39 research outputs found
Normal Forms for Symplectic Matrices
We give a self contained and elementary description of normal forms for
symplectic matrices, based on geometrical considerations. The normal forms in
question are expressed in terms of elementary Jordan matrices and integers with
values in related to signatures of quadratic forms naturally
associated to the symplectic matrix.Comment: 27 pages updated version, propositions 12 and 17 added, uniqueness of
normal form precise
The positive equivariant symplectic homology as an invariant for some contact manifolds
We show that positive -equivariant symplectic homology is a contact
invariant for a subclass of contact manifolds which are boundaries of Liouville
domains. In nice cases, when the set of Conley-Zehnder indices of all good
periodic Reeb orbits on the boundary of the Liouville domain is lacunary, the
positive -equivariant symplectic homology can be computed; it is generated
by those orbits. We prove a "Viterbo functoriality" property: when one
Liouville domain is embedded into an other one, there is a morphism (reversing
arrows) between their positive -equivariant symplectic homologies and
morphisms compose nicely. These properties allow us to give a proof of
Ustilovsky's result on the number of non isomorphic contact structures on the
spheres . They also give a new proof of a Theorem by Ekeland and
Lasry on the minimal number of periodic Reeb orbits on some hypersurfaces in
. We extend this result to some hypersurfaces in some negative
line bundles.Comment: Correction in the computations of the action, no modifications of the
result
Generalized Conley-Zehnder index
The Conley-Zehnder index associates an integer to any continuous path of
symplectic matrices starting from the identity and ending at a matrix which
does not admit 1 as an eigenvalue. We give new ways to compute this index.
Robbin and Salamon define a generalization of the Conley-Zehnder index for any
continuous path of symplectic matrices; this generalization is half integer
valued. It is based on a Maslov-type index that they define for a continuous
path of Lagrangians in a symplectic vector space , having
chosen a given reference Lagrangian . Paths of symplectic endomorphisms of
are viewed as paths of Lagrangians defined by their graphs
in and the
reference Lagrangian is the diagonal. Robbin and Salamon give properties of
this generalized Conley-Zehnder index and an explicit formula when the path has
only regular crossings. We give here an axiomatic characterization of this
generalized Conley-Zehnder index. We also give an explicit way to compute it
for any continuous path of symplectic matrices.Comment: arXiv admin note: substantial text overlap with arXiv:1201.372
On the minimal number of periodic orbits on some hypersurfaces in
We study periodic orbits on a nondegenerate dynamically convex starshaped
hypersurface in along the lines of Long and Zhu, but using
properties of the -equivariant symplectic homology. We prove that there
exist at least distinct simple periodic orbits on any nondegenerate
starshaped hypersurface in satisfying the condition that the
minimal Conley-Zehnder index is at least . The condition is weaker than
dynamical convexity.Comment: To appear in Annales de l'Institut Fourie
Two closed orbits for non-degenerate Reeb flows
We prove that every non-degenerate Reeb flow on a closed contact manifold
admitting a strong symplectic filling with vanishing first Chern class
carries at least two geometrically distinct closed orbits provided that the
positive equivariant symplectic homology of satisfies a mild condition.
Under further assumptions, we establish the existence of two geometrically
distinct closed orbits on any contact finite quotient of . Several examples
of such contact manifolds are provided, like displaceable ones, unit cosphere
bundles, prequantization circle bundles, Brieskorn spheres and toric contact
manifolds. We also show that this condition on the equivariant symplectic
homology is preserved by boundary connected sums of Liouville domains. As a
byproduct of one of our applications, we prove a sort of Lusternik-Fet theorem
for Reeb flows on the unit cosphere bundle of not rationally aspherical
manifolds satisfying suitable additional assumptions.Comment: Version 1: 33 pages. Version 2: minor corrections, to appear in
Mathematical Proceedings of the Cambridge Philosophical Societ
Coarse distance from dynamically convex to convex
Chaidez and Edtmair have recently found the first example of dynamically
convex domains in that are not symplectomorphic to convex domains
(called symplectically convex domains), answering a long-standing open
question. In this paper, we discover new examples of such domains without
referring to Chaidez-Edtmair's criterion. We also show that these domains are
arbitrarily far from the set of symplectically convex domains in
with respect to the coarse symplectic Banach-Mazur distance by using an
explicit numerical criterion for symplectic non-convexity.Comment: 18 pages, 7 figure