The Pentagram Map: A Discrete Integrable System

Abstract

The pentagram map is a projectively natural transformation defined on (twisted) polygons. A twisted polygon is a map from Z into RP 2 that is periodic modulo a projective transformation called the monodromy. We find a Poisson structure on the space of twisted polygons and show that the pentagram map relative to this Poisson structure is completely integrable. For certain families of twisted polygons, such as those we call universally convex, we translate the integrability into a statement about the quasi-periodic motion for the dynamics of the pentagram map. We also explain how the pentagram map, in the continuous limit, corresponds to the classical Boussinesq equation. The Poisson structure we attach to the pentagram map is a discrete version of the first Poisson structure associated with the Boussinesq equation. A research announcement of this work appeared in [16].

Triangular reductions of the 2D toda hierarchy

“…, v n ) is mapped to a point which is the intersection of two diagonals (v i−1 , v i+1 ) and (v i , v i+2 ). If n and k + 1 are co-prime, then, as shown in [13], the moduli space of n-gons in RP k is isomorphic, as algebraic varieties, to the space E k+1,n of n-periodic linear difference equations (1.1) V i = a V i+n = (−1) k V i .…”

“…In [13] it was shown that the pentagram map is a discrete complete integrable system, i.e. that the space of n-periodic lower-triangular operators (1.1) of order 3 is a Poisson manifold and a complete set of integrals in involution for the pentagram map was constructed.…”

Sobre reticulados de Coxeter-Toda

“…Usually, in the literature, the above bracket (2.25) is written as 26) where ∇f (X) := χ −1 (df (X)) is the gradient of f at X, i.e.,…”

“…Since T is invariant under projective transformations the map T can be extended to the projective plane on a wider class of generic n-gons called twisted n-gons on RP 2 . In [26] was proved that the pentagram map T is a discrete integrable system in the sense of Liouville-Arnold and it is a discretization of Boussinesq equation.…”

“…R. Schwartz [31] has founded other additional 2[n/2] invariants of T , called the monodromy invariants. In [26], considering the Poisson bracket (only listed the non-zero brackets) The complete integrability of the pentagram map T follows from the above proposition and the algebraic independence of the monodromy invariants. There exists another coordinate system on P n which allow see P n globally diffeomorphic to R 2n provided n is not divisible by 3.…”