A family of integrable transformations of centroaffine polygons: geometrical aspects

Abstract

Two polygons, (P1, . . . , Pn) and (Q1, . . . , Qn) in R 2 are c-related if det(Pi, Pi+1) = det(Qi, Qi+1) and det(Pi, Qi) = c for all i. This relation extends to twisted polygons (polygons with monodromy), and it descends to the moduli space of SL(2, R 2 )-equivalent polygons. This relation is an equiaffine analog of the discrete bicycle correspondence studied by a number of authors. We study the geometry of this relations, present its integrals, and show that, in an appropriate sense, these relations, considered for different values of the constants c, commute. We relate this topic with the dressing chain of Veselov and Shabat. The case of small-gons is investigated in detail.

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“…This proves items (1) and (4) of Theorem 8. Lemma 4.18 proves item (2). Item (3) follows from the well-known fact that 𝜀(𝑠) of formula (56) is "flat" at 𝑠 = 0 (all derivatives exist and vanish).…”

“…where 𝑎 2 = 1 − 𝑐 2 . The associated 𝑐-related curve 𝛿 = 𝑓𝛾 + 𝑐𝛾 ′ is non-periodic and stays bounded; it is self-Bäcklund with a parameter shift 𝛼 satisfying tanh(𝑢𝛼) = 𝑢 tan 𝛼, where 𝑢 = √ 1 − 𝑐 2 𝑐 , and the constant determinant is sin 𝛼.…”

“…Similarly to its continuous version, it is a completely integrable dynamical system. We refer to [2] for a detailed study; see also [33].…”

“…In this paper we are mostly interested in self-Bäcklund centroaffine curves, the curves 𝛾(𝑡) for which there exist 𝛼 ∈ (0, 𝜋) and a constant 𝑐 such that (2) [𝛾(𝑡), 𝛾(𝑡 + 𝛼)] = 𝑐 for all 𝑡.…”

“…We describe a discrete version of Bäcklund transformation on centroaffine polygons (this transformation is studied in detail in [2]). Theorem 10 presents some pairs (𝑛, 𝑘) for which non-trivial self-Bäcklund polygons do not exist, and some pairs for which they do.…”