1,540 research outputs found
Liouville-Arnold integrability of the pentagram map on closed polygons
International audienceThe pentagram map is a discrete dynamical system defined on the moduli space of polygons in the projective plane. This map has recently attracted a considerable interest, mostly because its connection to a number of different domains, such as classical projective geometry, algebraic combinatorics, moduli spaces, cluster algebras, and integrable systems. Integrability of the pentagram map was conjectured by Schwartz and proved by the present authors for a larger space of twisted polygons. In this article, we prove the initial conjecture that the pentagram map is completely integrable on the moduli space of closed polygons. In the case of convex polygons in the real projective plane, this result implies the existence of a toric foliation on the moduli space. The leaves of the foliation carry affine structure and the dynamics of the pentagram map is quasiperiodic. Our proof is based on an invariant Poisson structure on the space of twisted polygons. We prove that the Hamiltonian vector fields corresponding to the monodromy invariants preserve the space of closed polygons and define an invariant affine structure on the level surfaces of the monodromy invariants
Semitoric integrable systems on symplectic 4-manifolds
Let M be a symplectic 4-manifold. A semitoric integrable system on M is a
pair of real-valued smooth functions J, H on M for which J generates a
Hamiltonian S^1-action and the Poisson brackets {J,H} vanish. We shall
introduce new global symplectic invariants for these systems; some of these
invariants encode topological or geometric aspects, while others encode
analytical information about the singularities and how they stand with respect
to the system. Our goal is to prove that a semitoric system is completely
determined by the invariants we introduce
Polynomial cubic differentials and convex polygons in the projective plane
We construct and study a natural homeomorphism between the moduli space of
polynomial cubic differentials of degree d on the complex plane and the space
of projective equivalence classes of oriented convex polygons with d+3
vertices. This map arises from the construction of a complete hyperbolic affine
sphere with prescribed Pick differential, and can be seen as an analogue of the
Labourie-Loftin parameterization of convex RP^2 structures on a compact surface
by the bundle of holomorphic cubic differentials over Teichmuller space.Comment: 64 pages, 5 figures. v3: Minor revisions according to referee report.
v2: Corrections in section 5 and related new material in appendix
Quasiperiodic Motion for the Pentagram Map
The pentagram map is a projectively natural iteration defined on polygons,
and also on a generalized notion of a polygon which we call {\it 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 {\it universally convex}, we translate the
integrability into a statement about the quasi-periodic notion of the
pentagram-map orbits. We also explain how the continuous limit of the Pentagram
map is the classical Boissinesq equation, a completely integrable PDE.Comment: This note is a short announcement of arXiv:0810.560
Moment polytopes for symplectic manifolds with monodromy
A natural way of generalising Hamiltonian toric manifolds is to permit the
presence of generic isolated singularities for the moment map. For a class of
such ``almost-toric 4-manifolds'' which admits a Hamiltonian -action we
show that one can associate a group of convex polygons that generalise the
celebrated moment polytopes of Atiyah, Guillemin-Sternberg. As an application,
we derive a Duistermaat-Heckman formula demonstrating a strong effect of the
possible monodromy of the underlying integrable system.Comment: finally a revision of the 2003 preprint. 29 pages, 8 figure
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