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Keywords
(4)
boussinesq equation
Integrable System
Natural Transformation
poisson structure
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The Pentagram Map: A Discrete Integrable System
The Pentagram Map: A Discrete Integrable System,10.1007/s002200101075y,Communications in Mathematical Physics,Valentin Ovsienko,Richard Schwartz,Se
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The Pentagram Map: A Discrete Integrable System
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Citations: 6
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Valentin Ovsienko
,
Richard Schwartz
,
Serge Tabachnikov
The pentagram map is a projectively
natural transformation
defined on (twisted) polygons. A twisted polygon is a map from $${\mathbb Z}$$ into $${{\mathbb{RP}}^2}$$ 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 polygons, such as those we call universally convex, we translate the integrability into a statement about the quasiperiodic motion for the dynamics of the pentagram map. We also explain how the pentagram map, in the continuous limit, corresponds to the classical Boussinesq equation. The
Poisson structure
we attach to the pentagram map is a discrete version of the first
Poisson structure
associated with the Boussinesq equation. A research announcement of this work appeared in [16].
Journal:
Communications in Mathematical Physics  COMMUN MATH PHYS
, vol. 299, no. 2, pp. 409446, 2010
DOI:
10.1007/s002200101075y
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Citations
(6)
The Link Between Erectile and Cardiovascular Health: The Canary in the Coal Mine
(
Citations: 1
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David R. Meldrum
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Richard Evan Schwartz
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Published in 2010.