1,426 research outputs found
Automatic transversality in contact homology I: Regularity
This paper helps to clarify the status of cylindrical contact homology, a
conjectured contact invariant introduced by Eliashberg, Givental, and Hofer in
2000. We explain how heuristic arguments fail to yield a well-defined
homological invariant in the presence of multiply covered curves. We then
introduce a large subclass of dynamically convex contact forms in dimension 3,
termed dynamically separated, and demonstrate automatic transversality holds,
therby allowing us to define the desired chain complex. The Reeb orbits of
dynamically separated contact forms satisfy a uniform growth condition on their
Conley-Zehnder index under iteration, typically up to large action; see
Definition 1.15 These contact forms arise naturally as perturbations of
Morse-Bott contact forms such as those associated to -bundles. In
subsequent work, we give a direct proof of invariance for this subclass and,
when further proportionality holds between the index and action, powerful
geometric computations in a wide variety of examples.Comment: 68 pages, added more information about bad Reeb orbits, added a proof
of a beloved folk theorem concerning the factorization of multiply covered
curves, contains expository revisions helpfully suggested by the refere
Pseudo-Anosov flows in toroidal manifolds
We first prove rigidity results for pseudo-Anosov flows in prototypes of
toroidal 3-manifolds: we show that a pseudo-Anosov flow in a Seifert fibered
manifold is up to finite covers topologically equivalent to a geodesic flow and
we show that a pseudo-Anosov flow in a solv manifold is topologically
equivalent to a suspension Anosov flow. Then we study the interaction of a
general pseudo-Anosov flow with possible Seifert fibered pieces in the torus
decomposition: if the fiber is associated with a periodic orbit of the flow, we
show that there is a standard and very simple form for the flow in the piece
using Birkhoff annuli. This form is strongly connected with the topology of the
Seifert piece. We also construct a large new class of examples in many graph
manifolds, which is extremely general and flexible. We construct other new
classes of examples, some of which are generalized pseudo-Anosov flows which
have one prong singularities and which show that the above results in Seifert
fibered and solvable manifolds do not apply to one prong pseudo-Anosov flows.
Finally we also analyse immersed and embedded incompressible tori in optimal
position with respect to a pseudo-Anosov flow.Comment: 44 pages, 4 figures. Version 2. New section 9: questions and
comments. Overall revision, some simplified proofs, more explanation
Higher algebraic structures in Hamiltonian Floer theory I
This is the first of two papers devoted to showing how the rich algebraic
formalism of Eliashberg-Givental-Hofer's symplectic field theory (SFT) can be
used to define higher algebraic structures on the symplectic cohomology of open
symplectic manifolds. Using the SFT of Hamiltonian mapping tori we show how to
define a homotopy extension of the well-known Lie bracket on symplectic
cohomology. Apart from discussing applications to the existence of closed Reeb
orbits, we outline how the -structure is conjecturally related via
mirror symmetry to the extended deformation theory of complex structures.Comment: Results of arXiv:1310.6014 got merged into arXiv:1412.2682, now
entitled "Higher algebraic structures in Hamiltonian Floer theory" and
published in Advances in Geometry (DOI: 10.1515/advgeom-2019-0017).
Extensions of other announced results have been turned into an ongoing PhD
thesis projec
Peeling Bifurcations of Toroidal Chaotic Attractors
Chaotic attractors with toroidal topology (van der Pol attractor) have
counterparts with symmetry that exhibit unfamiliar phenomena. We investigate
double covers of toroidal attractors, discuss changes in their morphology under
correlated peeling bifurcations, describe their topological structures and the
changes undergone as a symmetry axis crosses the original attractor, and
indicate how the symbol name of a trajectory in the original lifts to one in
the cover. Covering orbits are described using a powerful synthesis of kneading
theory with refinements of the circle map. These methods are applied to a
simple version of the van der Pol oscillator.Comment: 7 pages, 14 figures, accepted to Physical Review
Visually building Smale flows in S3
A Smale flow is a structurally stable flow with one dimensional invariant
sets. We use information from homology and template theory to construct,
visualize and in some cases, classify, nonsingular Smale flows in the 3-sphere
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