13 research outputs found
An integer valued bi-invariant metric on the group of contactomorphisms of R^2n x S^1
In his 1992 article on generating functions Viterbo constructed a
bi-invariant metric on the group of compactly supported Hamiltonian
symplectomorphisms of R^2n. Using the set-up of arXiv:0901.3112 we extend the
Viterbo metric to the group of compactly supported contactomorphisms of R^2n x
S^1 isotopic to the identity. We also prove that the contactomorphism group is
unbounded with respect to this metric.Comment: 10 pages, final versio
Bi-invariant metrics on contactomorphism groups
Contact manifolds are odd-dimensional smooth manifolds endowed with a
maximally non-integrable field of hyperplanes. They are intimately related to
symplectic manifolds, i.e. even-dimensional smooth manifolds endowed with a
closed non-degenerate 2-form. Although in symplectic topology a famous
bi-invariant metric, the Hofer metric, has been studied since more than 20
years ago, it is only recently that some somehow analogous bi-invariant metrics
have been discovered on the group of diffeomorphisms that preserve a contact
structure. In this expository article I will review some constructions of
bi-invariant metrics on the contactomorphism group, and how these metrics are
related to some other global rigidity phenomena in contact topology which have
been discovered in the last few years, in particular the notion of orderability
(due to Eliashberg and Polterovich) and an analogue in contact topology (due to
Eliashberg, Kim and Poltorovich) of Gromov's symplectic non-squeezing theorem.Comment: Expository article, written in occasion of the 5-th IST-IME meeting
in S\~ao Paulo (July 2014) in honor of Prof. Orlando Lopes. To appear in the
S\~ao Paulo Journal of Mathematical Science
Givental's non-linear Maslov index on lens spaces
Givental's non-linear Maslov index, constructed in 1990, is a quasimorphism
on the universal cover of the identity component of the contactomorphism group
of real projective space. This invariant was used by several authors to prove
contact rigidity phenomena such as orderability, unboundedness of the
discriminant and oscillation metrics, and a contact geometric version of the
Arnold conjecture. In this article we give an analogue for lens spaces of
Givental's construction and its applications.Comment: 44 pages; v3: minor changes; v2: besides minor changes, we corrected
a mistake in Corollary 1.3(iv
Contact non-squeezing at large scale via generating functions
Using SFT techniques, Eliashberg, Kim and Polterovich (2006) proved that if
for some integer then there is no contact
squeezing in of the prequantization of the ball of
radius into the prequantization of the ball of radius . This result
was extended to the case of balls of radius and with by Chiu (2017) and the first author (2016), using
respectively microlocal sheaves and SFT. In the present article we recover this
general contact non-squeezing theorem using generating functions, a classical
method based on finite dimensional Morse theory. More precisely, we develop an
equivariant version, with respect to a certain action of a finite cyclic group,
of the generating function homology for domains of
defined by the second author (2011). A key role in the construction is played
by translated chains of contactomorphisms, a generalization of translated
points.Comment: 32 page
On iterated translated points for contactomorphisms of R^{2n+1} and R^{2n} x S^1
A point q in a contact manifold is called a translated point for a
contactomorphism \phi, with respect to some fixed contact form, if \phi (q) and
q belong to the same Reeb orbit and the contact form is preserved at q. The
problem of existence of translated points is related to the chord conjecture
and to the problem of leafwise coisotropic intersections. In the case of a
compactly supported contactomorphism of R^{2n+1} or R^{2n} x S^1 contact
isotopic to the identity, existence of translated points follows immediately
from Chekanov's theorem on critical points of quasi-functions and Bhupal's
graph construction. In this article we prove that if \phi is positive then
there are infinitely many non-trivial geometrically distinct iterated
translated points, i.e. translated points of some iteration \phi^k. This result
can be seen as a (partial) contact analogue of the result of Viterbo on
existence of infinitely many iterated fixed points for compactly supported
Hamiltonian symplectomorphisms of R^{2n}, and is obtained with generating
functions techniques in the setting of arXiv:0901.3112.Comment: 10 pages, revised version. I removed the discussion on linear growth
of iterated translated points, because it contained a mistake. To appear in
the International Journal of Mathematic