Knot theory of ℝ-covered Anosov flows: homotopy versus isotopy of closed orbits

Barthelmé, Thomas; Fenley, Sérgio R.

doi:10.1112/jtopol/jtt041

Cited by 9 publications

(5 citation statements)

References 21 publications

(44 reference statements)

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“…(3) That is to say, each pair of freely homotopic closed orbits is actually related by isotopy, so in particular the pair is the boundary of an immersed cylinder. We note that, however, each closed orbit is related to at most finitely many others by the pair being the boundary of an embedded cylinder [BF14]. (This latter relation is neither transitive nor reflexive.)…”

Section: New Reeb Flowsmentioning

confidence: 93%

Orbit growth of contact structures after surgery

Foulon¹,

Hasselblatt²,

Vaugon³

2021

Annales Henri Lebesgue

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This work is at the intersection of dynamical systems and contact geometry, and it focuses on the effects of a contact surgery adapted to the study of Reeb fields and on the effects of invariance of contact homology.We show that this contact surgery produces an increased dynamical complexity for all Reeb flows compatible with the new contact structure. We study Reeb Anosov fields on closed 3-manifolds that are not topologically orbit-equivalent to any algebraic flow; this includes many examples on hyperbolic 3-manifolds. Our study also includes results of exponential growth in cases where neither the flow nor the manifold obtained by surgery is hyperbolic, as well as results when the original flow is periodic. This work fully demonstrates, in this context, the relevance of contact homology to the analysis of the dynamics of Reeb fields.

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Section: New Reeb Flowsmentioning

confidence: 93%

Orbit growth of contact structures after surgery

Foulon¹,

Hasselblatt²,

Vaugon³

2021

Annales Henri Lebesgue

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“…Remark 4.10. We note that Theorem 4.1 provides new geometric tools for understanding the periodic orbits of Anosov flows, in particular regarding the knot theory of such periodic orbits, which there are many unanswered questions about [7]. More precisely, if γ is a periodic orbit of an Anosov flow with supporting bi-contact structure (ξ − , ξ + ), then γ is a Legendrian knot for both ξ − and ξ + .…”

Section: Contact and Symplectic Geometric Characterization Of Anosov ...mentioning

confidence: 99%

Symplectic Geometry of Anosov Flows in Dimension 3 and Bi-Contact Topology

Hozoori¹

2020

Preprint

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We give a purely contact and symplectic geometric characterization of Anosov flows in dimension 3 and set up a framework to systematically use tools from contact and symplectic geometry and topology in the study of Anosov dynamics. We also discuss some uniqueness results regarding the underlying (bi)-contact structures for an Anosov flow and give a characterization of Anosovity, based on Reeb flows.

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“…Anosov flows are an important class of dynamical system characterized by structural stability under C 1 -small perturbations (see [Ano63], [Ano67] and [Pla71]). Beyond their interesting dynamical properties there are evidences of an intricate and beautiful relationship with the topology of the manifold they inhabit (see the survey [Bar17] for classic and more recent developments [BF17], [BF15], [BF14], [BF13] by Fenley, Barbot and Barthelmé). Geometrically they are distinguished by the contracting and expanding behaviour of two invariant directions Definition 2.1.…”

Section: Anosov Flows Projectively Anosov Flows and Bi-contact Struct...mentioning

confidence: 99%

Surgery on Anosov flows using bi-contact geometry

Salmoiraghi¹

2021

Preprint

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We describe a Dehn type surgery along a Legendrian-transverse knot K in a bi-contact structure. We show that, when the bi-contact structure defines an Anosov flow, there is a strong connection between the Anosovity of the new flow and contact geometry. We give an application to the geodesic flow on the unit tangent bundle of an hyperbolic surface. In contrast to the existing Dehn type surgeries on contact Anosov flows, for a vast class of knots our procedure does not require any restriction on the slope of the twist to generate new contact Anosov flows. We finally show that there are connections between our construction and the ones defined by Handel-Thurston, Fried-Goodman and Foulon-Hasselblatt.

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Knot theory of ℝ-covered Anosov flows: homotopy versus isotopy of closed orbits

Cited by 9 publications

References 21 publications

Orbit growth of contact structures after surgery

Orbit growth of contact structures after surgery

Symplectic Geometry of Anosov Flows in Dimension 3 and Bi-Contact Topology

Surgery on Anosov flows using bi-contact geometry

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