Geodesic
The pentagram map is recurrent.
Experimental mathematics, Tome 10 (2001) no. 4, pp. 519-528
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Keywords: convex polygon, projective transformation, pentagram map, recurrent, volume form 
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Schwartz, Richard Evan. The pentagram map is recurrent.. Experimental mathematics, Tome 10 (2001) no. 4, pp. 519-528. http://geodesic.mathdoc.fr/item/EXMA_2001__10_4_50067/