## 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)

Zbl 0765.52004
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### References:

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