Polygonal dynamics on projective spaces exhibit collapsing to a limit point expressible via roots of d+1 degree polynomials, proven in select cases and conjectured generally, with applications to P1 systems and a new staircase cross-ratio dynamics.
Discrete Monodromy, Pentagrams, and the Method of Condensation
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abstract
This paper considers a simple geometric construction, called the Pentagram map. The pentagram map, performed on N-gons, gives rise to a birational mapping on the space of all N-gons. This paper finds what conjecturally are all the invariants for this map, and along the way relates the construction to the monodromy of 3rd order differential equations, and also to Dodgson's method of condensation for computing determinants.
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Collapsing in polygonal dynamics
Polygonal dynamics on projective spaces exhibit collapsing to a limit point expressible via roots of d+1 degree polynomials, proven in select cases and conjectured generally, with applications to P1 systems and a new staircase cross-ratio dynamics.