Notes of Richard Schwartz’ fifth Cambridge lecture 24-05-2017

The pentagram map and discrete integrable systems

Joint work with Valentin Ovsienko and Serge Tabachnikov

Start with a convex polygon. Draw diagonals between vertices at distance 2, they form a smaller polygon inside. Call this the pentagram map, although the number of sides is arbitrary.

The case of pentagons has been studied for centuries. Gauss’ Pentagrammon magnificum. Motzlan 1947.

In the non convex case, it becomes a projective construction (it is only generically defined). Hence it commutes with projective transformations. The space ${P_n}$ of ${n}$-gons up to projective transformation has dimension ${2n-8}$. The subset ${C_n}$ of convex polygons is homeomorphic to a ball (in fact, ${C_n}$ is a triangle bundle over ${C_{n-1}}$, as is seen by adding an extra vertex).

Let ${\phi:P_n\rightarrow P_n}$ denote the pentagram map.

Theorem 1 (Classical) If ${n=5,6}$, ${\phi}$ is periodic.

Let ${n=5}$. To a vertex, associate the cross-ratio of the sides and diagonals emanating from it. These numbers determine the polygon up to projective transformations. The cross-ratio of a vertex of ${\phi(P)}$ equals the cross-ratio of the 4 points along a diagonal, i.e. the cross-ratio of a vertex of ${P}$. Hence ${\phi^2=Id}$.

For ${n=6}$, I checked it using Mathematica.

For ${n\geq 7}$, no periodicity. Numerical experiments suggest that ${C_n}$ is foliated by ${\lfloor \frac{n-1}{2}\rfloor}$-tori, each with a natural flat structure. Whence the title. I will explain the origin of these tori.

1. Shrinking

Fact. The product of vertex cross-ratios ${\chi(P)}$ is ${\phi}$-invariant. Indeed, its log is the Hilbert length of ${\phi(P)}$ as a subset of ${P}$. View it as function of a point on a side. Complexify. Get a bounded holomorphic function on ${{\mathbb C}}$, it is constant. Checking on the regular polygon shows that ${\chi(\phi(P))=\chi(P)}$.

It implies that the diameter of ${\phi^j(P)}$ tends to 0.

Theorem 2 (Max Glick 2018) The shrink point is algebraic in the coordinates of the vertices.

Indeed, lift vertices to vectors in ${{\mathbb R}^3}$. Define an endomorphim of ${{\mathbb R}^3}$ by

$\displaystyle \begin{array}{rcl} T_P(A)=\sum_{i}\frac{det(V_{i-1},A,V_{i}}{det(V_{i-1},V_i,V_{i+1}}V_i, \end{array}$

Then ${T_{\phi(P)}=T_{P}}$.

A Poncelet polygon is a polygon which is inscribed in a conic and superscribed about a conic.

Poncelet polygons are fixed points of ${\phi}$.

2. Integrable systems

2.1. 4 dimensions

Let ${M}$ b a smooth closed 4-manifold, with a symplectic form ${\omega}$. To a smooth function ${f}$ on ${M}$, associate the Hamiltonian vector field ${H_f}$ such that ${\omega(H_f,V)}$. The flow of ${H_f}$ is tangent to the level sets of ${f}$.

Definition 3 Define the Poisson bracket by

$\displaystyle \begin{array}{rcl} \{f,g\}:=\omega(H_f,H_g). \end{array}$

Alternatively,

$\displaystyle \begin{array}{rcl} H_{\{f,g\}}=[H_f,H_g]. \end{array}$

Hence vanishing Poisson bracket means that the corresponding Hamiltonian flow commute. In non-degenerate cases, this implies that joint level curves are smooth surfaces with a natural map to ${{\mathbb R}^2}$. If the orbit of this action of ${{\mathbb R}^2}$ is closed, it must be a torus.

2.2. Higher dimensions

If, in dimension ${2n}$, ${n}$ functions Poisson-commute, one gets a map to ${{\mathbb R}^n}$ whose level sets are unions of orbits of an ${{\mathbb R}^n}$ action.

More generally, a Poisson bracket is a skew-symmetric map on vector fields, which is a derivation of the algebra of smooth functions and satisfies Jacobi identity.

I intend to turn ${P_n}$ into a symplectic manifold (in fact, merely indicate the Poisson bracket), in such a way that ${\phi}$ is symplectic. Then I will find ${n}$ Poisson-commuting functions.

F. Soloviev goes farther: he relates the dynamics of ${\phi}$ to Riemann surfaces, and so on.

3. Coordinates on ${P_n}$

Assign a cross-ratio to a flag, i.e. a vertex and an adjacent edge, ${\chi(e,v)=[u:v:w:x]}$ where ${u}$ is previous vertex, ${w}$ and ${x}$ are intersections of side ${e}$ with two next edges. This gives a map ${(I_1,\ldots,I_n):P_n\rightarrow{\mathbb R}^{2n}}$, this is too much for coordinates.

Define a twisted ${n}$-gon as a map ${\psi:{\mathbb Z}\rightarrow{\mathbb R} P^2}$ such that ${\psi(n+k)=M\psi(k)}$ for all ${k}$, for some projective transformation ${M}$. ${M}$ is called the monodromy. ${I_i}$‘s define coordinates on this enlarged space.

In terms of cross-ratio coordinates, ${trace(M)}$ and ${trace(M^{adj})}$ are rational functions, almost polynomials (only products of cross-ratios appear in the denominator).

3.1. Poisson structure

Define

$\displaystyle \begin{array}{rcl} \{I_0,I_2\}=I_0 I_2,\quad \{I_2,I_4\}=I_2 I_4,\quad \{I_1,I_3\}=-I_1 I_3,\quad \{I_3,I_5\}=I_3 I_5,\ldots \end{array}$

and all other Poissons brackets vanish. This defines a Poisson structure on twisted ${n}$-gons.