Cross-ratio Dynamics on Ideal Polygons

Abstract

Two ideal polygons, $(p_1,\ldots ,p_n)$ and $(q_1,\ldots ,q_n)$, in the hyperbolic plane or in hyperbolic space are said to be $\alpha $-related if the cross-ratio $[p_i,p_{i+1},q_i,q_{i+1}] = \alpha $ for all $i$ (the vertices lie on the projective line, real or complex, respectively). For example, if $\alpha = -1$, the respective sides of the two polygons are orthogonal. This relation extends to twisted ideal polygons, that is, polygons with monodromy, and it descends to the moduli space of Möbius-equivalent polygons. We prove that this relation, which is generically a 2-2 map, is completely integrable in the sense of Liouville. We describe integrals and invariant Poisson structures and show that these relations, with different values of the constants $\alpha $, commute, in an appropriate sense. We investigate the case of small-gons and describe the exceptional ideal pentagons and hexagons that possess infinitely many $\alpha $-related polygons.

A family of integrable transformations of centroaffine polygons: geometrical aspects

“…The next theorem, providing two additional integrals of the c-relation, is a discrete version of Proposition 3.4 in [16], concerning a continuous version of the c-relation on centroaffine curves. It is also an analog of Theorem 16 in [4].…”

“…As before, we do not dwell here on the question whether the relations from Proposition 3.12 are the only ones satisfied by the integrals when restricted to the moduli space of closed polygons (similarly to [4], we do expect this to be the case).…”

“…Remark 3.9. If one has s 2i−1 = 1 for all i, corresponding -as explained in Introduction -to the case studied in [4], then formula (19) simplifies. The integrals in [4] are given by the formulas…”

“…In this section we consider the moduli space X 5,S . This material is parallel to the one in Section 7.1.3 of [4]. The space X 5,S is 2-dimensional: the variables v 2i satisfy the Ptolemy-Plücker relation v 2 v 4 + v 8 s 3 = s 1 s 5 and its cyclic permutations.…”

“…The cross-ratio relation on projective polygons was thoroughly studied, starting with [11] and, more recently, in [4] and [3]. This is the second source of our motivation: many results in this paper have analogs in [4].…”