Pentagram Map, Twenty Years After

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Pentagram map, twenty years after Stony Brook, March 2014 1 Part 1. A survey of older results Pentagram Map of Richard Schwartz: P P T(P) T(P) A good reference (among many others): http://en.wikipedia. org/wiki/Pentagram_map 2 Example: if n=5 then T (P ) = P . (t1−t2)(t3−t4) Cross-ratio: [t1; t2; t3; t4] = : (t1−t3)(t2−t4) A D’ C’ B E A’ D C Exercise: if n = 6 then T 2(P ) = P . Let's watch an animation 3 Relevant moduli spaces: Cn, space of projective equivalence classes of closed n-gons in RP2 ; dim = 2n − 8; Pn, space of projective equivalence classes of twisted n-gons in RP2; dim = 2n: φ : Z ! RP2 s:t: φ(k + n) = M ◦ φ(k); 8k: M is the monodromy. 4 Theorem (OST 2010) The Pentagram Map is completely inte- grable on the space of twisted n-gons Pn: 1). There are 2[n=2] + 2 algebraically independent integrals; 2). There is an invariant Poisson structure of corank 2 if n is odd, and corank 4 if n is even, such that the integrals Poisson commute. In both cases, dim Pn − corank = 2(number of integrals − corank). 5 Arnold-Liouville integrability (Poisson set-up) Let Mp+2q have independent Poisson commuting ìntegrals' f1; : : : ; fp; fp+1; : : : ; fp+q where f1; : : : ; fp are Casimirs. One has the symplectic foliation given by f1 = const; f2 = const; : : : ; fp = const; and the Lagrangian subfoliation Fq given by fp+1 = const; fp+2 = const; : : : ; fp+q = const: The leaves carry a flat structure given by the commuting fields H ;:::;H : fp+1 fp+q If a (discrete or continuous time) dynamical system on M pre- serves all this, then the motion on the leaves is a parallel trans- lation. If the leaves are compact, they are tori, and the motion is quasi-periodic. 6 Poncelet-style Corollary. If a point is T -periodic then all points on the same leaf are periodic with the same period. ...... .... .... ... ... .. .. .. .. .. .. .. .. .. ..... ................................................................................................................................................................................................................................ ..

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On Generalizations of the Pentagram Map: Discretizations of Agd Flows
ON GENERALIZATIONS OF THE PENTAGRAM MAP: DISCRETIZATIONS OF AGD FLOWS GLORIA MARÍ BEFFA Abstract. In this paper we investigate discretizations of AGD flows whose projective realizations are defined by intersecting different types of subspaces m in RP . 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 m subspaces of RP . 1. Introduction The pentagram map is defined on planar, convex N-gons, a space we will denote by CN . The map T takes a vertex xn to the intersection of two segments: one is created by joining the vertices to the right and to the left of the original one, xn−1xn+1, the second one by joining the original vertex to the second vertex to its right xnxn+2 (see Fig. 1). These newly found vertices form a new N-gon. The pentagram map takes the first N-gon to this newly formed one. As surprisingly simple as this map is, it has an astonishingly large number of properties, see [16, 17, 18], [19], [14] for a thorough description. The name pentagram map comes from the following classical fact: if P 2 C5 is a pentagon, then T (P ) is projectively equivalent to P . Other relations seem to be 2 also classical: if P 2 C6 is a hexagon, then T (P ) is also projectively equivalent to P .
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