1,426 research outputs found

    Automatic transversality in contact homology I: Regularity

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    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 S1S^1-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

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    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

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    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 L∞L_{\infty}-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

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    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

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    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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