good model for the time evolution of the original curve 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 is singular for the pentagram map, then there exists an integer such that not singular for the iterate map.
Moreover, the pentagram map exhibit the Devron property. This means that if a polygon is singular for some iterate of the pentagram map, then it will also be singular for some iterate of the inverse map.

Generalizations

The definition of twisted polygons still makes sense in any projective space, under the action of the projective group. 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 and be a twisted polygon of in general position.

Short diagonal pentagram maps

The -th short diagonal hyperplane is uniquely defined by passing through the vertices. Generically, the intersection of consecutive hyperplanes uniquely defines a new point
Doing this for every vertex defines a new twisted polygon. This map, denoted by, is again projectively equivariant.

Generalized pentagram maps

The previous procedure can be generalized. Let be two sets of integers, respectively called the jump tuple and the intersection tuple. Define the -th hyperplane to be passing through the vertices. A new point is given by the intersection
The map is called a generalized pentagram map. It is conjectured that the maps are integrable for any, but that the general case is not.
Some of these maps are discretizations of higher dimensional counterpart of the Boussinesq equation in the KdV hierarchy.

Dented pentagram maps

Fix an integer. Consider the jump tuple, where the is at the -th place, and the intersection tuple. The dented pentagram map is. They are proved to be integrable.
For an integer, the deep dented pentagram map is the same map as before, but the number in the definition of is replaced by. This kind of pentagram maps are again integrable.

Corrugated polygons

A twisted polygon lying in is said to be corrugated if for any, the vertices span a projective two-dimensional plane. Such polygons are not in general position. A new point is defined by
The map 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 be integers. The pentagram map can also be generalized to the space of Grassmannians, which consists of -dimensional linear subspaces of an -dimensional vector space. When, the linear subspaces are lines, which retrieves the definition of projective spaces.
A point in is represented by an matrix such that its columns form a basis of. Consider the diagonal action of the general linear group on each column of. This defines an action on the Grassmannian, even though it's not faithfull. Hence, the polygons of and their moduli spaces are defined as before, after the change of underlying group.
Depending on the parity of, one can define linear subspaces spanned by some 's such that taking their intersection generically define a new point of. 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.

Works cited

Category:Projective geometry
Category:Dynamical systems
Category:Integrable systems
Category:Polygons