The pentagram integrals for Poncelet families. (English) Zbl 1309.37068
The pentagram map is a projectively natural map defined on the space of \(n\)-gons. It is now known to be a discrete integrable system. In this work, it is shown that the integrals of the pentagram map are constant along Poncelet families. That is, if \(P_1\) and \(P_2\) are two polygons in the same Poncelet family and \(f\) is a monodromy invariant for the pentagram map, then \(f(P_1) = f(P_2)\). The proof combines complex analysis with an analysis of the geometry of a degenerating sequence of Poncelet polygons.
MSC:
| 37K20 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions |
| 51M20 | Polyhedra and polytopes; regular figures, division of spaces |
| 14N05 | Projective techniques in algebraic geometry |
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