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The pentagram map is recurrent. (English) Zbl 1013.52003

Start with a strictly convex \(n\)-gon \(P\). Associate with each vertex \(v\) of \(P\) the half-plane bounded by the line through \(v\)’s two neighbors and not containing \(v\). Let \(P'\) be the intersection of the half-planes associated with the vertices of \(P\). The pentagram map takes the polygon \(P\) to \(P''\), the result of repeating the above operation a second time; there is a natural correspondence of vertices of \(P\) with vertices of \(P''\). This map commutes with any projective transformation, so the pentagram map \(T_n\) is considered as a map of projective equivalence classes of labeled, strictly convex \(n\)-gons. For \(n\geq 7\) the pentagram map does not have finite order.
In this paper it is proved that the map is recurrent, that is, the polygon \(P\) is an accumulation point of the sequence \((T_n^i(P))_i\). This verifies a conjecture of R. E. Schwartz [Exp. Math. 1, 71-81 (1992; Zbl 0765.52004)]. The ideas for the proof came largely from computer experimentation, but this is mentioned only briefly in the paper. The main tool of the proof is a smooth volume form on the space of \(n\)-gons.

MSC:

52A10 Convex sets in \(2\) dimensions (including convex curves)
52A38 Length, area, volume and convex sets (aspects of convex geometry)

Citations:

Zbl 0765.52004
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References:

[1] Arnol’d, V. I. 1978.Mathematical methods of classical mechanicsSecond editionNew York: Springer. [Arnol’d 1978], Graduate Texts in Math. 60
[2] Hilbert D., Geometry and the Imagination. (1950) · Zbl 0047.38806
[3] Schwartz R., Experiment. Math. 1 (1) pp 71– (1992)
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