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On generalizations of the pentagram map: discretizations of AGD flows
 Journal of Nonlinear Science: Volume 23, Issue
"... Abstract. In this paper we investigate discretizations of AGD flows whose projective realizations are defined by intersecting different types of subspaces in RPm. These maps are natural candidates to generalize the pentagram map, itself defined as the intersection of consecutive shortest diagonals o ..."
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Abstract. In this paper we investigate discretizations of AGD flows whose projective realizations are defined by intersecting different types of subspaces in RPm. These maps are natural candidates to generalize the pentagram map, itself defined as the intersection of consecutive shortest diagonals of a convex polygon, and a completely integrable discretization of the Boussinesq equation. We conjecture that the kAGD flow in m dimensions can be discretized using one k − 1 subspace and k − 1 different m − 1dimensional subspaces of RPm. 1.
On integrable generalizations of the pentagram map
 International Mathematics Research Notices (2014); doi: 10.1093/imrn/rnu044
"... Abstract. In this paper we prove that the generalization to RPn of the pentagram map defined in [3] 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. 1. ..."
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Abstract. In this paper we prove that the generalization to RPn of the pentagram map defined in [3] 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. 1.
Integrable cluster dynamics of directed . . .
"... The pentagram map was introduced by R. Schwartz more than 20 years ago. In 2009, V. Ovsienko, R. Schwartz and S. Tabachnikov established Liouville complete integrability of this discrete dynamical system. In 2011, M. Glick interpreted the pentagram map as a sequence of cluster transformations assoc ..."
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The pentagram map was introduced by R. Schwartz more than 20 years ago. In 2009, V. Ovsienko, R. Schwartz and S. Tabachnikov established Liouville complete integrability of this discrete dynamical system. In 2011, M. Glick interpreted the pentagram map as a sequence of cluster transformations associated with a special quiver. Using compatible Poisson structures in cluster algebras and Poisson geometry of directed networks on surfaces, we generalize Glick’s construction to include the pentagram map into a family of discrete integrable maps and we give these maps geometric interpretations.
ON THE INTEGRABILITY OF THE SHIFT MAP ON TWISTED PENTAGRAM SPIRALS
"... Abstract. In this paper we prove that the shift map defined on the moduli space of twisted pentagram spirals of type (N, 1) posses a Lax representation with an associated monodromy whose conjugation class is preserved by the map. We prove this by finding a coordinate system in the moduli space of tw ..."
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Abstract. In this paper we prove that the shift map defined on the moduli space of twisted pentagram spirals of type (N, 1) posses a Lax representation with an associated monodromy whose conjugation class is preserved by the map. We prove this by finding a coordinate system in the moduli space of twisted spirals, writing the map in terms of the coordinates and associating a natural parameterfree Lax representation. We then show that the map is invariant under the action of a 1parameter group on the moduli space of twisted (N, 1) spirals, which allow us to construct the Lax pair. 1.