Non-integrability vs. integrability in pentagram maps

JOURNAL OF GEOMETRY AND PHYSICS

卷:87

Article

Khesin, Boris¹  Soloviev, Fedor^2,3 

[1] Univ Toronto, Dept Math, Toronto, ON M5S 2E4, Canada

[2] Fields Inst, Toronto, ON, Canada

[3] Univ Montreal, CRM, Montreal, PQ, Canada

关键词: Integrable systems;    Pentagram maps;    Lax representation;    Discrete dynamics;    Arnold-Liouville theorem;   

DOI  :  10.1016/j.geomphys.2014.07.027

来源: Elsevier

【 摘 要 】

We revisit recent results on integrable cases for higher-dimensional generalizations of the 2D pentagram map: short-diagonal, dented, deep-dented, and corrugated versions, and define a universal class of pentagram maps, which are proved to possess projective duality. We show that in many cases the pentagram map cannot be included into integrable flows as a time-one map, and discuss how the corresponding notion of discrete integrability can be extended to include jumps between invariant tori. We also present a numerical evidence that certain generalizations of the integrable 2D pentagram map are non-integrable and present a conjecture for a necessary condition of their discrete integrability. (C) 2014 Elsevier B.V. All rights reserved.

【 授权许可】

Free