Liouville-Arnold integrability of the pentagram map on closed polygons. (English) Zbl 1315.37035

Ovsienko, Valentin; Schwartz, Richard Evan; Tabachnikov, Serge

Duke Math. J. 162, No. 12, 2149-2196 (2013).

Let \(P=v_1v_2\dots v_n\) be a convex polygon (\(n\)-gon, \(n \geq 5\)) in \({\mathbb R}^2\), and let \(v_{k+n}=v_n\) for all integers \(k\). For any four consecutive points \(v_{i-1},v_{i},v_{i+1},v_{i+2}\), define the new point \(w_i=(v_{i-1},v_{i+1})\cap(v_{i},v_{i+2})\), where \((v_j,v_k)\) stands for the line through \(v_j\) and \(v_k\). Thus we obtain a new \(n\)-gon \(T(P)=w_1w_2\dots w_n\) and the so-called pentagram map \(T: {\mathcal E}_n \rightarrow {\mathcal E}_n\), where \({\mathcal E}_n\) is the moduli space of projective equivalence classes of \(n\)-gons in the projective plane \({\mathbb R}P^2\).
For \(n=5\) or \(n=6\), the pentagram map is always periodic, more precisely \(T\) or \(T^2\) are equal to the identity map. It turns out that the orbit \(P,T(P),T^2(P),\dots\) of a general convex \(n\)-gon \(P\) undergoes quasi-periodic motion. The authors give a purely geometric proof of the following theorem.
Theorem: Almost every point of \({\mathcal E}_n\) lies on a \(T\)-invariant submanifold \(M\) of dimension \(d=n-4\) for \(n\) odd or \(d=n-5\) for even \(n\). The submanifold \(M\) is defined by a locally free action of \(d\) commuting and independent at every point vector fields over \(M\).
To prove this discrete version of an Arnold-Liouville’s type of theorem, some corner invariants \((x_{2i-1},x_{2i})\) for each point \(v_i\) have been introduced. These \(x_1,\dots, x_{2n}\) are the local coordinates in \({\mathcal E}_n\) to express \(T\) as certain rational functions, the Poisson brackets as \(\{ x_i, x_{i+2} \} = (-1)^i x_i x_{i+2}\) (otherwise vanishing), the commuting vector fields as polynomials, etc.

Reviewer: Angel Zhivkov (Sofia)

MSC:

37J35	Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
51A20	Configuration theorems in linear incidence geometry

Keywords:

pentagram map; complete integrability

References:

[1]	V. I. Arnol’d, Mathematical Methods of Classical Mechanics , 2nd ed., Grad. Texts in Math. 60 , Springer, New York, 1989.
[2]	V. Fock and A. Marshakov, Integrable systems, clusters, dimmers, and loop groups , in preparation. · Zbl 1417.37248
[3]	S. Fomin and A. Zelevinsky, Cluster algebras, I: Foundations , J. Amer. Math. Soc. 15 (2002), 497-529. · Zbl 1021.16017 · doi:10.1090/S0894-0347-01-00385-X
[4]	E. Frenkel, N. Reshetikhin, and M. A. Semenov-Tian-Shansky, Drinfeld-Sokolov reduction for difference operators and deformations of \(W\)-algebras, I: The case of Virasoro algebra , Comm. Math. Phys. 192 (1998), 605-629. · Zbl 0916.17020 · doi:10.1007/s002200050311
[5]	M. Gekhtman, M. Shapiro, S. Tabachnikov, and A. Vainshtein, Higher pentagram maps, weighted directed networks, and cluster dynamics , Electron. Res. Announc. Math. Sci. 19 (2012), 1-17. · Zbl 1278.37047 · doi:10.3934/era.2012.19.1
[6]	M. Gekhtman, M. Shapiro, and A. Vainshtein, Cluster Algebras and Poisson Geometry , Amer. Math. Soc., Providence, 2010. · Zbl 1217.13001
[7]	M. Glick, The pentagram map and Y-patterns , Adv. Math. 227 (2011), 1019-1045. · Zbl 1229.05021 · doi:10.1016/j.aim.2011.02.018
[8]	M. Glick, On singularity confinement for the pentagram map , preprint, [math.CO]. 1110.0868v2
[9]	B. Khesin and F. Soloviev, The pentagram map in higher dimensions and KdV flows , Electron. Res. Announc. Math. Sci. 19 (2012), 86-96. · Zbl 1257.37046 · doi:10.3934/era.2012.19.86
[10]	B. Khesin and F. Soloviev, Integrability of higher pentagram maps , in preparation. · Zbl 1280.37056
[11]	I. Krichever, Analytic theory of difference equations with rational and elliptic coefficients and the Riemann-Hilbert problem (in Russian), Uspekhi Mat. Nauk. 59 , no. 6 (2004), 111-150; English translation in Russian Math. Surveys 59 (2004), 1117-1154. · Zbl 1075.39012 · doi:10.4213/rm798
[12]	S. B. Lobb and F. W. Nijhoff, Lagrangian multiform structure for the lattice Gel’fand-Dikii hierarchy , J. Phys. A 43 (2010), no. 7., 11 pp. · Zbl 1184.37056 · doi:10.1088/1751-8113/43/7/072003
[13]	G. Mari-Beffa, On integrable generalizations of the pentagram map , preprint, [math.DS]. 1303.4295v1 · doi:10.1090/conm/285/04730
[14]	G. Mari-Beffa, On generalizations of the pentagram map: Discretizations of AGD flows , J. Nonlinear Sci. 23 (2013), 303-334. · Zbl 1358.37111
[15]	S. Morier-Genoud, V. Ovsienko, and S. Tabachnikov, \(2\)-frieze patterns and the cluster structure of the space of polygons , Ann. Inst. Fourier (Grenoble) 62 (2012), 937-987. · Zbl 1290.13014 · doi:10.5802/aif.2713
[16]	Th. Motzkin, The pentagon in the projective plane, with a comment on Napier’s rule , Bull. Amer. Math. Soc. 51 (1945), 985-989. · Zbl 0060.34103 · doi:10.1090/S0002-9904-1945-08488-2
[17]	V. Ovsienko, R. Schwartz, and S. Tabachnikov, Quasiperiodic motion for the pentagram map , Electron. Res. Announc. Math. Sci. 16 (2009), 1-8. · Zbl 1163.37019 · doi:10.3934/era.2009.16.1
[18]	V. Ovsienko, R. Schwartz, and S. Tabachnikov, The pentagram map: A discrete integrable system , Comm. Math. Phys. 299 (2010), 409-446. · Zbl 1209.37063 · doi:10.1007/s00220-010-1075-y
[19]	V. Ovsienko and S. Tabachnikov, Projective Differential Geometry Old and New: From the Schwarzian Derivative to the Cohomology of Diffeomorphism Groups , Cambridge Univ. Press, Cambridge, 2005. · Zbl 1073.53001
[20]	R. Schwartz, The pentagram map , Experiment. Math. 1 (1992), 71-81. · Zbl 0765.52004
[21]	R. Schwartz, The pentagram map is recurrent , Experiment. Math. 10 (2001), 519-528. · Zbl 1013.52003 · doi:10.1080/10586458.2001.10504671
[22]	R. Schwartz, Discrete monodromy, pentagrams, and the method of condensation , J. Fixed Point Theory Appl. 3 (2008), 379-409. · Zbl 1148.51001 · doi:10.1007/s11784-008-0079-0
[23]	R. Schwartz, S. Tabachnikov, Elementary surprises in projective geometry , Math. Intelligencer 32 (2010), 31-34. · Zbl 1204.51024 · doi:10.1007/s00283-010-9137-8
[24]	R. Schwartz and S. Tabachnikov, The pentagram integrals on inscribed polygons , Electron. J. Combin. 18 (2011), paper 171, 19 pp. · Zbl 1246.37091
[25]	F. Soloviev, Integrability of the pentagram map , to appear in Duke Math. J., preprint, [math.AG]. 1106.3950v3 · Zbl 1282.14061
[26]	A. Tongas and F. Nijhoff, The Boussinesq integrable system: Compatible lattice and continuum structures , Glasg. Math. J. 47 (2005), 205-219. · Zbl 1085.35125 · doi:10.1017/S0017089505002417