TY - JOUR

T1 - Quasiperiodic motion for the pentagram map

AU - Ovsienko, Valentin

AU - Schwartz, Richard

AU - Tabachnikov, Serge

PY - 2009

Y1 - 2009

N2 - The pentagram map is a projectively natural iteration defined on polygons, and also on a generalized notion of a polygon which we call twisted polygons. In this note we describe our recent work on the pentagram map, in which 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 in the sense of Arnold-Liouville. For certain families of twisted polygons, such as those we call universally convex, we translate the integrability into a statement about the quasi-periodic motion of the pentagram-map orbits. We also explain how the continuous limit of the pentagram map is the classical Boussinesq equation, a completely integrable P.D.E.

AB - The pentagram map is a projectively natural iteration defined on polygons, and also on a generalized notion of a polygon which we call twisted polygons. In this note we describe our recent work on the pentagram map, in which 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 in the sense of Arnold-Liouville. For certain families of twisted polygons, such as those we call universally convex, we translate the integrability into a statement about the quasi-periodic motion of the pentagram-map orbits. We also explain how the continuous limit of the pentagram map is the classical Boussinesq equation, a completely integrable P.D.E.

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U2 - 10.3934/era.2009.16.1

DO - 10.3934/era.2009.16.1

M3 - Article

AN - SCOPUS:64549151494

SN - 1079-6762

VL - 16

SP - 1

EP - 8

JO - Electronic Research Announcements of the American Mathematical Society

JF - Electronic Research Announcements of the American Mathematical Society

ER -