Orbit growth of contact structures after surgery

Foulon, Patrick; Hasselblatt, Boris; Vaugon, Anne

doi:10.5802/ahl.98

Cited by 8 publications

(24 citation statements)

References 49 publications

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“…Anosovity has been proven by Anosov [Ano63] (see also [FHV21]). We remark that Anosovity can be derived easily from Hozoori's criterion.…”

Section: Theorem 31 (Anososv) the Geodesic Flow On The Unit Tangent B...mentioning

confidence: 92%

“…Foulon-Hasselblatt-Vaugon show in [FHV21] that if the thickness of the surgery annulus in Foulon-Haseelblatt construction is fixed, there are infinite values of q such that the new flow is not Anosov. If the closed geodesic g is simple in S, this issue can be addressed considering a nested sequence of annuli containig K.…”

Section: Theorem 3 If the Flow Is Anosov The Surgery Of Theorem 2 Yie...mentioning

confidence: 99%

“…The geodesic flow on the unit tangent bundle of a hyperbolic surface S carries some remarkable geometric structure that can be interpreted in the context of bi-contact geometry. We follow the discussion of [FHV21]. Using the identification of UTH 2 with PSL(2, R) it is possible to show that there is a canonical framing consisting of the vector X that generates the flow, the periodic vector field W pointing in the fiber direction, and the vector field defined by H := [W, X].…”

Section: Geometric Structures On Utsmentioning

confidence: 99%

“…While the equivalence between Fried and Goodman operations shows that the former yields Anosov flows for any slope, Foulon-Hasselblatt surgery requires some restriction on the sign of the Dehn twist in order to generate hyperbolic flows. For instance it can be shown that performing it in the wrong direction weakens the hyperbolicity and might produce flows that are not Anosov (see [FHV21]). In the present work we introduce a new type of surgery on Anosov flows that overcomes this issue if applied to a geodesic flow Theorem 1.…”

Section: Introductionmentioning

confidence: 99%

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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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“…Anosovity has been proven by Anosov [Ano63] (see also [FHV21]). We remark that Anosovity can be derived easily from Hozoori's criterion.…”

Section: Theorem 31 (Anososv) the Geodesic Flow On The Unit Tangent B...mentioning

confidence: 92%

Section: Theorem 3 If the Flow Is Anosov The Surgery Of Theorem 2 Yie...mentioning

confidence: 99%

Section: Geometric Structures On Utsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Surgery on Anosov flows using bi-contact geometry

Salmoiraghi¹

2021

Preprint

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“…Definition of the shear and the perturbations. We define the shear along the surgery annulus C similarly to [FH13], [FHV19] and [Sal21]. Consider…”

Section: Lemma 51mentioning

confidence: 99%

Goodman surgery and projectively Anosov flows

Salmoiraghi¹

2022

Preprint

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We introduce a generalization of Goodman surgery to the category of projectively Anosov flows. This construction is performed along a knot that is simultaneously Legendrian and transverse for a supporting bi-contact structure. When the flow is Anosov, our operation generates the same flows of Goodman's construction. The Anosovity of the new flow is strictly connected to contact geometry. We use this relation to give a refinement to the sequence of surgeries coefficients producing Anosov flows in concrete examples. We give an interpretation of the bi-contact surgery in terms of contact-Legendrian surgery and admissible-inadmissible transverse surgery and we deduce some (hyper)tightness result for contact and transverse surgeries. Outside of the realm of Anosov flows, we produce new projectively Anosov flows on hyperbolic 3-manifolds with leaves of genus g > 1 in the invariant foliations.

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Accessibility and centralizers for partially hyperbolic flows

Fisher

Hasselblatt²

2021

Ergod. Th. Dynam. Sys.

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Stable accessibility of partially hyperbolic systems is central to their stable ergodicity, and we establish its $C^1$ -density among partially hyperbolic flows, as well as in the categories of volume-preserving, symplectic, and contact partially hyperbolic flows. As applications, we obtain on one hand in each of these four categories of flows the $C^1$ -density of the $C^1$ -stable topological transitivity and triviality of the centralizer, and on the other hand the $C^1$ -density of the $C^1$ -stable K-property of the natural volume in the latter three categories.

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Orbit growth of contact structures after surgery

Cited by 8 publications

References 49 publications

Surgery on Anosov flows using bi-contact geometry

Surgery on Anosov flows using bi-contact geometry

Goodman surgery and projectively Anosov flows

Accessibility and centralizers for partially hyperbolic flows

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