The Pentagram Map is Recurrent
نویسنده
چکیده
CONTENTS The pentagram map is defined on the space of convex n-gons
منابع مشابه
Glick’s Conjecture on the Point of Collapse of Axis-aligned Polygons under the Pentagram Maps
The pentagram map has been studied in a series of papers by Schwartz and others. Schwartz showed that an axis-aligned polygon collapses to a point under a predictable number of iterations of the pentagram map. Glick gave a different proof using cluster algebras, and conjectured that the point of collapse is always the center of mass of the axis-aligned polygon. In this paper, we answer Glick’s ...
متن کاملThe Pentagram Integrals for Poncelet Families
The pentagram map is now known to be a discrete integrable system. We show that the integrals for the pentagram map are constant along Poncelet families. That is, if P1 and P2 are two polygons in the same same Poncelet family, and f is a monodromy invariant for the pentagram map, then f(P1) = f(P2). Our proof combines complex analysis with an analysis of the geometry of a degenerating sequence ...
متن کاملThe pentagram map: A discrete integrable system
The pentagram map is a projectively natural transformation defined on (twisted) polygons. A twisted polygon is a map from Z into RP that is periodic modulo a projective transformation called the monodromy. We find a Poisson structure on the space of twisted polygons and show that the pentagram map relative to this Poisson structure is completely integrable. For certain families of twisted polyg...
متن کاملHigher pentagram maps, weighted directed networks, and cluster dynamics
The pentagram map was introduced by R. Schwartz about 20 years ago [25]. Recently, it has attracted a considerable attention: see [11, 16, 17, 20, 21, 22, 26, 27, 28, 29, 30] for various aspects of the pentagram map and related topics. On plane polygons, the pentagram map acts by drawing the diagonals that connect second-nearest vertices and forming a new polygon whose vertices are their consec...
متن کاملOn Integrable Generalizations of the Pentagram Map
In this paper we prove that the generalization to RP of the pentagram map defined in [4] is invariant under certain scalings for any n. This property allows the definition of a Lax representation for the map, to be used to establish its integrability.
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عنوان ژورنال:
- Experimental Mathematics
دوره 10 شماره
صفحات -
تاریخ انتشار 2001