1 Introduction
The use of surgery has tremendously advanced our understanding of Anosov flows on
$3$
-manifolds. In the groundbreaking work of Handel and Thurston [Reference Handel and Thurston19], the first example of non-algebraic transitive Anosov flow was constructed performing surgery on a geodesic flow. Inspired by Handel and Thurston’s work, Goodman [Reference Goodman17] defined a Dehn type surgery near a closed orbit and constructed the first examples of an Anosov flow on a hyperbolic 3-manifold.
In the present work, we introduce a new type of Dehn surgery on Anosov flows using a supporting bi-contact structure. Mitsumatsu [Reference Mitsumatsu25] first noticed that the generating vector field of an Anosov flow belongs to the intersection of a pair of transverse contact structures
$(\xi _-,\xi _+)$
rotating towards each other along the flow and asymptotic to the stable and unstable foliations. These pairs are called bi-contact structures and the associated flows are called projectively Anosov flows. Projectively Anosov flows are more abundant than Anosov flows: it can be shown indeed that every closed 3-manifolds admits a projectively Anosov flow, while not every 3-manifold admits an Anosov flow.
A knot K in a bi-contact structure is called Legendrian-transverse if its tangents are contained in one of the contact structures and they are transverse to the other one. There are plenty of Legendrian-transverse knots in a bi-contact structure supporting an Anosov flow. Recently, Hozoori [Reference Hozoori20] has shown that if a flow is Anosov with orientable weak invariant foliations, there are supporting bi-contact structures such that near any periodic orbit
$\gamma $
, there is always a Legendrian-transverse knot isotopic to
$\gamma $
. Additionally, there are many Legendrian-tranverse knots that are not isotopic to any closed orbit. Examples of these knots can be found on the torus fibre of an Anosov suspension flow or considering the circles of the Seifert fibration on the unit tangent bundle of an hyperbolic surface.
Let M be a 3-manifold equipped with a bi-contact structure
$(\xi _-,\xi _+)$
defining an Anosov flow and let K be a knot that is Legendrian for
$\xi _-$
and is transverse to
$\xi _+$
. For a
$q\in \mathbb {Z}$
, we introduce a bi-contact surgery, which is a special type of
$(1,q)$
-Dehn surgery along an annulus
$A_0$
tangent to the flow and containing K (see Figure 1 for a comparison with Goodman surgery). More precisely, this operation consists in cutting the manifold M along
$A_0$
and gluing back the two sides of the cut adding a
$(1,-q)$
-Dehn twist along the core of the annulus. (See remark Theorem 5.7 for the minus sign ahead of the coefficient q.) This operation produces a new manifold
$\tilde {M}$
and a pair of plane field distributions
$(\tilde {\xi }_-,\tilde {\xi }_+)$
on the new manifold. In general, the new plane field distributions does not define a bi-contact structure. However, we have the following theorem.
Theorem 1.1. (See Theorem 5.11 for the proof)
For
$q\in \mathbb {N}$
, the bi-contact surgery produces a new pair
$(\tilde {\xi }_-,\tilde {\xi }_+)$
of plane field distributions which is a bi-contact structure.
Figure 1 On the left, the surgery annulus C in Goodman surgery near a closed orbit
$\gamma $
. The intersecting surfaces are the weak stable and unstable leaves containing
$\gamma $
. The left and the right edge of every strip are identified. The trajectories of the Anosov flow (black curves) are transverse to C. After surgery, the endpoints of the semi-trajectories that hit the annulus are identified with the staring points of the semi-trajectories on the other side of the annulus. This identification is achieved by a shear map that is a Dehn twist (in the picture, this map identifies the vertical segment on one side of C with the curve on the other side of C). On the right, the surgery annulus
$A_0$
used in the bi-contact surgery. The solid black lines are the trajectories of the new Anosov flow, while the dashed lines are those of the original flow. Note that while C is transverse to the Anosov flow,
$A_0$
is tangent to the flow (colour online).
Since being defined by a bi-contact structure is not a sufficient condition to be Anosov, it is not immediately clear if the bi-contact structures produced in Theorem 1.1 support an Anosov flow. The following result shows that the Anosovity of the new flow is strictly connected to bi-contact geometry.
Theorem 1.2. (See Theorem 5.11)
Let
$\phi ^t$
be a volume preserving Anosov flow. There is a bi-contact structure
$(\xi _-,\xi _+)$
defining
$\phi ^t$
such that if a bi-contact surgery along a Legendrian-transverse knot K yields a new bi-contact structure
$(\tilde {\xi }_-,\tilde {\xi }_+)$
, then
$(\tilde {\xi }_-,\tilde {\xi }_+)$
defines an Anosov flow.
In particular, in the hypothesis of Theorem 1.2, a
$(1,q)$
-bi-contact surgery yields new Anosov flows for every
$q\in \mathbb {N}$
.
The construction of Theorem 1.1 naturally fits into the framework set up by Hozoori in [Reference Hozoori21], where the author gives a contact geometric characterization of the bi-contact structures that define Anosov flows. Using this characterization, we give a proof of Theorem 1.2 that relies only on the properties of the Reeb dynamics of an underlying bi-contact structure. (In particular, we do not use the cone field criterion of hyperbolicity.)
The construction of Theorem 1.1 gives new insight into the properties of the flows produced by Goodman surgery in virtue of the following equivalence.
Theorem 1.3. (See Theorem 7.1 for a more precise statement)
Let K be a Legendrian- transverse knot in a bi-contact structure defining an Anosov flow. Suppose also that K is isotopic and close to a closed orbit of the Anosov flow. Any Anosov flow generated by a
$(1,q)$
-bi-contact surgery along a tangent annulus
$A_0$
containing K is orbit equivalent to an Anosov flow generated by
$(1,q)$
-Goodman surgery along a transverse annulus C containing K.
Goodman’s operation consists in cutting an Anosov flow along a transverse annulus C in a neighbourhood of a closed orbit and gluing back the two sides of the cut adding a
$(1,q)$
-Dehn twist (Figure 1). Goodman proved that such an annulus comes with a preferred direction in the sense that there is a sign of the twist that always produces hyperbolicity. For example, along an annulus with positive preferred direction, a
$(1,q)$
-Goodman surgery produces an Anosov flow for every integer
$q>0$
.
For a fixed transverse annulus with positive preferred direction, it is not known in general if a
$(1,q)$
-Goodman surgery with
$q<0$
generates hyperbolicity. Theorems 1.2 and 1.3 allow us to study the Anosovity of the flows generated by surgeries performed in the direction opposite to the preferred one using tools from contact geometry. Let C be an embedded transverse annulus with positive preferred direction centred on a Legendrian-transverse knot K (non-necessarily isotopic to a closed orbit). To a flow-box neighbourhood
$\Lambda $
of C (
$\Lambda $
is constructed by flowing the transverse annulus C using the Anosov flow), we associate a positive real number k depending just on the behaviour of the contact structure
$\xi _+$
on
$\partial \Lambda $
. This number k can be interpreted as the slope of the characteristic foliation induced by
$\xi _+$
on
$\partial \Lambda $
. For a volume preserving Anosov flow, we have the following theorem.
Theorem 1.4. (Theorem 6.3)
Let
$k>0$
be the slope of the characteristic foliation induced by
$\xi _+$
on
$\partial \Lambda $
. For every
$q>-k$
, a
$(1,q)$
-Goodman surgery along C produces an Anosov flow.
Let
$\phi ^t$
be a volume preserving Anosov flow with weak orientable invariant foliations. Suppose that there is an embedded quasi-transverse annulus
$ C_{-\infty }$
bounded on one side by a Legendrian-transverse knot K and on the other side by a closed orbit
$\gamma $
of the flow. We have the following corollary of Theorem 1.4.
Corollary 1.5. (See Corollary 7.3.)
There is a sequence of nested annuli
$C_{-1}\subset C_{-2}\subset \cdots \subset C_{p}\subset \cdots \subset C_{-\infty }$
with
$p\in \mathbb {Z}^-$
. Each
$C_{p}$
is transverse to the Anosov flow and bounded on one side by the Legendrian-transverse knot K. An annulus
$C_p$
of the sequence has the property that a
$(1,q)$
-Goodman surgery along
$C_{p}$
yields an Anosov flow for
$q\geq p$
. The annulus
$C_{-\infty }$
is quasi-transverse, and it is bounded by K on one side and by the closed orbit
$\gamma $
on the other side.
In particular, given an annulus
$C_{p},\:p\in \mathbb {Z}^-$
of the sequence, if we want to construct an annulus
$C_{p-1}$
such that a
$(1,p-1)$
-Goodman surgery produces an Anosov flow, we need to extend
$C_{p}$
towards the closed orbit
$\gamma $
(see Figure 12).
We use Theorem 1.4 and its corollaries to study surgeries in the context of contact Anosov flows that are Anosov flows preserving a contact form. The condition of being contact Anosov has a number of remarkable geometric consequences (see [Reference Fang13, Reference Hamenstädt18, Reference Liverani22]) including exponential decay of correlations. In the context of
$3$
-manifolds, Barbot has shown that every contact Anosov flow is skewed
$\mathbb {R}$
-covered. (An Anosov flow is
$\mathbb {R}$
-covered if the stable (or the unstable) weak foliation lifts to a foliation in the universal cover which has leaf space homeomorphic to
$\mathbb {R}$
.) The relation between these two classes of flows is expected to be stronger.
Conjecture 1.6. (Barbot–Barthelmé)
If
$\phi ^t$
is a positively skewed
$\mathbb {R}$
-covered Anosov flow, then
$\phi ^t$
is orbit equivalent to a contact Anosov flows. (Marty [Reference Marty24] posted a proof of this statement after the completion of the present work.)
For almost half of a century since the seminal work of Anosov, the only known examples of contact Anosov flows were the geodesic flows of Riemannian or Finsler manifolds. Foulon and Hasselblatt [Reference Foulon and Hasselblatt14] showed that it is not only possible to construct new examples of contact Anosov flows performing Goodman surgery on a geodesic flow, but also that it could be done producing hyperbolic manifolds.
The construction of Foulon and Hasselblatt uses a special class of arbitrarily thin transverse annuli with positive preferred direction containing a knot L that is Legendrian for the contact structure preserved by the geodesic flow. Such annuli are located far from a closed orbit and the set of negative surgery coefficients that yield a contact Anosov flow is not known. If the knot L is associated to a simple closed geodesic there is an embedded quasi-transverse annulus containing L and Corollary 1.5 applies. As a consequence, we have the following theorem.
Theorem 1.7. (See Theorem 8.2)
Choose a closed orbit
$\gamma $
in the geodesic flow on the unit tangent bundle of a hyperbolic surface S. If
$q<0$
, a
$(1,q)$
-Goodman surgery generates a flow that is orbit equivalent to a contact Anosov flow if and only if the orbit
$\gamma $
is a lift of a simple closed geodesic on S. (Our proof of Anosovity does not rely on Barbot’s version of the cone field criterion [Reference Barbot5] and can be used to give an alternate proof of [Reference Foulon and Hasselblatt14, Theorem 4.3].)
Theorem 1.7 represents the counterpart in the category of contact Anosov flows of results recently proven by Bonatti and Iakovoglou [Reference Bonatti and Iakovoglou11, Theorem 1] and Marty [Reference Marty23, Theorem J] in the category of
$\mathbb {R}$
-covered Anosov flows. Using Theorem 1.7 and work of Asaoka, Bonatti and Marty [Reference Asaoka, Bonatti and Marty4], we prove a version of Conjecture 1.6 for a single surgery along a closed orbit of the geodesic flow. Let
$\phi ^t$
be the geodesic flow with the orientation that makes it a positively skewed
$\mathbb {R}$
-covered Anosov flow.
Theorem 1.8. (See Theorem 9.5)
Any positive skewed
$\mathbb {R}$
-covered Anosov flow obtained by a single surgery along a closed orbit of a geodesic flow is orbit equivalent to a positive contact Anosov flow.
Note that Theorem 1.7 has a natural interpretation in the context of contact and symplectic geometry. Foulon and Hasselblatt construction is an example of a contact surgery in the sense that given a Legendrian knot L in a contact 3-manifold
$(M,\eta )$
, where
$\eta $
is the contact form preserved by an Anosov flow
$\phi ^t$
(in particular,
$\eta $
is not part of a supporting bi-contact structure for
$\phi ^t$
), it produces a new contact 3-manifold performing a Dehn type surgery along L. Historically, a
$(1,-1)$
-contact surgery is called Legendrian surgery. Legendrian surgeries preserve tightness [Reference Wand31] and have a natural interpretation from a symplectic point of view. Foulon and Hasselblatt show [Reference Foulon and Hasselblatt14] that positive Goodman surgeries (therefore, not Legendrian) always generate contact Anosov flows. If the geodesic is filling, the new manifold is hyperbolic. Since a filling geodesic is non-simple, a consequence of Theorem 1.7 is the impossibility of generating a positive contact Anosov flow on a hyperbolic manifold performing a single Legendrian surgery on a geodesic flow. In contrast, it is possible to generate a positive contact Anosov flow performing Legendrian surgery along any simple closed geodesic. This operation yields a graph manifold and the associated flows are orbit equivalent to those described by Handel and Thurston in [Reference Handel and Thurston19] for a shear with
$a_j<0$
and short enough geodesic
$\gamma _j$
.
2 Anosov flows, projectively Anosov flows and bi-contact structures
Anosov flows are an important class of dynamical system characterized by structural stability under
$\mathcal {C}^1$
-small perturbations (see [Reference Anosov1, Reference Anosov2, Reference Plante29]). Beyond their interesting dynamical properties, there is evidence of an intricate and beautiful relationship with the topology of the manifold they inhabit (see the survey [Reference Barthelmé8] for classic and more recent developments [Reference Barbot and Fenley6, Reference Barbot and Fenley7, Reference Barthelmé and Fenley9, Reference Barthelmé and Fenley10] by Fenley, Barbot and Barthelmé). Geometrically, they are distinguished by the contracting and expanding behaviour of two invariant directions
Definition 2.1. Let M be a closed manifold and
$\phi ^t:M \rightarrow M$
a
$\mathcal {C}^1$
flow on M. The flow
$\phi ^t$
is called Anosov if there is a splitting of the tangent bundle
$TM=E^{uu} \oplus E^{ss} \oplus \langle X \rangle $
preserved by
$D\phi ^t$
, and positive constants A and B such that
$$ \begin{align*}\lVert D \phi^t(v^u) \rVert\geq Ae^{Bt} \lVert v^u\rVert\quad\text{for any } v^u \in E^{uu},\: t\geq 0,\end{align*} $$
$$ \begin{align*}\lVert D \phi^{t}(v^s) \rVert\leq Ae^{-Bt} \lVert v^s\rVert\quad\text{for any } v^s \in E^{ss},t\geq 0.\end{align*} $$
Here,
$\lVert \cdot \rVert $
is induce by a Riemmanian metric on
$TM$
. We call
$E^{uu}$
and
$E^{ss}$
respectively the strong unstable bundle and the strong stable bundle.
Classic examples of Anosov flows are the geodesic flow on the unit tangent bundle of a hyperbolic surface and the suspension flows of hyperbolic linear automorphisms of the torus.
The definition above has remarkable geometric consequences. Anosov showed that the distributions
$E^{ss}$
and
$E^{uu}$
are uniquely integrable and the associated foliations are denoted by
$\mathcal {F}^{ss}$
and
$\mathcal {F}^{uu}$
. Moreover, the weak stable bundle
$E^s=E^{ss} \oplus \langle X \rangle $
and the weak unstable bundle
$E^u=E^{uu} \oplus \langle X \rangle $
are also uniquely integrable and the codimension one associated foliations are dented with
$\mathcal {F}^{s}$
and
$\mathcal {F} ^{u}$
(see Figure 2).
Figure 2 On the left, the tangent bundle in a neighbourhood of a flow line of an Anosov flow. The oblique plane fields define the bi-contact structure. The vertical plane is the leaf of the unstable weak foliation
$\mathcal {F}^{u}$
. On the right, the normal bundle
$TM/\langle X \rangle $
(colour online).
Mitsumatsu [Reference Mitsumatsu25] first noticed that an Anosov flow with orientable weak invariant foliations is tangent to the intersection of two transverse contact structures (see also Eliashberg and Thurston [Reference Eliashberg and Thurston12]). We will call such pairs bi-contact structures. However, the converse statement is not true and there are bi-contact structures that do not define Anosov flows.
2.1 Contact structures and Reeb flows
A co-oriented contact structure is a plane field distribution that is maximally non-integrable in the sense that it can be described as the kernel of a
$\mathcal {C}^1$
1-form satisfying the relation
$\alpha \wedge d\alpha \neq 0$
. By Frobenious theorem, contact structures can be thought of as polar opposite of foliations: there is not a subsurface S such that
$TS=\ker \alpha $
, even in a neighbourhood of a point. Contact structures come in two types, positive and negative. A positive contact structure is a plane field distribution
$\xi _+$
described by a
$\mathcal {C}^1$
1-form satisfying the relation
$\alpha _+\wedge d\alpha _+>0$
. A negative contact structure
$\xi _-$
is described instead by a
$\mathcal {C}^1$
1-form such that
$\alpha _-\wedge d\alpha _-<0$
.
An important property of contact structures is that they do not have local invariants. Indeed, by Darboux theorem, positive (negative) contact structures are all locally contactomorphic to the positive (negative) standard contact structure in
$\mathbb {R} ^3$
described by
$\xi _{std}^+=\ker dz-y\:dx$
and
$\xi _{std}^-=\ker dz+y\:dx$
. Therefore, we can locally picture a positive contact (negative) structure as a plane field whose plane rotates counterclockwise (clockwise) along the x-axis. Associated to the defining contact forms, there is an important class of flows called Reeb flows. Given a contact form
$\alpha $
, we define the Reeb vector field
$R_{\alpha }$
as the unique vector field satisfying the equations
$$ \begin{align*}\alpha(R_{\alpha})=1,\quad d\alpha(R_{\alpha},\cdot)=0.\end{align*} $$
These relations imply that
$R_{\alpha }$
is transverse to
$\ker \alpha $
and
$\mathcal {L}_{R_{\alpha }}\alpha =0$
(
$R_{\alpha }$
preserves
$\alpha $
).
We conclude this section introducing a very important class of vector fields that is well studied and has remarkable geometric properties [Reference Foulon and Hasselblatt14, Reference Liverani22].
Definition 2.2. A Reeb flow that is also an Anosov flow is called a contact Anosov flow.
In other words, a contact Anosov flow is an Anosov flow preserving a contact form.
2.2 Bi-contact structures and projectively Anosov flows
We now give an example of a pair of opposite and transverse contact structures that does not define an Anosov flow.
Example 2.3. We construct a bi-contact structure on
$T^3$
using a recipe introduced by Mitsumatsu in [Reference Mitsumatsu25, Reference Mitsumatsu26]. Consider the contact forms defined on
$T^2\times I$
,
$$ \begin{align*}\alpha_{n}=\cos(2n\pi z)\,dx-\sin(2n\pi z)\,dy,\end{align*} $$
$$ \begin{align*}\alpha_{-m}=\cos(2m\pi z)\,dx+\sin(2m\pi z)\,dy.\end{align*} $$
They are not transverse to each other on the tori defined by
$\{z=0\}$
,
$\{z=\tfrac 14\}$
,
$\{z=\tfrac 12\}$
,
$\{z=\tfrac 34\}$
. If we introduce a perturbation
$\epsilon (z)\: dz$
such that
$\epsilon (z)$
is a function that does not vanish on the tori, the contact forms
$\alpha _+=\alpha _{n}+\epsilon (z)\: dz$
and
$\alpha _-=\alpha _{-m}$
define transverse contact structures of opposite orientations.
As Plante and Thurston showed in [Reference Plante and Thurston30], the fundamental group of a manifold that admits an Anosov flow grows exponentially. Since the ambient manifold is
$T^3$
, any flow defined by the pair of contact structures
$(\ker \alpha _-,\ker \alpha _+)$
is not Anosov.
Definition 2.4. (Mitsumatsu [Reference Mitsumatsu25])
Let M be a closed manifold and
$\phi ^t:M \rightarrow M$
a
$C^1$
flow on M. The flow
$\phi ^t$
is called projectively Anosov if there is a splitting of the projectified tangent bundle
$TM/ \langle X \rangle =\mathcal {E}^u \oplus \mathcal {E}^s$
preserved by
$D\phi ^t$
and positive constants A and B such that
$$ \begin{align*}\frac{\lVert D \phi^t(v^u) \rVert}{ \lVert D \phi^t(v^s) \rVert}\geq Ae^{Bt} \frac{\lVert v^u\rVert}{\lVert v^s\rVert}\quad\text{for any } v^u \in \mathcal{E}^u \text{ and }v^s \in \mathcal{E}^s,\:t\geq 0. \end{align*} $$
Here,
$\lVert \cdot \rVert $
is induce by a Riemmanian metric on
$TM$
. We call
$\mathcal {E}^u$
and
$\mathcal {E}^s$
respectively the unstable bundle and the stable bundle.
The invariant bundles
$\mathcal {E}^u$
and
$\mathcal {E}^s$
induce invariant plane fields
$E^u$
and
$E^s$
on M. These plane fields are continuous and integrable, but unlike the Anosov case, the integral manifolds may not be unique (see [Reference Eliashberg and Thurston12]). However, when they are smooth, they also are uniquely integrable. We call these flows regular projectively Anosov (see [Reference Asaoka3, Reference Noda27, Reference Noda and Tsuboi28] for a complete classification).
Definition 2.5. We say that a flow
$\phi ^t$
is defined (or supported) by a bi-contact structure
$(\xi _-,\xi _+)$
if the generating vector field X belongs to the intersection
$\xi _-\cap \xi _+$
.
The following result gives a geometric characterization of the class of projectively Anosov flows.
Theorem 2.6. (Mitsumatsu [Reference Mitsumatsu25])
Let X be a
$\mathcal {C}^1$
vector field on M. Then, X is projectively Anosov if and only if it is defined by a bi-contact structure.
Remark 2.7. We refer to [Reference Hozoori20, Reference Hozoori21] for a more complete overview of the connection between bi-contact geometry, symplectic geometry and projectively Anosov flows, and for a precise discussion on the regularity of the plane fields, bundles and foliations involved.
2.3 Reeb dynamics of a bi-contact structure defining an Anosov flows
We now present a characterization of Anosov flows due to Hoozori [Reference Hozoori21] that uses the Reeb dynamics of the underling bi-contact structure.
Definition 2.8. (Hozoori [Reference Hozoori21])
Consider a bi-contact structure
$(\xi _-,\xi _+)$
on M and a supported vector field X. We say that X is dynamically positive (negative) if at every point
$p\in M$
, its image into
$TM/\langle X \rangle $
lies in the interior of the region defined by the stable and unstable bundle. More precisely, we say that X is dynamically positive if its image into
$TM/\langle X \rangle $
lies in the interior of the region defined by the stable and unstable bundle in clock-wise order and considering the flow pointing into the page (see Figure 2).
Theorem 2.9. (Hozoori [Reference Hozoori21])
Let
$\phi ^t$
be a projectively Anosov flow on M. The following are equivalent.
-
(1) The flow
$\phi ^t$
is Anosov. -
(2) There is a pair
$(\alpha _-,\alpha _+)$
of positive and negative contact forms defining
$\phi ^t$
such that the Reeb vector field of
$\alpha _+$
is dynamically negative. -
(3) There is a pair
$(\alpha _-,\alpha _+)$
of positive and negative contact forms defining
$\phi ^t$
such that the Reeb vector field of
$\alpha _-$
is dynamically positive.
In particular, we have the following sufficient condition of Anosovity.
Corollary 2.10. Let
$(\ker \alpha _-=\xi _-,\xi _+)$
be a bi-contact structure such that
$R_{\alpha _-}\in \xi _-$
. The flow
$\phi ^t$
defined by
$(\xi _-,\xi _+)$
is Anosov.
It is important to remark that under these circumstances, the knowledge of the position of stable and unstable invariant direction in
$TM/\langle X \rangle $
is not needed since under this condition,
$R_{\alpha _-}$
is automatically dynamically positive.
A volume preserving Anosov flow is an Anosov flow preserving a continuous volume form. It is known that if the flow is
$\mathcal {C}^k$
, such a volume form is automatically
$\mathcal {C}^k$
(see [Reference Hozoori20] for more references).
Hozoori shows [Reference Hozoori20] the following characterization of bi-contact structures defining volume preserving Anosov flows.
Theorem 2.11. (Hozoori [Reference Hozoori20])
Let
$\phi ^t$
be a projectively Anosov flow on M. Here,
$\phi ^t$
is a volume preserving Anosov flow if and only if there are pairs of positive and negative contact forms defining
$\phi ^t$
such that
$R_{\alpha _-}\in \ker \alpha _+$
and
$R_{\alpha _+}\in \ker \alpha _-$
.
At the time of writing this work, it is not known if a general Anosov flow is always supported by a bi-contact structure
$(\xi _-,\xi _+)$
such that
$R_{\alpha _-}\in \ker \alpha _+$
.
2.4 Flexibility of bi-contact structures and structural stability
The interplay between structural stability of an Anosov flow and the flexibility of its underlying bi-contact structure can be used to construct
$\mathcal {C}^1$
-paths of Anosov flows.
Lemma 2.12. Suppose that M is equipped with a pair of contact structures
$(\ker \alpha _-=\xi _-,\xi _+)$
defining an Anosov flow
$\phi ^t$
such that the Reeb vector field
$R_{\alpha _-}$
of
$\alpha _-$
belongs to
$\xi _+$
. An isotopy of
$\xi _+$
along the flow-lines of
$R_{\alpha _-}$
defines a path of orbit equivalent Anosov flows.
Proof. Assume that
$R_{\alpha _-}\in \xi _+$
. An isotopy of
$\xi _+=\ker \alpha _+$
as above defines a
$\mathcal {C}^1$
-path of bi-contact structures
$(\xi _-,\ker (\alpha _+)_t)_{t\in [0,1]}$
such that
$R_{\alpha _-}\in (\xi _+)_t=\ker (\alpha _+)_t$
in N. The statement follows from Theorem 2.9 and the
$\mathcal {C}^1$
-structural stability of Anosov flows.
In particular, all the flows along the path are orbit equivalent with an orbit equivalence isotopic to the identity. The above result will be used in the proof of Theorems 5.6 and 6.4.
3 The geodesic flow on the unit tangent bundle of a hyperbolic surface
In this section, we introduce the prototype of an Anosov flow: the geodesic flow on the unit tangent bundle of a hyperbolic surface (see [Reference Anosov1]). This object is particularly important in our context because it is naturally supported by a bi-contact structure: it can be considered the motivating example for constructing a bi-contact theory of Anosov flow. Moreover, for a long time, it was the only known example of a contact Anosov flow.
3.1 Geometric structures on UTS
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 [Reference Foulon, Hasselblatt and Vaugon15]. Using the identification of UT
$\mathbb {H}^2$
with
$\operatorname {{\mathrm {PSL}}}(2,\mathbb {R})$
, it is possible to show that there is a canonical framing consisting of the vector field X that generates the flow, the periodic vector field V pointing in the fibre direction and the vector field defined by
$H:=[V,X]$
. This frame satisfies the following relations:
$$ \begin{align} [V,X]=H,\;\;\;\;[H,X]=V,\;\;\;\;[H,V]=X. \end{align} $$
A consequence of the structure equations is that the strong stable and unstable bundles
$E^{\pm }$
are spanned by the vectors
$e^{\pm }=V\pm H$
. It is not difficult to show that equation (3.1) implies the existence of three 1-forms
$\alpha _-, \alpha _+$
and
$\beta _+$
defining mutually transverse contact structures (see Figure 3 and [Reference Foulon, Hasselblatt and Vaugon15] for more details). They are defined by
$$ \begin{align*}\beta_+(V)=0=\beta_+(H),\;\;\;\;\alpha_+(X)=0=\alpha_+(V),\;\;\;\;\alpha_-(X)=0=\alpha_-(H),\end{align*} $$
$$ \begin{align*}\beta_+(X)=1,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\alpha_+(H)=1,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\alpha_-(V)=1,\end{align*} $$
$$ \begin{align*}d\beta_+(X,\cdot)=0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;d\alpha_+(H,\cdot)=0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;d\alpha_-(V,\cdot)=0.\end{align*} $$
Figure 3 On the top right, a Birkhoff torus associated to simple closed geodesic
$\gamma $
. On the right side, the contact structures described in §3.2. The plane field transverse to the arrows is 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 special knot L is represented by the dashed line (colour online).
The relations above show that the vector fields X, H and V are respectively the Reeb vector fields of the contact forms
$\beta _+$
,
$\alpha _+$
and
$\alpha _-$
. In particular, the geodesic flow is the Reeb flow of
$\beta _+$
. The pair of contact forms
$(\alpha _-,\alpha _+)$
define a bi-contact structure that supports X.
3.2 Structures on a Birkhoff torus and Legendrian knots
Let S
$\;=\mathbb {H}^2 /\Gamma $
be a hyperbolic surface and let
$\gamma $
be a closed geodesic. The lift of
$\gamma $
to the unit tangent bundle of
$\gamma $
is an immersed Birkhoff torus UT
$\gamma $
with two closed orbits
$\{\gamma _1,\gamma _2\}$
, in the sense that UT
$ \gamma $
is an immersed torus transverse to the geodesic flow in UT
$\gamma \setminus \{\gamma _1,\gamma _2\}$
and tangent to the geodesic flow along
$\{\gamma _1,\gamma _2\}$
. If the geodesic
$\gamma $
is also simple, UT
$ \gamma $
is an embedded torus.
We now describe a special knot L on the Birkhoff torus that plays a very important role in numerous applications. This knot is defined by the angle
$\theta ={\pi }/{2}$
on each fibre along
$\gamma $
and has the following remarkable properties (see Figures 2 and 4).
-
(1) L is a Legendrian knot for the contact structure
$\eta _+$
preserved by the flow. -
(2) L is transverse to the weak stable and unstable foliations.
-
(3) L is a Legendrian-transverse knot with respect to the bi-contact structure
$(\ker \alpha _-,\ker \alpha _+)$
.
Lemma 3.1. Let
$\phi ^t$
be the geodesic flow on the unit tangent bundle
$UTS$
of an oriented hyperbolic surface S. In a neighbourhood
$N=A_0\times (-\epsilon ,\epsilon )$
of a Birkhoff annulus associated to a simple closed geodesic, there is a coordinate system
$(w,s,v)$
such that the natural contact forms
$( \alpha _-,\alpha _+,\beta _+)$
have the following expressions:
$$ \begin{align*}\alpha_-=dw+v\,ds;\end{align*} $$
$$ \begin{align*}\alpha_+=e^{({1}/{2})v^2}(\cos w\,ds-\sin w\,dv);\end{align*} $$
$$ \begin{align*}\beta_+=e^{({1}/{2})v^2}(-\sin w\,ds-\cos w\,dv).\end{align*} $$
Here,
$A_0$
is constructed as follows. Consider the Legendrian-transverse knot L constructed in §3.2 and let
$A_0$
be an annulus tangent to the flow
$\phi ^t$
of X constructed by flowing L (using the flow
$\phi ^t$
). Let
$(s,v)$
be a coordinate system on
$A_0$
such that the s-curves are parallel to
$\partial A_0$
. We extend
$(s,v)$
to a coordinate system
$(s,v,w)$
on N using the flow of
$R_{\alpha _-}$
. (This can always be done since, by construction,
$R_{\alpha _-}$
is transverse to
$A_0$
.)
Figure 4 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
$(\ker \alpha _+,\ker \alpha _-)$
(colour online).
Proof. Let
$( \alpha _-,\alpha _+,\beta _+)$
be the contact forms defined above. It is enough to show that if
$V=R_{\alpha _-}$
,
$H=R_{\alpha _+}$
and
$X=R_{\beta _+}$
, we have
$$ \begin{align*}\beta_+(V)=\beta_+(H)=\alpha_-(X)=\alpha_-(H)=\alpha_+(X)=\alpha_+(V)=0.\end{align*} $$
This shows that the contact forms
$( \alpha _-,\alpha _+,\beta _+)$
define a local model of the geodesic flow near a quasi-transverse torus, as described in §3.2. An elementary calculation gives the following expressions for the Reeb vector fields of
$( \alpha _-,\alpha _+,\beta _+)$
in N:
$$ \begin{align*}V=R_{\alpha_-}=\frac{\partial}{\partial w};\end{align*} $$
$$ \begin{align*}H=R_{\alpha_+}=e^{-({1}/{2})v^2}\bigg(\!\cos w\:\frac{\partial}{\partial s}-\sin w\:\frac{\partial}{\partial v}-v\cos w \:\frac{\partial}{\partial w}\bigg); \end{align*} $$
$$ \begin{align*}X=R_{\beta_+}=e^{-({1}{2})v^2}\bigg({-}\sin w\:\frac{\partial}{\partial s}-\cos w\:\frac{\partial}{\partial v}+v\sin w \:\frac{\partial}{\partial w}\bigg). \end{align*} $$
We check that the vector field
$V=R_{\alpha _-}={\partial }/{\partial w}$
satisfies the requirements of the (unique) Reeb vector field of
$\alpha _-$
,
$$ \begin{align*}\alpha_-(V)=(dw+v\:ds)\bigg(\frac{\partial}{\partial w}\bigg)=1,\end{align*} $$
and since
$d\alpha _-=dv\wedge \,ds$
, we have
$d\alpha _-(V,\cdot )=0$
. Moreover, since
$\alpha _+$
and
$\beta _+$
do not have a
$dw$
-term,
$\alpha _+(V)=0$
and
$\beta _+(V)=0$
. We leave to the reader to check the other identities.
4 Foulon–Hasselblatt contact surgery
In this section, we introduce a construction of Foulon and Hasselblatt that will play a fundamental role in the definition of the bi-contact surgery. Remarkably, the bi-contact surgery will allow us to extend the classic application of Foulon and Hasselblatt construction to a broader class of (contact) Anosov flows.
Let
$\alpha $
be a
$\mathcal {C}^k$
contact form with Reeb vector field
$R_{\alpha }$
, and consider a Legendrian knot L for
$\xi =\ker \alpha $
. Foulon and Hasselblatt describe in [Reference Foulon and Hasselblatt14] a family of contact surgeries of the Dehn type that generates a new
$\mathcal {C}^k$
contact form
$\tilde {\alpha }$
in the new manifold
$\tilde {M}$
and a
$\mathcal {C}^{k-1}$
vector field
$R_{\tilde {\alpha }}$
such that:
-
(1)
$\tilde {\xi }=\ker \tilde {\alpha }$
is isotopic to a contact structure obtained from
$\xi =\ker \alpha $
by classic contact surgery and such that
$\tilde {\alpha }=\alpha $
outside a neighbourhood of an annulus C transverse to the flow of
$R_{\alpha }$
; -
(2)
$R_{\tilde {\alpha }}$
is the Reeb vector field of
$\tilde {\alpha }$
. Moreover,
$R_{\tilde {\alpha }}$
is collinear to
$R_{\alpha }$
in
$M\setminus C$
.
In other words, the Foulon and Hasselblatt construction is a contact surgery that allows us to have some control on the direction of the resulting Reeb vector field.
4.1 Definition of the contact surgery
In the following, we recall the main features of the Foulon–Hasselblatt construction (see [Reference Foulon and Hasselblatt14, Reference Foulon, Hasselblatt and Vaugon15] for more details). Given a 3-manifold with a contact structure
$\xi =\ker \alpha $
and a Legendrian knot L, there is a coordinate system
$$ \begin{align*}(t,s,w)\in N= (-\delta,\delta)\times S^1 \times (-\epsilon,\epsilon),\end{align*} $$
where the parameters
$(s,w)$
are defined on the surgery annulus
$C=\{0\}\times S^1 \times (-\epsilon ,\epsilon )$
(see Figure 5). More precisely,
$s \in S^1$
is the parameter of L and w belongs to some interval
$(-\epsilon ,\epsilon )$
. The transverse parameter t is such that the Reeb vector field of
$\gamma $
satisfies
$R_\alpha ={\partial }/{\partial t}$
; therefore, N is a flow-box chart for
$R_\alpha $
. In this coordinate system, a contact form defining a contact structure takes the particularly simple expression
$\gamma =dt+w\, ds$
.
Figure 5 The surgery annulus in Foulon–Hasselblatt construction.
The surgery can be thought of as first cutting the manifold M along the annulus C and then gluing back the two sides of the cut in a different way. In particular, we glue the point that, on one side of the cut, is described by coordinates
$(s,w)$
to that described by coordinates
$(s+f(w),w)$
on the other side. Here,
$f:[-\epsilon ,\epsilon ]\rightarrow S^1$
is a non-decreasing function such that
$f(-\epsilon )=0$
and
$f(\epsilon )=2\pi q$
, and satisfying a number of additional requirements. The transition map
$F:C\rightarrow C,\; (s,w)\rightarrow (s+f(w),w)$
is often called shear. Since
$$ \begin{align*}F_*\frac{\partial}{\partial t}=\frac{\partial}{\partial t},\end{align*} $$
the shear defines a smooth vector field from
$R_\alpha $
. Note that if we restrict our attention to the vector field
$R_{\alpha }$
, Foulon–Hasselblatt construction can be interpreted as a Goodman surgery on
$R_\alpha $
.
Let us denote with
$(\alpha )^-$
the contact form defined on one side of the flow-box chart and with
$(\alpha )^+$
the contact form on the other side, we easily see that
$$ \begin{align*}F^*(\alpha)^+=(\alpha)^-+wf'(w)\,dw; \end{align*} $$
therefore, the shear defines a smooth vector field from
$R_\alpha $
, but does not define a smooth contact structure on the new manifold. This issue is addressed by Foulon and Hasselblatt introducing a deformation that yields to a 1-form
$(\tilde {\alpha })^{+}=(\alpha )^+-dh$
of class
$\mathcal {C}^1$
and
$dh$
is the differential of the following function:
$$ \begin{align*}h(t,w)=\unicode{x3bb}(t) \int_{-\epsilon}^{w}xf'(x)\,dx.\end{align*} $$
Here,
$\unicode{x3bb} :\mathbb {R}\rightarrow [0,1]$
is a
$\mathcal {C}^1$
bump function with support in some interval
$(0,\delta )$
with
$\delta>0$
that takes value
$1$
in a neighbourhood of
$0$
and takes value
$0$
in a neighbourhood of
$\delta $
. With these choices, we have
$F^*(\alpha -dh)^+=(\alpha )^-$
. Therefore,
$\tilde {\alpha }$
is a well-defined 1-form of class
$\mathcal {C}^1$
in
$\tilde {M}$
. Note that the deformation depends on the shear and it is not immediately clear if the plane field distribution
$\ker \tilde {\alpha }$
still defines a contact structure. The authors show that if we choose
$0<\epsilon < {\delta }/{2\pi q}$
, this is in fact the case. Moreover, the Reeb vector field of
$\tilde {\alpha }$
takes the form
$$ \begin{align*}R_{\tilde{\alpha}}=\frac{R_\alpha}{1- dh(R_\alpha)}.\end{align*} $$
4.2 Application to contact Anosov flows
The Foulon–Hasselblatt construction is purely contact geometric, it does not require the Reeb vector field of
$\alpha $
to be Anosov and it can be performed in a neighbourhood of any Legendrian knot.
If applied to a geodesic flow on the unit tangent bundle of an hyperbolic surface, we have the following theorem.
Theorem 4.1. (Foulon–Hasselblatt–Vaugon [Reference Foulon, Hasselblatt and Vaugon15])
Select a closed geodesic
$\gamma $
on a hyperbolic surface S and consider the geodesic flow on
$UTS$
. Consider the knot L defined by the angle
$\theta ={\pi }/{2}$
on each fibre along
$\gamma $
. Let
$2\epsilon $
be the width of the surgery annulus.
-
(1) The
$(1,q)$
-Dehn surgery along L defined in §4.1 does produce an Anosov flow if
$q>0$
regardless of the width of the surgery annulus. -
(2) It does not produce an Anosov if
$-q/\epsilon $
is large enough, that is, if either
$q<0$
is fixed and
$\epsilon $
is small enough or if
$\epsilon>0$
is fixed and
$q<0$
with
$|q|$
big enough.
The following result shows that the geodesic
$\gamma $
can be chosen in such a way that the result of the surgery is hyperbolic.
Theorem 4.2. (Calegari/Folklore [Reference Foulon and Hasselblatt14])
Let
$\gamma $
a closed, filling geodesic in an hyperbolic surface S. Then, the complement of its image in the unit tangent bundle of S is hyperbolic.
Note that the Birkhoff torus corresponding to a closed, filling geodesic is self-intersecting. However, for
$q>0$
, we can choose
$\epsilon>0$
small enough to ensure that the surgery annulus
$C_{\epsilon }$
is embedded in M.
5 Bi-contact surgery on Anosov flows
We introduce a new type of Dehn surgery on a projectively Anosov flow with weak orientable invariant foliations. Our construction is defined in a neighbourhood N of a knot K that is simultaneously Legendrian for
$\xi _-$
and transverse for
$\xi _+$
. Furthermore, we require that in N, the Reeb vector field of a contact form
$\alpha _-$
such that
$\ker \alpha _-=\xi _-$
is contained in
$\xi _+$
. As Hozoori shows in [Reference Hozoori20], if the flow is Anosov and volume preserving, there is a supporting bi-contact structure
$(\ker \alpha _-= \xi _-, \ker \alpha _+=\xi _+)$
that satisfies the above property everywhere. However, if the flow is just Anosov, Hozoori [Reference Hozoori20] shows that the condition
$R_{\alpha _-}\in \ker \alpha _+=\xi _+$
can be always achieved in a neighbourhood of a closed orbit.
Theorem 5.1. (Hozoori [Reference Hozoori20])
Let
$\phi ^t$
be a
$\mathcal {C}^1$
-Hölder Anosov flow with orientable weak invariant foliations and let
$\gamma $
be a periodic orbit. There is a pair of contact forms
$(\alpha _-,\alpha _+)$
, such that
$(\ker \alpha _-,\ker \alpha _+)$
is a supporting bi-contact structure for
$\phi ^t$
and, in a neighbourhood N of
$\gamma $
, the Reeb vector field
$R_{\alpha _-}$
of
$\alpha _-$
satisfies the condition
$\alpha _+(R_{\alpha _-})=0$
.
We use the following corollary to construct a Legendrian-transverse knot from a closed orbit
$\gamma $
.
Corollary 5.2. Let
$\gamma $
be a closed orbit of a flow defined by a bi-contact structure
$(\ker \alpha _- =\xi _-,\ker \alpha _+=\xi _+)$
such that in a neighbourhood N of
$\gamma $
, the Reeb vector field
$R_{\alpha _-}$
belongs to
$\ker \alpha _+=\xi _+$
. Let
$\psi ^w$
be the flow of
$R_{\alpha _-}$
. The knot
$K=\psi ^w(\gamma )$
is a Legendrian-transverse knot for
$0<w$
sufficiently small.
Definition 5.3. Consider a closed orbit
$\gamma $
and a pair of contact forms
$(\alpha _-,\alpha _+)$
such that in a neighbourhood N of
$\gamma $
, the Reeb vector field
$R_{\alpha _-}$
belongs to
$\ker \alpha _+=\xi _+$
. We call a Legendrian-transverse push-off of
$\gamma $
a knot K that is simultaneously Legendrian for
$\xi _-=\ker \alpha _-$
and transverse for
$\xi _+=\ker \alpha _+$
, and it is obtained by translating
$\gamma $
using the flow of
$R_{\alpha _-}$
(see Figure 6).
Figure 6 Legendrian-transverse push-off of a closed orbit. The vertical arrows represent
$R_{\alpha _- }$
. In black, the Anosov flow (colour online).
5.0.1 Sketch of the construction of a bi-contact surgery
Let K be a Legendrian- transverse knot in a bi-contact structure
$(\xi _-,\xi _+)$
defining a vector field X with flow
$\phi ^t$
. Suppose that in a neighbourhood N of K, we have
$R_{\alpha _-}\in \xi _+$
for some contact form
$\alpha _-$
such that
$\ker \alpha _-=\xi _-$
. Since K is Legendrian for
$\xi _-$
, we can apply Foulon–Hasselblatt contact surgery to the pair
$(\alpha _-,R_{\alpha _-})$
using a shear F on a surgery annulus
$A_0$
constructed by translating the knot K using the flow
$\phi ^t$
. Note that with the above assumptions,
$A_0$
is transverse to
$R_{\alpha _-} $
and tangent to X. After surgery, we have a new contact form
$\tilde {\alpha }_-$
with Reeb vector field
$R_{\tilde {\alpha }_-}$
that is collinear to
$R_{\alpha _-}$
in
$M\setminus A_0$
. Consider now the contact structure
$\ker \alpha _+=\xi _+$
transverse to the knot K. If the surgery coefficient q is positive, we will show that there is a new contact form
$\tilde {\alpha }_+$
such that the Reeb vector field
$R_{\tilde {\alpha }_-}$
belongs to
$\ker \tilde {\alpha }_+=\tilde {\xi }_+$
. Suppose now that the initial defining bi-contact structure is such that
$R_{\alpha _-}\in \ker \alpha _+$
everywhere (therefore, the supported flow is Anosov by Corollary 2.10. The new bi-contact structure
$(\ker \tilde {\alpha }_-,\ker \tilde {\alpha }_+)$
has the same property; therefore, the supported vector field
$\tilde {X}$
is Anosov.
Remark 5.4. The condition
$R_{\alpha _-}\in \ker \alpha _+$
everywhere can be always achieved in a volume preserving Anosov flow (see Theorem 2.11). At the time of writing this work, it is not known if the condition can be achieved on any Anosov flow with orientable weak foliations.
5.1 Definition of the bi-contact surgery
We first construct a special coordinate system in a neighbourhood N of the Legendrian-transverse knot K, where two supporting contact forms
$(\alpha _-,\alpha _+)$
take a very simple expression.
Lemma 5.5. Suppose M is a 3-manifold endowed with a projectively Anosov flow
$\phi ^t$
defined by a bi-contact structure
$(\xi _-,\xi _+)$
and consider a knot K that is simultaneously Legendrian for
$\xi _-$
and transverse for
$\xi _+$
. Suppose also that there is a contact form
$\alpha _-$
defining
$\xi _-$
such that the Reeb vector field of
$\alpha _-$
is contained in
$\xi _+$
in a neighbourhood
$N=A_0\times (-\epsilon ,\epsilon )$
of K, where
$A_0$
is an annulus tangent to the Anosov flow and
$\epsilon>0$
. (
$A_0$
is also tangent to the contact structure along its core K.) We can choose N equipped with coordinates
$(s,v,w)$
and a contact form
$\alpha _+$
supporting
$\xi _+$
such that in N, we have
$$ \begin{align*}\alpha_-=dw+v\,ds,\end{align*} $$
$$ \begin{align*}\alpha_+=ds-b(s,v,w)\,dv \quad \textit{with} \;\;\alpha_+=ds \;\;on \;\;A_0 .\end{align*} $$
Here, s and v are coordinates on
$A_0$
such that
$s\in S^1$
is the parameter describing K and
$v\in (-\delta ,\delta )$
with
$\delta \in \mathbb {R}^+$
, while
$w\in (-\epsilon ,\epsilon )$
is the transverse parameter to
$A_0$
defined by the flow of
$R_-$
.
Proof. The construction of the surgery annulus
$A_0$
is straightforward. We start with the Legendrian-transverse knot
$K:S^1\rightarrow M$
and we define the tangent annulus
$$ \begin{align*}A_0=\bigcup_{t\in [-\tau,\tau]}\phi^t(K)\end{align*} $$
spanned by the flow
$\phi ^t$
for
$t\in [-\tau ,\tau ]$
with
$\tau>0$
. Let
$\alpha _-$
be the 1-form specified in the statement of the lemma. Since K is Legendrian, we can choose a small enough
$\tau $
such that the Reeb vector field
$R_{\alpha _-}$
is transverse to
$A_0$
(see Figure 7).
Figure 7 The surgery annulus
$A_0$
spanned by the flow. The dashed line is the Legendrian-transverse knot K. The vertical arrows represent
$R_{\alpha _- }$
. In black, the Anosov flow (colour online).
Let
$\psi ^w$
be the flow of the Reeb vector field
$R_{\alpha _-}$
of
$\alpha _-$
. There is an auxiliary coordinate system
$(s,u,w)$
in a neighbourhood of A such that
$$ \begin{align*}\alpha_-=dw+a(s,u)\,ds.\end{align*} $$
Since K is a Legendrian knot, we have
$a(s,0)=0$
and since
${\partial }/{\partial u}a(s,u)\neq 0$
, the transformation of coordinates
$$ \begin{align*}(s,u)\rightarrow (s,a(s,u))=:(s,v)\end{align*} $$
is non-singular and therefore we have (see [Reference Foulon and Hasselblatt14])
$$ \begin{align*}\alpha_-=dw+v\,ds.\end{align*} $$
Since the flow lines of
$R_{\alpha _-}$
are Legendrian for
$\xi _+$
by hypothesis and the v-curves are transverse to
$\xi _+$
in a neighbourhood N of K, there is a contact form
$\alpha _+$
supporting
$\xi _+$
that in N can be written
$$ \begin{align*}\alpha_+=ds-b(s,v,w)\,dv.\end{align*} $$
In general, the v-curves do not describe the same codimension-two foliation described by the flowlines of X. However, these codimension-two foliations coincide on the surgery annulus. Therefore,
$b(s,v,0)=0$
and
$\alpha _+=ds$
.
Lemma 5.6. In the hypothesis of Lemma 5.5, there is a neighbourhood
$\Lambda '\subset N$
of K such that the contact form
$\alpha _+^0=ds-b(s,v,w)\,dv$
can be isotoped along the flow lines of
$R_{\alpha _-}$
to a contact form independent on the s- and v-coordinate. In particular, we can write
$$ \begin{align*}\alpha_+^1=ds-h(w)\,dv.\end{align*} $$
Proof. Since
$\xi _+$
is a positive contact structure,
$b(s,v,w)$
is strictly increasing with w. Therefore,
$b(s,v,0)=0$
implies that
$b(s,v,w)>0$
for
$w>0$
. Choose a smooth function
$h:\mathbb {R}\rightarrow \mathbb {R}$
such that
$h(w)$
is strictly increasing with w and
$h(0)=0$
. For every point
$(s,v)\in A_0$
and k small enough, there is function
$g_k:A_0\rightarrow \mathbb {R}$
such that
$g_k(s,v)>0$
and
$b(s,v,g_k(s,v))=h(k)$
. The graph of
$g_k$
, namely
$A_{g_k}=\{(s,v,w)|(s,v)\in A_0, w=g_k(s,v)\}$
, is an annulus above
$A_0$
. More generally, for
$w\in [0,k]$
, we have a family of annuli
$A_{g_w}=\{(s,v,w)|(s,v)\in A_0, w=g_w(s,v)\}$
, where
$g_w(s,v)$
is such that
$b(s,v,g_w(s,v))=h(w)$
. Now, let
$\mathcal {F}_{A_{g_w}}$
be the characteristic foliation induced by
$\xi _+ $
on
$A_{g_w}$
for
$w\in [0,k]$
. Since
$h(w)$
is independent on the s- and v-coordinate, the projection of
$\mathcal {F}_{A_{g_w}}$
along the w-curves onto
$A_0$
is a foliation of
$A_0$
that is invariant in the s- and v-direction. Consider the union
$\Lambda =\bigcup _{w\in [0,k]}A_{g_w} $
of all the annuli
$A_{g_w}$
for
$w\in [0,k]$
and the subset
$\Lambda '\subset \Lambda $
with
$\Lambda '=\bigcup _{w\in [0,k]}A^{\prime }_{g_w} $
, where
$A^{\prime }_{g_w}=\{(s,v,w)|(s,v)\in A^{\prime }_0, w=g_w(s,v)\}$
and
$A^{\prime }_0$
is defined by
$v\in [-\delta _1,\delta _1]\subset (-\delta ,\delta )$
. Let
$\psi ^t$
be a smooth isotopy with support in
$\Lambda $
along the flow-lines of
$R_{\alpha _-}$
such that
$\psi ^1(A^{\prime }_{g_w})$
is the set
$A^{\prime }_w=\{(s,v,w)|(s,v)\in A^{\prime }_0, w=\mathrm {cost}\}$
. Consider on
$A_w=\psi ^1(A_{g_w})$
the foliation
$\mathcal {F}_{A_w}=\psi ^1(\mathcal {F}_{A_{g_w}})$
and let
$\mathcal {F}_{A^{\prime }_w}$
be its restriction on
$A_w'$
. Since the isotopy has been realized deforming each annulus
$A_{g_w}$
along the w-curves, the projection of
$\mathcal {F}_{A_w}$
on
$A_0$
is independent on the s- and v-coordinate. Since
$A^{\prime }_w$
is parallel to
$A^{\prime }_0$
, the foliation
$\mathcal {F}_{A^{\prime }_w}$
(not just its projection) is invariant in the s- and v-coordinate. Note that by construction,
$\mathcal {F}_{A_w}$
is the characteristic foliation induced on
$A_w$
by the contact structure
$\psi ^1_*\xi _+$
. Since
$\mathcal {F}_{A^{\prime }_w}$
on each
$A^{\prime }_w,\;w\in [0,k]$
is independent on the s- and v-coordinates, there is a contact form defining
$\psi ^1_*\xi _+$
on
$\Lambda '$
that is independent on the s- and v-coordinates. In particular, we can write
$\psi ^1_*\xi _+=\ker \alpha _+^1$
with
$\alpha _+^1=ds-h(w)\,ds$
.
Remark 5.7. Consider a path of projectively Anosov flows defined by an isotopy of the contact structure
$\xi _+$
along the flow-lines of
$R_{\alpha _-}$
as in Lemma 5.6. Every flow along the path is Anosov. In particular, any two flows
$\phi _0^t$
and
$\phi _1^t$
in the path are orbit equivalent by a diffeomorphism isotopic to the identity (see Theorem 2.12).
5.1.1 Deformations and gluings
Our procedure can be thought of as first cutting the manifold M along the annulus
$A_0$
and then gluing back the two sides of the cut in a different way. In particular, we glue the points described by the coordinates
$(s,v)$
on the side of the cut where
$w\leq 0$
to points
$(s+f(v),v)$
on the side of the cut where
$w\geq 0$
. More precisely, this can be stated as the following remark.
Remark 5.8. (Notation)
Let
$w\in [-\epsilon ,\epsilon ]$
be the parameter of the flow generated by
$R_{\alpha _-}$
. In the following, by
$A_0^-$
, we mean the surgery annulus as a subset of the flow-box chart
$(-\epsilon ,0]\times A_0$
, while by
$A_0^+$
, we mean the surgery annulus as a subset of the flow-box chart
$[0,\epsilon )\times A_0$
. Moreover,
$(\alpha _-)|_{A_0^-}$
is the restriction of
$\alpha _-$
to
$A_0^-$
, and
$(\alpha _-)^-$
is the restriction of
$\alpha _-$
to
$(-\epsilon ,0]\times A_0$
. Similarly,
$(\alpha _-)^+$
is the restriction of
$\alpha _-$
to
$(0,\epsilon ]\times A_0$
.
We define
$$ \begin{align*}f:\mathbb{R}\rightarrow S^1,\;\;\;v\rightarrow f(v),\end{align*} $$
where
$f:\mathbb {R}\rightarrow [0,2\pi ]$
monotone in
$(-\delta ,\delta )$
, smooth and such that
$f((-\infty ,-\delta ))=0$
,
$f([\delta ,\infty ))=-2q\pi $
, where
$q\in \mathbb {Z}$
, and we define the shear map
$$ \begin{align*}F:A^-_0\rightarrow A^+_0,\; (s,v)\rightarrow (s+f(v),v).\end{align*} $$
Remark 5.9. Note that for a
$(1,q)$
-bi-contact surgery with
$q>0$
, the function f used to define the shear is non-increasing, while the function used in §4 to define a
$(1,q)$
-Foulon–Hasselblatt surgery is non-decreasing. This is because Foulon–Hasselblatt construction was originally performed along a transverse annulus C, while the Legendrian-transverse surgery is performed on an annulus
$A_0$
tangent to the flow. With this convention, if
$K\subset C\cap A_0$
, the two operations produce homeomorphic manifolds for the same q.
As noticed in §4.1, the shear
$F:A^-_0\rightarrow A^+_0$
defines a smooth new
$3$
-manifold
$\tilde {M}$
, but it does not preserve the contact form
$\alpha _-$
since on the surgery annulus
$A_0$
,
$$ \begin{align*}F^*(\alpha_-)|_{A_0^+}=dw+v\:d(s+f(v))=(\alpha_-)^-+vf'(v)\,ds.\end{align*} $$
Here, by
$F^*(\alpha _-)|_{A_0^+}$
, we mean the image of
$(\alpha _-)|_{A_0^+}$
under the pull-back map
$F^*$
.
An analogous calculation shows that
$\alpha _+$
is not preserved either since
$$ \begin{align*}F^*(\alpha_+)|_{A_0^+}=F^*\,ds=d(s+f(v))=ds+f'(v)\,dv=(\alpha_+)|_{A_0^-}+f'(v)\,dv.\end{align*} $$
The strategy is to deform each of the contact structures independently on
$M\setminus A_0$
in such a way that they remain transverse to each other and after the application of the shear, they define two new (transverse) contact structures (see Figure 8).
Figure 8 On the top left, the bi-contact structure of the original flow. On the bottom left, the bi-contact structure after deformation. On the right, the corresponding flow-lines on the two sides of the surgery annulus
$A_0$
(colour online).
The deformation that we apply to
$\alpha _-$
is the same one used by Foulon and Hasselblatt to define their contact surgery.
We define
$(\tilde {\alpha }_-)^-=(\alpha _-)^-$
in the flow-box charts
$(-\epsilon ,0]\times A_0$
and
$(\tilde {\alpha }_-)^+=(\alpha _-)^+\,-dh$
in
$[0,\epsilon )\times A_0$
. Here,
$h:(-\delta ,\delta )\times [0,\epsilon )\rightarrow \mathbb {R} $
is such that
Here,
$\unicode{x3bb} _1:\mathbb {R}:\rightarrow [0,1]$
is a smooth bump function supported in
$[0,\epsilon )$
such that
$\unicode{x3bb} _1(0)=1$
and
$| \unicode{x3bb} _1' |<{1}/{\epsilon }+\iota $
with
$\iota $
arbitrarily small. Note that in a neighbourhood of
$A_0^+$
, we have [Reference Foulon, Hasselblatt and Vaugon15]
$$ \begin{align*}\tilde{\alpha}_-=dw+v\,ds- f'(v)v\,dv;\end{align*} $$
therefore,
$$ \begin{align*}F^*((\alpha_-)^+-dh)|_{A_0^+}=(\alpha_-)|_{A_0^-}\end{align*} $$
and we have a well-defined 1-form on the new manifold
$\tilde {M}$
.
The deformation that we apply to
$\alpha _+$
is defined as follows. We set
$(\tilde {\alpha }_+)^-=(\alpha _+)^-$
and
$(\tilde {\alpha }_+)^+=(\alpha _+)^+-\sigma $
, and consider the 1-form
where
$ \unicode{x3bb} _2: \mathbb {R}\rightarrow [0,1] $
is a smooth bump function, with support in
$[0,\epsilon )$
that takes the value
$1$
in a neighbourhood of
$0$
and value
$0$
in a neighbourhood of
$\epsilon $
. More explicitly,
Note that since
$f'(v)$
has support in
$(-\delta ,\delta )$
, the 1-form
$\sigma $
is smooth and vanishing in the complement of
$A_0\times [0,\epsilon )$
.
We check that on the surgery annulus
$A_0$
, we have
$F^*((\alpha _+)^+-\sigma )|_{A_0^+}=(\alpha _+)|_{A_0^+}$
. A direct computation shows that
therefore,
$\tilde {\alpha }_+$
is a well-defined
$1$
-form on
$\tilde {M}.$
Remark 5.10. The plane field distribution
$\tilde {\xi }_+=\ker (\tilde {\alpha }_+)$
and the plane field distribution
$\tilde {\xi }_-=\ker (\tilde {\alpha }_-)$
are transverse since
$\tilde {\xi }_+$
contains
$R_{\tilde {\alpha }_-}$
that is a vector field always transverse to
$\tilde {\xi }_-=\ker (\tilde {\alpha }_-)$
.
The following statement encompasses Theorems 1.1 and 1.2 in the introduction.
Theorem 5.11. Let
$(\xi _-=\ker \alpha _-,\xi _+=\ker \alpha _+)$
be a bi-contact structure such that in a neighbourhood of a Legendrian-transverse knot K, we have
$R_{\alpha _-}\subset \ker \alpha _+$
. If
$q>0$
, a bi-contact
$(1,q)$
-surgery along K produces a new bi-contact structure
$(\tilde {\xi }_-=\ker \tilde {\alpha }_-,\tilde {\xi }_+=\ker \tilde {\alpha }_+)$
. Moreover,
$R_{\tilde {\alpha }_-}\subset \ker \tilde {\alpha }_+$
.
Proof. We first show that the
$(1,q)$
-surgery produces a 1-form
$\tilde {\alpha }_-$
that is contact for every
$q\in \mathbb {Z}$
. This is done using the contact surgery introduced by Foulon and Hasselblatt in [Reference Foulon and Hasselblatt14], and interpreting the surgery annulus
$A_0$
as a transverse annulus to the Reeb vector field of
$\alpha _-$
. We use a slightly different argument to that used by Foulon and Hasselblatt. In particular, we show that we do not need any bound on the derivative of f in the definition of the shear in §5.1.1. As shown in [Reference Foulon and Hasselblatt14], we have
$$ \begin{align*}\tilde{\alpha}\wedge \,d\tilde{\alpha}=\bigg(-1+\frac{\partial h}{\partial w}\bigg)\,dV,\end{align*} $$
with
$dV=ds\wedge \,dv\wedge \, dw$
. Therefore, the condition
$|{\partial h}/{\partial w}|<1$
ensures that
$\tilde {\alpha }$
is contact.
$$ \begin{align*}\bigg|\frac{\partial h}{\partial w}\bigg|=\bigg|\unicode{x3bb}^{\prime}_1 \int_{-\delta}^{v}xf'(x)\;dx\bigg|,\end{align*} $$
integrating by parts and since
$f(-\delta )=0$
, we get
$$ \begin{align*}\bigg|\frac{\partial h}{\partial w}\bigg|=\bigg|\unicode{x3bb}^{\prime}_1 \bigg(vf(v)+\delta f(-\delta)-\int_{-\delta}^{v}f(x)\;dx\bigg)\bigg|= \bigg|\unicode{x3bb}^{\prime}_1 \bigg(vf(v)-\int_{-\delta}^{v}f(x)\;dx\bigg)\bigg|\end{align*} $$
since
$|\unicode{x3bb} ^{\prime }_1|<{1}/{\epsilon }+\iota $
with
$\iota $
arbitrarily small,
$-\delta <v<\delta $
(by definition of v),
$|f(v)|<2\pi |q|$
(by definition of f) and
$|\int _{-\delta }^{v}f(x)\,dx|\leq |\int _{-\delta }^{\delta }f(x)\,dx)|< 4\pi \delta |q|$
, we have
$$ \begin{align*}\bigg|\frac{\partial h}{\partial w}\bigg|\leq \bigg(\frac{1}{\epsilon}+\iota\bigg)(\delta 2\pi|q|+4\delta\pi|q| )= \bigg(\frac{1}{\epsilon}+\iota\bigg)(6\delta\pi|q| ),\end{align*} $$
and since
$\iota $
is arbitrarily small, we have
$$ \begin{align*}\bigg|\frac{\partial h}{\partial w}\bigg|\leq \frac{\delta}{\epsilon}(6\pi|q| +1).\end{align*} $$
This shows that the 1-form
$\tilde {\alpha }_-$
is contact if
$$ \begin{align} 0<\delta <\frac{\epsilon}{6\pi|q| +1}. \end{align} $$
Remark 5.12. Note that condition (5.2) holds independently of the sign of q and the positivity of
$\alpha _-$
. Moreover,
$\delta $
can be chosen arbitrarily small, in particular, the tangent surgery annulus
$A_0$
can be chosen arbitrarily thin.
We now show that for
$q>0$
, the 1-form
$\tilde {\alpha }_+$
defines a contact form on
$\tilde {M}$
. The 1-form
$\tilde {\alpha }_+$
defines a positive contact form if and only if [Reference Eliashberg and Thurston12]
Since
$b'(w)>0$
by the contact condition and
$\unicode{x3bb} _2'(w)\leq 0$
by the definition of the bump function
$\unicode{x3bb} _2$
, the inequality is always satisfied if
$f'(v)\leq 0$
. Therefore, the family of
$(1,q)$
-Dehn surgeries produces bi-contact structures for
$q>0$
.
Finally, since
$\tilde {\alpha }_+=ds-(b(w)+\unicode{x3bb} _2(w)\:f'(v))\,dv$
and
$R_{\tilde {\alpha }_+}$
is parallel to
$R_{\alpha _+}$
on
$\tilde {M}\setminus A_0$
, we have
$R_{\tilde {\alpha }_+}\in \ker \tilde {\alpha }_+ $
.
The proof of Theorem 5.11 shows that there is a choice of the direction of the twist that strengthens the contact condition of the contact structure transverse to K. Performing the surgery in the opposite direction may result in a violation of the contact condition. An analogous phenomenon emerges in Goodman surgery (see [Reference Barbot5, Reference Foulon, Hasselblatt and Vaugon15, Reference Handel and Thurston19]), where there is a choice of the direction of the shear that weakens the hyperbolicity of the flow. The following result shows that there is indeed a strong connection between the Anosovity of the new flows produced by the bi-contact surgery and contact geometry.
Theorem 5.13. Let M be a
$3$
-manifold endowed with a bi-contact structure
$(\ker \alpha _-,\ker \alpha _+)$
such that
$R_{\alpha _-}\in \ker \alpha _+$
everywhere. If the Legendrian-transverse surgery yields a new bi-contact structure, it yields an Anosov flow.
Proof. The proof is a straightforward application of Hoozori criterion of Anosovity. Since by construction, we have
$R_{\tilde {\alpha } _-}\subset \tilde {\xi }_+$
, the Reeb vector field
$R_{\tilde {\alpha }}$
is dynamically positive everywhere (Corollary 2.10).
6 New Anosov flows for negative surgery coefficients
We now give a sufficient condition for a bi-contact
$(1,q)$
-surgery to yield an Anosov flow when the operation is performed in the direction that weakens the hyperbolicity. Let K be a Legendrian-transverse knot in a bi-contact structure
$(\xi _-=\ker \alpha _-,\xi _+=\ker \alpha _+) $
such that
$R_{\alpha _-}\subset \xi _+$
everywhere. The condition we give for a bi-contact
$(1,q)$
-surgery to yield an Anosov flow is purely contact geometric and involves the behaviour of
$\xi _+$
along the boundary of a neighbourhood
$\Lambda $
of K. To this end, we introduce the following definition.
Definition 6.1. Let
$\Lambda $
be a neighbourhood of a Legendrian-transverse knot K such that the leaves of the characteristic foliation
$\mathcal {F}_{\partial \Lambda }$
induced by
$\xi _+$
on
$\partial \Lambda $
are simple closed curves in the homology class
$r[\mu ]+s[K]$
, where
$\mu $
bounds a disk in
$\Lambda $
. We call
$k={s}/{r}>0$
the slope of the characteristic foliation
$\mathcal {F}_{\partial \Lambda }$
.
Remark 6.2. Let N be a neighbourhood of a Legendrian-transverse knot K as in Lemma 5.5. After an isotopy of the
$\xi _+$
along the flow-lines of
$\alpha _-$
that induces an orbit equivalence isotopic to the identity, we assume that
$\alpha _+=ds-h(w)\,dv$
by Lemma 5.6.
Theorem 6.3. Let K be a Legendrian-tansverse knot in a bi-contact structure
$(\xi _-=\ker \alpha _-,\xi _+=\ker \alpha _+) $
such that
$R_{\alpha _-}\subset \xi _+$
everywhere. There is a neighbourhood
$\Lambda \subset N$
of K such that if
$k>0$
is the slope of the characteristic foliation
$\mathcal {F}_{\partial N}$
induced by
$\xi _+$
on
$\partial N$
, a bi-contact
$(1,q)$
-surgery with
$q>-k$
yields an Anosov flow.
Proof. Let
$\phi ^t$
be an Anosov flow supported by
$(\ker \alpha _-=\xi _-,\xi _+)$
such that
$R_{\alpha _-}\in \xi _+$
. Call
$\psi ^w$
the flow of
$R_{\alpha _-}$
. Consider a flow-box neighbourhood
$\Lambda \subset N$
of
$K=K_0$
bounded at the bottom by the surgery annulus
$A_{K_0}=A_0$
, bounded on the sides by the annuli (transverse to the flow
$\phi ^t$
)
$C_{\mathrm{in}}$
and
$C_{\mathrm{out}}$
constructed by flowing the boundary components of
$A_0$
using the flow of
$R_{\alpha _-}$
, and bounded on the top by an annulus (see Figure 9)
$$ \begin{align*}A_{K_\epsilon}=\bigcup_{t\in [-\tau,\tau]}\phi^t(\psi^{\epsilon}(K)).\end{align*} $$
In particular,
$A_{K_\epsilon }$
is constructed as follows. Let
$K_\epsilon =\psi ^\epsilon (K)$
be the knot obtained by translating K using the flow
$\psi ^w$
for a (small) time
$\epsilon $
. The annulus
$A_{K_\epsilon }$
is constructed translating the knot
$\psi ^\epsilon (K)$
using the Anosov flow
$\phi ^t$
for
$t\in [-\tau ,\tau ]$
. Moreover, we can assume that
$\mathcal {F}_{\partial \Lambda }$
is composed by closed curves. This is possible since
$\mathcal {F}_{\partial \Lambda }$
is a linear foliation since it is independent on the s-coordinate. If the slope is irrational, after a
$\mathcal {C}^1$
-small perturbation along the flow-lines of
$R_{\alpha _-}$
, we can assume that
$\mathcal {F}_{\partial \Lambda }$
has closed leaves. Let
$k={s}/{r}>0$
be the slope of
$\mathcal {F}_{\partial \Lambda }$
. Let p be the integral part of
$-k$
if
$-k$
is not an integer and set
$p=-k+1$
if
$-k$
is an integer. Since
$p>-k$
, we can choose a function
$f_p:[-\delta ,\delta ]\rightarrow S^1$
with
$f^{\prime }_p(-\delta )=f^{\prime }_p(\delta )=0$
such that the curve
$\mathfrak {f} _p=\{(v,s)\in A_0|v\in [-\delta ,\delta ],\:s=f_p(v)\}$
, defined by the graph of
$f_p$
, has the following properties:
-
(1) the projection
$\pi _w(\mathfrak {f} _p)$
of
$\mathfrak {f} _p$
into
$A_{K_\epsilon }$
along the w-curves intersects transversely the curves of the characteristic foliation
$\mathcal {F}_{A_{K_\epsilon }}$
obtained by restricting
$\mathcal {F}_{\partial \Lambda }$
to
$A_{K_\epsilon};$
-
(2) the foliation on
$\partial \Lambda $
constructed by replacing
$\mathcal {F}_{A_{K_\epsilon }}$
with a family of curves parallel to
$\pi _w(\mathfrak {f} _p)$
has slope p.
The shear map
$$ \begin{align*}F:A_0\rightarrow A_0,\quad (s,v)\rightarrow (s+f_p(v),v)\end{align*} $$
induces a
$(1,p)$
-Dehn surgery. Similarly to §5.1.1, we set
On
$A_{K_\epsilon }$
, the characteristic foliation induced by
$\xi _+=\ker (ds-b(w)\:dv)$
is directed by
$v=({\partial }/{\partial s})b(g_{K_\epsilon }(v))+{\partial }/{\partial v}+C_1(v){\partial }/{\partial w}$
, where
$g_{K_\epsilon }(v):[-\delta ,\delta ]\rightarrow [0,\epsilon ]$
is such that
$A_{K_\epsilon }=\{v\in [-\delta ,\delta ],w=g_{K_\epsilon }(v)\}$
and some
$C_1:[-\delta ,\delta ]\rightarrow \mathbb {R}$
. Since the curve
$\pi _w(\mathfrak {f} _p)$
defined above positively intersects
$\mathcal {F}_{A_{K_\epsilon }}$
, we have
$b(g_{K_\epsilon }(v))>f^{\prime }_p(v)$
. Moreover,
$b(0)=0$
on
$A_0$
and
$b'(w)>0$
in
$\Lambda $
. Therefore, there is an annulus
$A_{\mathfrak {f} _p}$
between
$A_{K_\epsilon }$
and
$A_0$
such that the characteristic foliation
$\mathcal {F}_{A_{\mathfrak {f} _p}}$
induced by
$\xi _+$
on
$A_{\mathfrak {f} _p}$
is directed by
$u=({\partial }/{\partial s})f^{\prime }_p(v)+{\partial }/{\partial v}+C_2(v){\partial }/{\partial w}$
with
$C_2:[-\delta ,\delta ]\rightarrow \mathbb {R}$
. We have
$A_{\mathfrak {f} _q}=\{v\in [-\delta ,\delta ],w=h_{\mathfrak {f} _p}(v)\}$
with
$h_{\mathfrak {f} _p}:[-\delta ,\delta ]\rightarrow [0,\epsilon ]$
such that
$b(h_{\mathfrak {f} _p}(v))=f^{\prime }_p(v)$
. We now define the smooth function
$\unicode{x3bb} _p:\Lambda \rightarrow [0,1]$
. (Since
$\alpha _+$
is invariant in the s-direction,
$\unicode{x3bb} _p$
can be chosen to be constant along the coordinate s, so we will write for simplicity
$\unicode{x3bb} _p(v,w)$
instead of
$\unicode{x3bb} _p(s,v,w)$
.) We set
$\unicode{x3bb} _p(v,w)=0$
near
$A_{K_\epsilon }$
. Let
$\Lambda '\subset \Lambda $
be the portion of
$\Lambda $
between
$A_0$
and
$A_{\mathfrak {f} _p}$
. We set
$\unicode{x3bb} _p(v,w)=1$
on a subset
$\Lambda "\subset \Lambda $
that contains
$\Lambda '$
. Therefore, we have
$\unicode{x3bb} _p(v,w)=1$
near
$A_{\mathfrak {f} _q}$
and
$\unicode{x3bb} _p(v,w)=0$
near
$A_{K _\epsilon }$
. Since in the region
$\Lambda \setminus \Lambda "$
we have
$b(w)>f^{\prime }_p(v)$
, we can define
$\unicode{x3bb} _p:\Lambda \rightarrow [0,1]$
such that
In particular, we can choose
$\unicode{x3bb} _p(v,w)<-{b(w)}/{f^{\prime }_p(v)}$
with
$ {\partial \unicode{x3bb} _p}/{\partial w}<-{b'(w)}/{f^{\prime }_p(v)}$
. With these choices, we have
which is equivalent to say that
$\tilde {\alpha }_+$
is contact. Since the w-curves are flow lines of
$R_{\alpha _-}$
, Anosovity follows by Hozoori’s criterion.
Figure 9 On the right, the positive contact structure rotates along the w axis. As usual, the Legendrian-transverse knot is denoted by a dashed line. On the left, the annuli
$A_{K_w}$
spanned by the flow (colour online).
6.1 Bi-contact surgery near a closed orbit
We now show that if the knot K is a Legendrian-transverse push off of a closed orbit
$\gamma $
of the flow, it is possible to perform the bi-contact surgery with any integral surgery coefficient. First, we give a version of Lemma 5.5 in a neighbourhood of a closed orbit
$\gamma $
.
Lemma 6.4. Suppose M is a 3-manifold endowed with a projectively Anosov flow
$\phi ^t$
defined by a bi-contact structure
$(\ker \alpha _-=\xi _-,\xi _+)$
and consider a knot K that is a Legendrian-transverse push-off of a closed orbit
$\gamma $
. Suppose also that the Reeb vector field of
$\alpha _-$
is contained in
$\xi _+$
in a neighbourhood
$N=A_0\times (-\epsilon ,\epsilon )$
of K, where
$A_0$
is an annulus tangent to the Anosov flow and
$\epsilon>0$
. We can choose N equipped with coordinates
$(s,v,w)$
and a contact form
$\alpha _+$
supporting
$\xi _+$
such that in N, we have
$$ \begin{align*}\alpha_-=dw+v\,ds,\end{align*} $$
$$ \begin{align*}\alpha_+=(\overline{\epsilon}-w)\,ds-w\,dv.\end{align*} $$
Here, s and v are coordinates on
$A_0$
such that
$s\in S^1$
is the parameter describing K and
$v\in (-\delta ,\delta )$
with
$\delta \in \mathbb {R}^+$
, while
$w\in (-\epsilon ,\epsilon )$
is the transverse parameter to
$A_0$
defined by the flow of
$R_-$
. Finally,
$\overline {\epsilon }\in (0,\epsilon )$
.
Proof. The proof is similar to that of Lemma 5.5. Let
$\psi ^w$
be the flow of
$R_{\alpha _-}$
. Since K is a Legendrian-transverse push-off of
$\gamma $
, we can write
$K=\psi ^{-\overline {\epsilon }}(\gamma )$
for some
$\overline {\epsilon }>0$
. Since
$\gamma $
is a closed orbit of the flow,
$\gamma $
is a knot that is Legendrian for both
$\xi _-$
and
$\xi _+$
. Similarly to the proof of Lemma 5.5, we can write
$$ \begin{align*}\alpha_+=b_1(s,v,w)\:ds-b_2(s,v,w)\,dv,\end{align*} $$
with
$b_2(s,v,0)=0$
and
$b_1(s,v,\overline {\epsilon })=0$
since
$\gamma $
is Legendrian for
$\xi _+$
. Using the same argument of Lemma 5.6, we can isotope
$\xi _+$
along the w-curves obtaining the desired expression for
$\alpha _+$
.
Corollary 6.5. Let
$\gamma $
be a closed orbit of a flow defined by a bi-contact structure
$(\xi _-=\ker \alpha _-,\xi _+=\ker \alpha _+) $
such that
$R_{\alpha _-}\subset \xi _+$
and let K be a Legendrian-transverse push-off of
$\gamma $
. For every
$p\in \mathbb {Z}^-$
, there is a flow-box neighbourhood
$\Lambda _p$
of K such that a bi-contact
$(1,q)$
-surgery along K yields an Anosov flow for every
$q>p$
.
Proof. Consider the flow-box neighbourhood
$\Lambda $
as in the proof of Theorem 6.3. Since
$\psi ^{\overline {\epsilon }}(K)=\gamma $
, we have that if w approaches
$\overline {\epsilon }$
, the contact form
$\alpha _+$
approaches the plane field
$\ker (-\overline {\epsilon }\,dv)$
. In particular, the slope of the characteristic foliation induced by
$\alpha _+$
on
$\Lambda $
can be made arbitrarily large taking w close enough to
$\overline {\epsilon }$
. We finally use Theorem 6.3 and the fact that either p is the integral part of
$-k$
or
$p=-k+1$
(with k the slope of the characteristic foliation induced by
$\xi _+$
on
$\Lambda $
defined in the proof of Theorem 6.3).
Remark 6.6. In a neighbourhood N of a closed orbit
$\gamma $
of a flow defined by a bi-contact structure
$(\xi _-=\ker \alpha _-,\xi _+=\ker \alpha _+) $
such that
$R_{\alpha _-}\subset \xi _+$
, there is a sequence of nested flow-box neighbourhoods of the Legendrian-transverse knot K namely
$\Lambda _{-1}\subset \Lambda _{-2}\subset \cdots \subset \Lambda _{p}\subset \cdots \subset \Lambda _{-\infty }$
,
$p\in \mathbb {Z}^-$
such that a bi-contact
$(1,q)$
-surgery yields an Anosov flow for every
$q\geq p$
. Here,
$$ \begin{align*}\Lambda_{p}=\bigcup_{w\in [0,\epsilon_p]}A_{K_w},\end{align*} $$
and
$$ \begin{align*}A_{K_w}=\bigcup_{t\in [-\tau,\tau]}\phi^t(\psi^{w}(K))\end{align*} $$
with
$\epsilon _p$
sufficiently close to
$\overline {\epsilon }$
. Moreover,
$$ \begin{align*}\Lambda_{-\infty}=\overline{\bigcup_{w\in [0,\overline{\epsilon})}A_{K_w}}.\end{align*} $$
All the neighbourhoods are bounded on one side by
$A_0$
. For
$p\in \mathbb {Z}^-$
, the neighbourhood
$\Lambda _p$
is a flow-box (see Figure 10), while
$\Lambda _{-\infty }$
is not a flow-box, since it contains the closed orbit
$\gamma $
.
Figure 10 The positive contact structure
$\xi _+$
in the proximity of a closed orbit. The curves are the leaves of the characteristic foliation induced by
$\xi _+$
on
$A_{\epsilon }$
(colour online).
7 Relations with Goodman surgery
In this section, we study the relationship between the bi-contact surgery and the surgery introduced by Goodman [Reference Goodman17]. At a first glance, the two operations look very different. First of all, the surgery annulus used in the bi-contract surgery is tangent to the generating vector field X, while in Goodman’s construction, we use an annulus transverse to the flow. Second, in the bi-contact surgery construction, the deformation of the bi-contact structure induces a three-dimensional perturbation of the flow in a neighbourhood of the surgery annulus. However, in Goodman’s surgery, the flow lines of the new flow coincide with the old ones outside the (transverse) surgery annulus. Despite these differences, the flows generated by bi-contact surgery and those generated by Goodman’s construction are orbit equivalent.
Theorem 7.1. Suppose that K is a Legendrian-transverse push-off of a closed orbit in a bi-contact structure
$(\xi _-,\xi _+)$
defined by two contact forms
$(\alpha _-,\alpha _+)$
such that
$R_{\alpha _-}$
is contained in
$\xi _+$
. An Anosov flow generated by a bi-contact
$(1,q)$
-surgery along the tangent annulus
$A_0$
is orbit equivalent to an Anosov flow generated by a
$(1,q)$
-Goodman surgery along a transverse annulus C.
Proof. As in the proof of Theorem 6.3, let
$\phi ^t$
be an Anosov flow supported by
$(\ker \alpha _-=\xi _-,\xi _+)$
such that
$R_{\alpha _-}\in \xi _+$
. Call
$\psi ^w$
the flow of
$R_{\alpha _-}$
. Let N be a neighbourhood of a Legendrian knot K with a coordinate system
$(s,v,w)$
defined as in Lemma 5.5, where the contact form
$\alpha _+$
can be assumed (by Lemma 5.6) to be independent on the s-coordinate. Consider a flow-box neighbourhood
$\Lambda \subset N$
of
$K=K_0$
bounded at the bottom by the tangent surgery annulus
$A_{K_0}=A_0$
, bounded on the sides by the transverse annuli
$C_{\mathrm{in}}$
and
$C_{\mathrm{out}}$
constructed by flowing the boundary components of
$A_0$
using the flow of
$R_{\alpha _-}$
, and bounded on the top, by a tangent annulus (see Figure 9)
$$ \begin{align*}A_{K_\epsilon}=\bigcup_{t\in [-\tau,\tau]}\phi^t(\psi^{\epsilon}(K)).\end{align*} $$
Let
$\mathcal {F}_{\mathrm{out}}$
be the foliation obtained by projecting on
$C_{\mathrm{out}}$
(using the flow of X) the w-curves defined on
$C_{\mathrm{in}}$
. Consider now a bi-contact
$(1,q)$
-surgery along the tangent annulus
$A_0$
with shear
$$ \begin{align*}F:A_0\rightarrow A_0,\;\;\;(v,s)\rightarrow (v,s+f(v))\end{align*} $$
with
$f(-\delta )=0$
and
$f(\delta )=-2\pi q$
. Let
$\tilde {X}$
be the new vector field supported by the new bi-contact structure
$(\tilde {\xi }_-,\tilde {\xi }_+)$
. Note that after the surgery procedure the flow-box
$\Lambda $
is transformed in another flow-box
$\tilde {\Lambda }$
, bounded by the same annuli as
$\Lambda $
. Let
$\tilde {\mathcal {F}}_{\mathrm{out}}$
be the foliation on
$C_{\mathrm{out}}$
obtained by projecting (using the flow of
$\tilde {X}$
) the w-curves defined on
$C_{\mathrm{in}}$
. We want to show that a leaf
$\tilde {l}_{\mathrm{out}}$
of
$\tilde {\mathcal {F}}_{\mathrm{out}}$
has the same endpoints of a leaf
$l_{\mathrm{out}}$
of
$\mathcal {F}_{\mathrm{out}}$
, but they differ by a Dehn twist along the core of
$C_{\mathrm{out}}$
. Consider a segment
$l_{\mathrm{in}}$
of a segment of a w-curve with endpoints
$p\in \partial A_0$
and
$q\in \partial A_{K_\epsilon }$
, and let
$l_{\mathrm{out}}$
and
$\tilde {l}_{\mathrm{out}}$
be its projection into
$C_{\mathrm{out}}$
using respectively the flow of X and
$\tilde {X}$
. Since on
$A_{K_\epsilon }$
the vector field X and
$\tilde {X}$
coincide,
$l_{\mathrm{out}}$
and
$\tilde {l}_{\mathrm{out}}$
share the same endpoint on the intersection of
$\partial C_{\mathrm{out}}$
and
$\partial A_{K_\epsilon }$
. Since the foliation induced by
$\tilde {X}$
on
$A_0$
is directed by
$\tilde {W}_0=f'(v)\:{\partial }/{\partial s}+{\partial }/{\partial v}$
, while the one induced by X on
$A_0$
is directed by
$W_0={\partial }/{\partial v}$
, a segment of a flow-line of
$\tilde {X}$
on
$A_0$
starting from the point
$p\in \partial A_0$
differs from that of X (staring from p) by a full Dehn twist along the core of
$A_0$
; therefore, the flow-line segments directed by
$W_0$
and
$\tilde {W}_0$
share also the endpoint on the intersection of
$\partial A_0$
and
$\partial C_{\mathrm{out}}$
. Thus,
$\tilde {l}_{\mathrm{out}}$
and
$l_{\mathrm{out}}$
have the same endpoints. Since
$\tilde {\Lambda }$
and
$\Lambda $
are both toroidal flow-boxes with the same transverse and tangent faces, and since the flow-line of
$\tilde {X}$
on
$A_0$
differs from that of X by a full Dehn twist along the core of
$A_0$
,
$\tilde {l}_{\mathrm{out}}$
must differ from
$l_{\mathrm{out}}$
by a full Dehn twist along the core of
$C_{\mathrm{out}}$
. Consider the flow obtained by collapsing to a point each of the flow-line segments of
$\tilde {X}$
in
$\tilde {\Lambda }$
. The resulting flow is clearly orbit equivalent to
$\tilde {X}$
and also it is orbit equivalent to a flow obtained by removing
$\tilde {\Lambda }$
and gluing
$C_{\mathrm{in}}$
to
$C_{\mathrm{out}}$
using a map that identifies each segment of w-curves
$l_{\mathrm{in}}$
on
$C_{\mathrm{in}}$
to the corresponding
$\tilde {l}_{\mathrm{out}}$
on
$C_{\mathrm{out}}$
. Call
$\tilde {X}_\delta $
the family of flows obtained by bi-contact surgery for different choices of
$\delta \in [\tilde {\delta },0)$
(
$2\delta $
represents the width of the surgery annulus
$A_0$
). Since this family defines a smooth path of Anosov flows, they are all orbit equivalent by an orbit equivalence isotopic to the identity. Taking
$\delta $
small enough, we can make the annuli
$C_{\mathrm{in}}$
and
$C_{\mathrm{out}}$
to be
$\mathcal {C}^1$
-close to the annulus C (this is the annulus transverse to the flow and bounded by
$K=K_0$
on the bottom and by
$K_\epsilon $
on the top). However,
$l_{\mathrm{in}}$
is
$\mathcal {C}^1$
-close to
$l_{\mathrm{out}}$
. Since
$l_{\mathrm{in}}$
is a w-segment,
$l_{\mathrm{out}}$
is
$\mathcal {C}^1$
-close to a w-segment and since
$\tilde {l}_{\mathrm{out}}$
is obtained adding a Dehn twist to
$l_{\mathrm{out}}$
and the knot
$K_w,\;w\in [0,\epsilon ]$
is Legendrian (the fact that
$K_w$
is Legendrian is relevant since if we take
$\delta $
small, we can assume that the foliation in annuli of
$\Lambda $
and
$\tilde {\Lambda }$
(defined in Remark 6.6) are
$\mathcal {C}^1$
-close), the identification of
$l_{\mathrm{in}}$
with
$\tilde {l}_{\mathrm{out}}$
(for
$\delta $
small enough) gives a flow that is
$\mathcal {C}^1$
-close to one constructed by cutting X along C and gluing back the two sides
$C^-$
and
$C^+$
of C using the shear map
$$ \begin{align*}G:C^-\rightarrow C^+,\;\;\;(w,s)\rightarrow (w,s+g(w)),\end{align*} $$
with
$g:\mathbb {R}\rightarrow [0,2\pi ]$
, smooth, and such that
$g(0)=0$
and
$g(\epsilon )=2\pi q$
. The procedure just described is Goodman surgery (see Figure 11).
Figure 11 The flow-box N. For simplicity, we depicted the top layer
$A_{\epsilon }$
as flat. The two vertical segments on the right side of the left picture represent two leaves of
$\mathcal {F}$
, while on the right, two leaves of the new foliation
$\mathcal {\tilde {F}}$
are represented. A negative Dehn twist on
$A_0$
(represented by the curve on the bottom of the picture on the right) corresponds to a positive Dehn twist on
$C_{\mathrm{out}}$
. Therefore, a flow constructed by a
$(1,q)$
-bi-contact surgery along
$A_0$
is orbit equivalent to a flow generated by
$(1,q)$
-Goodman surgery along
$C_{\mathrm{out}}$
(colour online).
Corollary 7.2. Suppose that K is a Legendrian-transverse knot in a bi-contact structure
$(\xi _-=\ker \alpha _-,\xi _+)$
such that
$R_{\alpha _-}\in \xi _+$
everywhere. Choose
$p\in \mathbb {Z}^-$
. There is a transverse annulus
$C_{p}$
such that a
$(1,q)$
-Goodman surgery on
$C_{p}$
yields an Anosov flow for every
$q\geq p$
.
Proof. For every flow-box
$\Lambda _p$
as in Remark 6.6, consider the core annulus
$C_p$
of
$\Lambda _p$
,
$$ \begin{align*}C_p=\bigcup_{w\in [0,\epsilon_p]} K_w, \end{align*} $$
where
$K_w =\psi ^w(K)$
and use the proof of Theorem 7.1.
Corollary 7.3. In the hypothesis of Theorem 7.2, there is a sequence of nested annuli
$C_{-1}\subset C_{-2}\subset \cdots \subset C_p\subset \cdots \subset C_{-\infty }$
bounded on one side by the Legendrian- transverse knot K. Every
$C_p$
is the core of
$\Lambda _p$
and it is transverse to the flow if
$p\in \mathbb {Z}$
. The annulus
$C_{-\infty }$
is quasi-transverse and it is bounded by K on one side and by the closed orbit
$\gamma $
on the other side (see Figure 12). A
$(1,q)$
-Goodman surgery along
$C_p$
produces an Anosov flow for every
$q\geq p$
.
Figure 12 If a
$(1,p)$
-Goodman surgery along an annulus
$C_p$
produces an Anosov flow for a fixed
$p<0$
, a
$(1,p-1)$
-Goodman produces an Anosov flow on an annulus
$C_{p-1}$
obtained by expanding
$C_p$
towards the closed orbit. The curve on the left represents a
$(1,-1)$
Goodman surgery, while the curve on the right is associated to a
$(1,-2)$
Goodman surgery (colour online).
8 Surgery along simple closed geodesics in a geodesic flow
Let X be the generating vector field of the geodesic flow on the unit tangent bundle of an oriented hyperbolic surface S and let
$\beta _+$
be the contact form preserved by
$X=R_{\beta _+}$
. Select a closed geodesic
$\gamma $
(eventually self intersecting) on S and consider the knot L defined by the angle
$\theta ={\pi }/{2}$
on each fibre along
$\gamma $
. Here, L is a Legendrian knot for
$\ker \beta _+$
. Let
$C_{\epsilon }=S^1\times [-\epsilon , \epsilon ]\subset $
UT
$\gamma $
be a transverse annulus with
$\epsilon $
small enough to ensure that
$C_{\epsilon }$
is not self intersecting. As shown by Foulon and Hasselblatt for any
$q>0$
, their construction ensures that a
$(1,q)$
-Goodman surgery produces a contact Anosov flow.
If L is associated to a simple closed geodesic
$\gamma $
, a recent result of Marty [Reference Marty23] shows that Goodman surgery produces a positively skewed
$\mathbb {R}$
-covered Anosov flow regardless of the sign of the surgery if and only if the closed orbit is the lift of a simple closed geodesic. We now prove a counterpart of Marty’s result in the contact category.
We prove the following lemma showing that while the Foulon and Hasselblatt construction requires an arbitrarily thin annulus to produce a contact flow for positive surgery coefficients, in the case of negative surgery coefficients, this requirement is not necessary.
Lemma 8.1. Suppose that
$R_{\beta _+}$
is the Reeb vector of a contact form
$\beta _+$
defining a positive contact structure and let L be a Legendrian knot for
$\ker \beta _+$
. For
$q<0$
, the Foulon and Hasselblatt construction along a transverse embedded annulus C produces a new contact form
$\tilde {\beta }_+$
with Reeb vector field
$R_{\tilde {\beta }_+}$
regardless of the thickness of the surgery annulus.
Proof. Foulon and Hasselblatt show that the vector field
$\tilde {X}$
obtained by Goodman surgery with shear
$$ \begin{align*}G:C\rightarrow C,\;\;\;(s,w)\rightarrow (w,s+g(w)),\end{align*} $$
along C preserves a positive contact form
$$ \begin{align*}\tilde{\beta}_+=dt+w\,ds- dh\end{align*} $$
with
$$ \begin{align*}h(t,w)= \unicode{x3bb}(t) \int_{-\epsilon}^{w}xg'(x)\,dx;\end{align*} $$
hence,
$$ \begin{align*}\tilde{\beta}_+=dt+w\,ds-\unicode{x3bb}'(t)\int_{-\epsilon}^{w}xg'(x)\,dx\,dt-\unicode{x3bb}(t)wg'(w)\,dw.\end{align*} $$
They indeed show that the contact condition
$$ \begin{align} \tilde{\beta}_+\wedge d\tilde{\beta}_+=\bigg(1-\frac{\partial h}{\partial t}\bigg)\:dV>0 \end{align} $$
is satisfied regardless of the sign of the Dehn twist if the surgery annulus is sufficiently thin. Note that if
$$ \begin{align} \frac{\partial h}{\partial t}=\unicode{x3bb}'(t)\int_{-\epsilon}^{w}xg'(x)\,dx\,dt\leq 0, \end{align} $$
the contact condition (8.1) is satisfied independently of the thickness of the surgery annulus. If
$g'(w)<0$
and
$g:[-\epsilon ,\epsilon ]\rightarrow S^1$
is odd, we have
$\int _{-\epsilon }^{w}xg'(x)\,dx\geq 0$
for
$w\in (-\epsilon , \epsilon )$
and since
$\unicode{x3bb} '(t)\leq 0$
for
$t>0$
, inequality (8.2) is satisfied. This corresponds to
$(1,q)$
-Foulon–Hasselblatt surgery with
$q<0$
.
Theorem 8.2. Let
$\phi ^t$
be the geodesic flow on the unit tangent bundle
$UTS$
of an oriented hyperbolic surface S. Let
$C\subset $
UT
$\gamma $
be a quasi-transverse annulus associated to a simple closed geodesic. For every
$p<0$
, there is an embedded transverse annulus
$C_p\subset C$
centred on L (as defined in §3.2) such that a
$(1,p)$
-Goodman surgery along
$C_p$
produces a contact Anosov flow.
Proof. We first show that for every
$p\in \mathbb {Z}^-$
, there are choices such that the bi-contact surgery along a tangent annulus
$A_0$
containing L produces an Anosov flow. Then, we show that these flows are orbit equivalent to flows constructed by
$(1,p)$
-Foulon–Hasselblatt surgery along an annulus C transverse to the geodesic flow. The first part of the proof is essentially the same as Theorem 7.1. The difference is that in this case, we are not allowed to perform any isotopy to transform the natural form
$\alpha _+$
in a contact form independent on the s and v coordinates since such an isotopy would modify the vector field X defined by
$(\alpha _-,\alpha _+)$
in a vector field
$X'$
that a priori does not preserve
$\beta _+$
. Nevertheless, Theorem 3.1 shows that
$\alpha _+$
and
$\alpha _-$
are independent on the s coordinates and this is enough for our purpose. More precisely, consider the
$(s,v,w)$
-coordinate system in a neighbourhood N of UT
$\gamma $
as in Theorem 3.1. Let
$\phi ^t$
be the geodesic flow supported by
$( \alpha _-,\alpha _+)$
, the natural contact forms. Call
$\psi ^w$
the flow of
$R_{\alpha _-}$
. Consider a flow-box neighbourhood
$\Lambda \subset N$
of
$L=L_0$
bounded at the bottom by the annulus
$$ \begin{align*}A_{L_{-\epsilon}}=\bigcup_{t\in [-\tau,\tau]}\phi^t(\psi^{-\epsilon}(L)),\end{align*} $$
bounded on the sides by the annuli (transverse to the flow
$\phi ^t$
)
$C_{\mathrm{in}}$
and
$C_{\mathrm{out}}$
constructed by flowing the boundary components of
$A_0$
using the flow of
$R_{\alpha _-}$
, and bounded on the top by an annulus (see Figure 9)
$$ \begin{align*}A_{L_\epsilon}=\bigcup_{t\in [-\tau,\tau]}\phi^t(\psi^{\epsilon}(L)).\end{align*} $$
Note that in the proof of Theorem 7.1, the bottom of
$\Lambda $
was bounded by the surgery annulus
$A_0$
, while here,
$A_0$
is in between
$A_{L_-\epsilon }$
and
$A_{L_\epsilon }$
. This is because we want to construct a transverse annulus C centred on L. We can achieve this performing a bi-contact surgery along
$A_0$
splitting the difference as in [Reference Foulon and Hasselblatt14]. This method consists of introducing half of the deformation of the bi-contact structure on the side of the annulus where
$w<0$
and half on the side of the annulus where
$w>0$
. Since by Theorem 3.1
$\xi _+$
does not depend on the s-coordinate, the characteristic foliation induced by
$\xi _+$
on
$\partial \Lambda $
does not depend on the s-coordinate. By Theorem 6.6, for every
$p\in \mathbb {Z}$
, there is a neighbourhood
$\Lambda _p$
and a new pair of contact structures
$(\tilde {\xi }_-,\tilde {\xi }_+)_q$
on the manifold
$\tilde {M}_p$
obtained by performing bi-contact
$(1,p)$
-surgery along
$A_0$
in the manifold M. Note that
$\Lambda _p$
and the pair
$(\tilde {\alpha }_-,\tilde {\alpha }_+)_p$
can be chosen to be independent of the s-coordinate. By Theorem 7.1, these flows are orbit equivalent to flows obtained by
$(1,p)$
-Goodman surgery with shear
$$ \begin{align*}G:C\rightarrow C,\;\;\;(s,w)\rightarrow (w,s+g(w)),\end{align*} $$
where we chose
$g'(w)<0$
. Since L is Legendrian for
$\ker \beta _+$
and
$\beta _+$
is also not dependent on s and
$\ker \beta _-$
contains the w-curves, we can apply the Foulon–Hasselblatt construction with surgery annulus
$$ \begin{align*}C=\bigcup_{w\in[-\epsilon,\epsilon]} K_{L_\epsilon}\end{align*} $$
and coordinates
$(s,w,t)$
, where
$(s,w)$
are defined on C and t is the parameter given by the geodesic flow (the Reeb flow of
$\beta _+$
). By Theorem 8.1, the resulting flow is contact regardless of the thickness of C. Therefore, for every
$q<0$
, the resulting flow is Anosov and contact (see Figure 13).
Figure 13 On the left, a neighbourhood of a Legendrian knot L associated to a simple closed geodesic on a geodesic flow. The curve on the left is the Dehn twist corresponding to a (negative) Legendrian-transverse surgery (note that the curve has positive slope). On the left, the characteristic foliations induced by
$\xi _+$
on
$A_{L_{\epsilon }}$
and
$A_{L_{-\epsilon }}$
. The curve on the right is the Dehn twist corresponding to the induced Goodman surgery (colour online).
9 Skewed
$\mathbb {R}$
-covered Anosov flows generated by surgery are contact
In this section, we prove a version of Conjecture 1.6 for surgeries on a closed orbit of a geodesic flow on the unit tangent bundle of an hyperbolic surface S. This result is a consequence of Theorem 8.1 and the work of Asaoka, Bonatti and Marty [Reference Asaoka, Bonatti and Marty4] that we recall in the following subsection.
9.1 Partial sections, multiplicity and linking numbers
Let M be an oriented 3-manifold equipped with a smooth flow
$\phi ^t$
. We call a subset P of M a partial section if it is the image of a smooth immersion
$\iota :\hat {P}\rightarrow M$
, where
$\hat {P}$
is a compact surface and the restriction of
$\iota $
to the interior of
$\hat {P}$
is an embedding transverse to the flow and
$\iota (\partial \hat {P})$
is a finite union of closed orbits of
$\phi ^t$
. The immersion
$\iota $
lifts to the manifold
$M_{\partial P}$
obtained by blowing up along
$\iota (\partial \hat {P})=\partial P$
. A closed orbit in
$\partial P$
is the image
$\iota (\hat {\gamma })$
of a boundary component
$\hat {\gamma }$
of
$\partial \hat {P}$
. We denote the lift of
$\iota (\hat {\gamma })$
to the blow-up with
$\gamma ^*$
and we call it the boundary component immersed in
$\gamma $
. Let M be an oriented 3-manifold equipped with an Anosov flow with orientable invariant foliations, an immersed boundary component
$\gamma ^*$
has two invariants: the multiplicity
$\mathrm {mult}(\gamma ^*)$
and the linking number
$\mathrm {link}(\gamma ^*)$
defined as follows:
Here,
$\unicode{x3bb} _{\gamma }$
is a curve homotopic to a lift of
$\gamma $
in the blow-up manifold (a parallel), while
$\mu _{\gamma }$
is a curve homotopic to a fibre of the projection
$\pi _{\partial P}:M_{\partial P}\rightarrow M$
(a meridian). If a closed orbit
$\gamma $
is the image of just one boundary components of
$\hat {P}$
, we can write
$\mathrm {mult}(\gamma ^*)=\mathrm {mult}(\gamma )$
and
$\mathrm {link}(\gamma ^*)=\mathrm {link}(\gamma )$
. We say that a boundary component
$\gamma ^*$
is positive (negative) if
$\mathrm {mult}(\gamma ^*)>0$
(
$\mathrm {mult}(\gamma ^*)<0$
). A partial section P is said to be positive (negative) when all its boundary components are positive (negative).
Theorem 9.1. (Asaoka–Bonatti–Marty [Reference Asaoka, Bonatti and Marty4])
If
$\phi ^t$
is positively skewed
$\mathbb {R}$
-covered, it does not admit a negative partial section.
In [Reference Asaoka, Bonatti and Marty4], the authors also study how the multiplicity of a boundary component of a partial section varies after a
$(1,q)$
-Goodman surgery.
Theorem 9.2. (Asaoka–Bonatti–Marty [Reference Asaoka, Bonatti and Marty4])
A
$(1,q)$
-Goodman surgery along a closed orbit in
$\partial P$
increases the multiplicity of a boundary component
$\gamma ^*$
by
$+ q\: \mathrm {link}(\gamma ^*)$
.
Example 9.3. Given a non-simple closed geodesic, we have an associated Birkhoff annulus C with self intersection. There is a process called Fried desingularization (see [Reference Fried16]) that allows us to obtain from C an immersed partial section P as defined above. The boundary components
$\gamma _1$
and
$\gamma _2$
of a partial section C associated to a (simple or non-simple) closed geodesic have both multiplicity
$1$
and
$\mathrm {link}(\gamma ^*)=p$
, where p is the number of self intersections of the geodesic. By Theorem 9.2, a
$(1,q)$
-Goodman surgery with
$q<0$
along
$\gamma _1$
changes the positivity of only one boundary component. The resulting partial section has two boundary components with opposite sign.
Example 9.4. In [Reference Marty23, §1.5.20], Marty describes a method to produce partial sections associated to a closed orbit
$\gamma $
that is the lift of non-simple closed geodesics on an orientable surface of negative curvature. These partial sections have multiple negative boundary components and the only positive boundary component is that associated to
$\gamma $
with
$\mathrm {mult}(\gamma )=1$
and
$\mathrm {link}(\gamma )>0$
.
We are now ready to prove the main result of this section. Let
$\phi ^t$
be a geodesic flow with the orientation that makes it a positive skewed
$\mathbb {R}$
-covered Anosov flow.
Theorem 9.5. Let
$\phi ^t$
be the geodesic flow on the unit tangent bundle of an oriented hyperbolic surface. Any positive skewed
$\mathbb {R}$
-covered Anosov flow obtained by a single Goodman surgery along a closed orbit
$\gamma $
that is the lift of a simple closed geodesic is orbit equivalent to a positive contact Anosov flow.
Proof. As a consequence of Theorem 8.1, it is enough to show that a
$(1,q)$
-Goodman surgery along a non-simple closed geodesic does not produce a positive skewed
$\mathbb {R}$
-covered Anosov flow if
$q<0$
. Given a closed orbit
$\gamma $
associated to a non-simple geodesic, we chose a partial section as described in Example 9.4. Since
$\mathrm {mult}(\gamma )=1$
and
$\mathrm {link}(\gamma )>0$
, by Theorem 9.2, a
$(1,q)$
-Goodman surgery along
$\gamma $
with
$q<0$
yields a negative partial section. By Theorem 9.1, the new flow cannot be positive skewed
$\mathbb {R}$
-covered.
Remark 9.6. One important ingredient in proving Theorem 9.5 is the fact that a
$(1,q)$
-Goodman surgery along a closed non-simple geodesic
$\gamma $
in a positive skewed
$\mathbb {R}$
-cover Anosov flow does not produce a positive skewed
$\mathbb {R}$
-covered Anosov flow if
$q<0$
. We now state a counterpart of this statement in the contact category.
Proposition 9.7. It is not possible to construct a contact Anosov flow performing Foulon–Hasselblatt construction if the closed geodesic is non-simple and
$q<0$
.
Sketch of the proof.
The idea is the following. To a neighbourhood N of L, Theorem 1.4 associates a positive number k (the slope of the characteristic foliation induced by
$\xi _+$
on the neighbourhood
$\Lambda $
) with the property that a
$(1,q)$
-Goodman surgery produces a contact Anosov flow if
$k+q>0$
. A different neighbourhood
$N'$
is associated to a different
$k'$
. For instance, a sequence of nested neighbourhoods
$ N'\subset N" \subset N"' $
is associate to a sequence of increasing slopes
$k"'>k">k'$
. For a fixed K, we denote with
$\overline {k}$
the largest of all the possible values of k. If the geodesic is closed and simple, there is an infinite sequence of nested neighbourhoods as in Remark 6.3 and
$\overline {k}=\infty .$
If the geodesic is non-simple, we claim that
$\overline {k}\leq 1$
. A neighbourhood N of L with largest slope
$\overline {k}$
contains the closed orbit
$\gamma $
in its boundary
$\partial N$
. As a curve in
$\partial N$
, it has the form
$r[\mu ]+s[\unicode{x3bb} ]$
, where
$\unicode{x3bb} $
is homotopic to L. Since also
$\gamma $
is homotopic to L, we have
$s=1$
. Since the geodesic is non-simple, the associated closed orbit intersects C and the slope of the characteristic foliation is finite at least in a neighbourhood of the point of self intersection. Therefore,
$\overline {k}\neq \infty $
, implying that
$\overline {k}={1}/{r}$
. Finally, since
$r\in \mathbb {Z}$
, we have
$\overline {k}\leq 1$
. This shows that there does not exist a neighbourhood N of L such that
$k+q>0$
if
$q<0$
.
Acknowledgements
The author would like to express his gratitude to Tali Pinsky for introducing him to this subject and for her support throughout the development of this project. He also would like to thank Thomas Barthelmé, Surena Hozoori, Anne Vaugon, Théo Marty and Martin Mion-Mouton for the illuminating conversations and their patience in explaining to the author various subtleties of the subject. Moreover, the author would like to thank the anonymous referee whose precious suggestions improved the quality and clarity of the paper.
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