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Surgery on Anosov flows using bi-contact geometry

Federico Salmoiraghi

Abstract

Using bi-contact geometry, we define a new type of Dehn surgery on an Anosov flow with orientable weak invariant foliations. The Anosovity of the new flow is strictly connected to contact geometry and the Reeb dynamics of the defining bi-contact structure. This approach gives new insights into the properties of the flows produced by Goodman surgery and clarifies under which conditions Goodman's construction yields an Anosov flow. Our main application gives a necessary and sufficient condition to generate a contact Anosov flow by Foulon-Hasselblatt Legendrian surgery on a geodesic flow. In particular we show that this is possible if and only if the surgery is performed along a simple closed geodesic. As a corollary we have that any (positive) skewed R-covered Anosov flow obtained by surgery on a closed orbit of a geodesic flow is orbit equivalent to a (positive) contact Anosov flow.

Surgery on Anosov flows using bi-contact geometry

Abstract

Using bi-contact geometry, we define a new type of Dehn surgery on an Anosov flow with orientable weak invariant foliations. The Anosovity of the new flow is strictly connected to contact geometry and the Reeb dynamics of the defining bi-contact structure. This approach gives new insights into the properties of the flows produced by Goodman surgery and clarifies under which conditions Goodman's construction yields an Anosov flow. Our main application gives a necessary and sufficient condition to generate a contact Anosov flow by Foulon-Hasselblatt Legendrian surgery on a geodesic flow. In particular we show that this is possible if and only if the surgery is performed along a simple closed geodesic. As a corollary we have that any (positive) skewed R-covered Anosov flow obtained by surgery on a closed orbit of a geodesic flow is orbit equivalent to a (positive) contact Anosov flow.

Paper Structure

This paper contains 22 sections, 34 theorems, 64 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Acknowledgment
  3. Anosov flows, projectively Anosov flows and bi-contact structures
  4. Contact structures and Reeb flows.
  5. Bi-contact structures and projectively Anosov flows.
  6. Reeb dynamics of a bi-contact structure defining an Anosov flows
  7. The geodesic flow on the unit tangent bundle of a hyperbolic surface
  8. Geometric structures on UTS
  9. Structures on a Birkhoff torus and Legendrian knots
  10. Foulon--Hasselblatt contact surgery
  11. Definition of the contact surgery
  12. Application to contact Anosov flows
  13. bi-contact surgery on Anosov flows
  14. Sketch of the construction of a bi-contact surgery
  15. Definition of the bi-contact surgery
  16. ...and 7 more sections

Key Result

Theorem 1

Let $K$ be a Legendrian-transverse knot in a bi-contact structure $(\xi_-,\xi_+)$ defining a volume preserving Anosov flow. There is a $(1,q)$-Dehn surgery along an annulus $A_0$ tangent to the flow that yields new bi-contact structures for every $q\in \mathbb{N}$.

Figures (12)

  • Figure 1: On the left, the tangent bundle in a neighborhood of a flow line of an Anosov flow. In red and blue are depicted the plane fields defining the bi-contact structure. The vertical plane is the the leaf of the unstable weak foliation $\mathcal{F}^{u}$. On the right, the normal bundle $TM/\langle X \rangle$.
  • Figure 2: On the top right, a Birkhoff torus associated to simple closed geodesic $\gamma$. On the right side, the contact structures described in \ref{['Bir']}. In grey, the contact structure $\eta_+=\ker\beta_+$ preserved by the geodesic flow. The closed orbits $\gamma^+$ and $\mathfrak{\gamma^-}$ are Legendrian knots for the bi-contact structure $(\xi_-,\xi_+)$ that generates the geodesic flow. The contact structure $\xi_+=\ker\alpha_+$ is represented in red while the $\xi_-=\ker\alpha_-$, is represented in blue. The special knot $L$ is represented by the dashed line.
  • Figure 3: On the left, the foliation on an embedded Birkhoff torus induced by the stable and unstable weak foliations. On the right, the characteristic foliations induced by the bi-contact structure $(\ker\alpha_+,\ker\alpha_-)$.
  • Figure 4: The surgery annulus in Foulon-Hasselblatt construction.
  • Figure 5: Legendrian-transverse push-off of a closed orbit. In blue $R_{\alpha_- }$. In black, the Anosov flow.
  • ...and 7 more figures

Theorems & Definitions (63)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Corollary 1
  • Corollary 2
  • Conjecture 1.1: Barbot--Barthelmé
  • Theorem 5
  • Theorem 6
  • Definition 2.1
  • ...and 53 more