# Surgery on Anosov flows using bi-contact geometry

Federico Salmoiraghi  Department of Mathematics
Queen’s University
Kingston, Canada<EMAIL_ADDRESS>
###### Abstract.

Using bi-contact geometry, we define a new type of Dehn surgery in an Anosov
flow with orientable weak invariant foliations. The Anosovity of the new flow
is strictly tied to contact geometry and the Reeb dynamics of the defining bi-
contact structure. This approach gives new insights into the properties of the
flows produced by Goodman surgery. To a closed orbit in a geodesic flow we
associate a negative number $\overline{q}$ and we show that a $(1,q)$-Goodman
surgery generates a contact Anosov flow for every $q>\overline{q}$. We
explicitly compute $\overline{q}$ for simple closed geodesics showing that a
$(1,q)$-Goodman surgery produces a contact Anosov flow regardless of the sign
of the twist if and only if the geodesic is simple. This extends a result of
Foulon and Hasselblatt [FH13] and it can be interpreted as the counterpart in
the contact category of results proven recently by Bonatti-Iakovoglou [BI20]
and Marty [Mar21] in the category of $\mathbb{R}$-covered Anosov flows.

This research was supported by the Israel Science Foundation (grant No.
51/4051)

##  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 [HT80] the
first example of non-algebraic transitive Anosov flow was constructed
performing surgery on a geodesic flow. Inspired by Handel and Thurston work,
Goodman [Goo83] introduced a Dehn type surgery associated to a closed orbit
and used it to produce the first examples of Anosov flows in hyperbolic three
manifolds.

In the present work we introduce a new type of Dehn surgery on an Anosov using
a supporting bi-contact structure. Mitsumatsu [Mit95] first noticed that the
generating vector field of an Anosov flow with weak invariant foliation
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 pA flow while not every
3-manifold admits an Anosov flows.

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 is plenty of Legendrian-transverse knots in
a bi-contact structure supporting an Anosov flows. Recently Hozoori [Hoz21]
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 a Legendrian-transverse knot isotopic to $\gamma$.
Additionally there are many examples of Legendrian-tranverse knots that are
not isotopic to any closed orbit. From now on we assume that $K$ is a knot
that is Legendrian for $\xi_{-}$ and transverse to $\xi_{+}$.

######  Theorem 1.

Let $K$ be a Legendrian-transverse knot in a bi-contact structure
$(\xi_{-},\xi_{+})$ defining a volume preserving Anosov flow. There is a
$(1,q)$-Dehn surgery along an annulus $A_{0}$ tangent to the flow that yields
new bi-contact structures for every $q\in\mathbb{N}$.

Theorem 1 can be extended to more general bi-contact structures. For example
it also holds near a closed orbit of a general Anosov flow with orientable
weak invariant foliations.

Throughout the rest of the paper we will refer to the construction of Theorem
1 as Legendrian-transverse surgery. For $q<0$ the smooth plane field
distribution produced by the construction of Theorem 1 is not in general a bi-
contact structure.

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 support an Anosov flow. The following result shows that the
Anosovity of the new flow is strictly tied to bi-contact geometry.

######  Theorem 2.

Let $K$ be a Legendrian-transverse knot in a bi-contact structure defining a
volume preserving Anosov flow. If a Legendrian-transverse surgery yields a bi-
contact structure it yields an Anosov flow.

The construction of Theorem 1 naturally fits in the framework set up by
Hozoori in [Hoz20] where is given a symplectic characterization of the bi-
contact structures that define Anosov flows. Using this characterization we
give a proof of Theorem 2 that relies only on the properties of the Reeb
dynamics of an underlying bi-contact structure.

The construction of Theorem 1 gives new insight into the properties of the
flows produced by Goodman surgery in virtue of the following equivalence.

######  Theorem 3.

There are choices such that a flow generated by a Legendrian-transverse
$(1,q)$-surgery on a volume preserving Anosov flow $\phi^{t}$ is orbit
equivalent to a flow constructed by $(1,q)$-Goodman surgery on $\phi^{t}$.

Goodman’s operation consists in cutting an Anosov flow along a transverse
annulus $C$ and gluing back the two sides of the cut adding a $(1,q)$-Dehn
twist. 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 $q>0$. Note that
Theorem 1 is consistent with Goodman’s condition.

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. Theorem 2 and Theorem 3 allows 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,
centered on a Legendrian-transverse knot $K$. To a neighborhood $N$ of $C$ we
associate a positive real number $k$ depending just on the behaviour of the
contact structure $\xi_{+}$ on $\partial N$. This number can be interpreted as
the slope of the characteristic foliation induced by $\xi_{+}$ on $\partial
N$. For a volume preserving Anosov flow we have the following.

######  Theorem 4.

Let $k>0$ be the slope of the characteristic foliation induced by $\xi_{+}$ on
$\partial N$. For every $q>-k$ a $(1,q)$-Goodman surgery along $C$ produces an
Anosov flow.

Let $C_{q}$ be a transverse annulus with positive preferred direction such
that a $(1,q)$-Goodman surgery for a fixed $q<0$ produces an Anosov flow.
Foulon, Hasselblatt and Vaugon show in [FHV21] that for a sufficiently thin
annulus $C\subset C_{q}$ centered on $K$, a $(1,q)$-Goodman surgery along $C$
produces a flow that is not Anosov. We show that the situation radically
changes after a local perturbation of the flow.

######  Corollary 5.

Suppose that $C_{q}$ is an annulus with positive preferred direction
transverse to a volume preserving Anosov flow $\phi^{t}$. Suppose also that a
$(1,q)$-Goodman surgery along $C_{q}$ produces an Anosov flow for a fixed
$q<0$. For an arbitrarily thin annulus $C\subset C_{q}$ there is a flow
$\phi_{t}^{\prime}$ orbit equivalent to $\phi_{t}$ such that a $(1,q)$-Goodman
surgery along $C$ generates an Anosov flow. The flow-lines of
$\phi^{\prime}_{t}$ coincide with the ones of $\phi_{t}$ outside an
arbitrarily small flow-box neighborhood $N$ of $C_{q}$.

Foulon, Hasselblatt and Vaugon furthermore show in [FHV21] that in any Anosov
flow for a fixed embedded transverse annulus $C$ with positive preferred
direction there is a large enough $q<0$ such that a $(1,q)$-Goodman surgery
produces a non Anosov flow. Let $\phi^{t}$ be a volume preserving Anosov flow
with weak orientable invariant foliations. If 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 $\gamma$ we have
the following.

######  Corollary 6.

There is a nested sequence $\\{C_{r}\\}_{r\in\mathbb{Z}^{-}}$ of transverse
annuli with positive preferred direction bounded by $K$ on one side and such
that a $(1,r)$-Goodman surgery along $C_{r}$ yields an Anosov flow. Moreover
we have

$C_{-\infty}=\overline{\bigcup_{r\in\mathbb{Z}^{-}}C_{r}}.$

In particular, given an annulus $C_{r}$ of the sequence, we obtain the annulus
$C_{r-1}$ extending $C_{r}$ towards the closed orbit $\gamma$.

We use Theorem 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 [Ham93], [Liv04] and [Fan09]) including exponential decay of
correlations. In the context of $3$-manifold, Barbot has shown that every
contact Anosov flow is skewed $\mathbb{R}$-covered. The relation between these
two classes of flows is expected to be stronger.

######  Conjecture 1.1 (Barbot–Barthelmé [Bar05], [Bar05]).

If $\phi^{t}$ is a (positively) skewed $\mathbb{R}$-covered Anosov flow, then
it is orbit equivalent to a contact Anosov flows.111Marty [Mar23] 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 [FH13] 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 therefore the set of negative
surgery coefficients that yields 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 6 applies. As a consequence we
have the following.

######  Theorem 7.

Chose a closed orbit $\gamma$ in the geodesic flow on the unit tangent bundle
of a hyperbolic surface $S$. For every $q\in\mathbb{Z}$ 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 $S$.

Theorem 7 represents the counterpart in the category of contact Anosov flows
of results recently proven by Bonatti-Iakovoglou [BI20] (Therem 1) and Marty
[Mar21] (Theorem J) in the category of $\mathbb{R}$-covered Anosov flows and
it is consistent with Conjecture 1.1.

Further directions 1. Let $\phi^{t}$ be the geodesic flow with the orientation
that makes it a positive skewed $\mathbb{R}$-covered Anosov flow. For a non-
simple closed geodesic Marty [Mar21] shows that there is a $\overline{p}<0$
such that every $(1,p)$-Goodman surgery with $p<\overline{p}$ generates an
Anosov flow that is not positively skewed $\mathbb{R}$-covered. We have the
following interpretation of Marty’s result: if the geodesic is not simple
there is not a quasi-transverse embedded annulus containing $L$ and Corollary
6 does not apply.

A consequence of Theorem 4 is a counterpart of Marty result in the contact
category. Let $(\xi_{-},\xi_{+})$ be the natural bi-contact structure
supporting the geodesic flow. To a neighborhood $N$ of $L$ we associated a
positive real number $k$ with the property that a $(1,q)$-Goodman surgery
produces a contact Anosov flow if $q>-k$. A different neighborhood
$N^{\prime}$ is associated to a different $k^{\prime}$. For instance a
sequence of nested neighborhoods $N^{\prime}\subset N^{\prime\prime}\subset
N^{\prime\prime\prime}$ is associate to a sequence of increasing slopes
$k^{\prime\prime\prime}>k^{\prime\prime}>k^{\prime}$. For a fixed $L$ we
denote with $\overline{k}$ the upper bound of all the possible values of $k$.
If the knot $L$ is associated to a simple closed geodesic there is a quasi-
transverse embedded annulus centered on $L$ and $\overline{k}=\infty$. If a
knot $L$ is induced to a non simple closed geodesic
$\overline{k}\in\mathbb{R}^{+}$ we have the following.

######  Corollary 8.

Let $\overline{q}<0$ be the integral part of $-\overline{k}$. For every
$q>\overline{q}$ a $(1,q)$-Goodman surgery along a closed orbit that is the
lift of a non simple closed geodesic produces a flow that is orbit equivalent
to a contact Anosov flow.

It is natural to ask if there is a relation between the integer $\overline{p}$
described in Marty’s work and the integer $\overline{q}$ defined above
(eventually for sequences of surgeries). Shannon shows (Theorem A in [Sha20])
that if the flow is transitive the orbit equivalence class of the Anosov flows
generated by Goodman surgery is determined by the surgery coefficient and the
closed orbit $\gamma$. As a consequence, in the case
$\overline{p}=\overline{q}$ for a closed geodesic, every (positively) skewed
$\mathbb{R}$-covered Anosov flow obtained by surgery on that geodesic is orbit
equivalent to a contact Anosov flow.

Further directions 2. The work of Hozoori [Hoz20] and more recently the work
of Cieliebiak, Lazarev, Massoni and Moreno [KC22] sets up the framework to
define Floer invariants for projectively Anosov flows and Anosov flows. It is
natural to ask if the bi-contact surgery introduced in this paper can be used
to understand the effect of surgery on these novel Floer invariants.

###  1.1. Acknowledgment

The author would like to express his gratitude to Tali Pinsky for introducing
him to the 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 subtlety of the
subject.

##  2\. Anosov flows, projectively Anosov flows and bi-contact structures

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

######  Definition 2.1.

Let $M$ be a closed manifold and $\phi^{t}:M\rightarrow M$ a $C^{1}$ flow on
$M$. The flow $\phi^{t}$ is called Anosov if there is a splitting of the
tangent biundle $TM=E^{uu}\oplus E^{ss}\oplus\langle X\rangle$ preserved by
$D\phi^{t}$ and positive constants $A$ and $B$ such that

$\lVert D\phi^{t}(v^{u})\rVert\geq Ae^{Bt}\lVert
v^{u}\rVert\;\;\;\;\;\;\text{for any}\;v^{u}\in E^{uu}$ $\lVert
D\phi^{t}(v^{s})\rVert\leq Ae^{-Bt}\lVert v^{s}\rVert\;\;\;\;\text{for
any}\;v^{s}\in E^{ss}.$

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
$\mathcal{F}^{u}$.

$\mathcal{F}^{u}$ at 36 188 $\mathcal{F}^{s}$ at 280 10 $X$ at 3 85

Figure 1. The tangent bundle in a neighborhood of a flow line of an Anosov
flow. In red and blue is depicted the plane fields defining the bi-contact
structure. The vertical plane is the the leaf of the unstable weak foliation
$\mathcal{F}^{u}$.

Mitsumatsu [Mit95] first noticed that an Anosov flow with orientable weak
invariant foliations is tangent to the intersection of two transverse contact
structures (see also Eliashberg-Thurston [ET98]). 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 as the following example
shows.

######  Example 2.2.

We construct a bi-contact structure on $T^{3}$ using a recipe introduced by
Mitsumatsu in [Mit95] and [Mit02]. Consider the contact forms defined on
$T^{2}\times I$

$\alpha_{n}=cos(2n\pi z)dx-sin(2n\pi z)dy,$ $\alpha_{-m}=cos(2m\pi
z)dx+sin(2m\pi z)dy.$

They are not transverse to each other on the tori defined by $\\{z=0\\}$,
$\\{z=\frac{1}{4}\\}$, $\\{z=\frac{1}{2}\\}$, $\\{z=\frac{3}{4}\\}$. If we
introduce a perturbation $\epsilon(z)\>dz$ such that $\epsilon(z)$ is a
function that doesn’t vanish on the tori, the contact structures
$\alpha_{+}=\alpha_{n}+\epsilon(z)\>dz$ and $\alpha_{-}=\alpha_{-m}$ are
transverse.

As Plante and Thurston showed in [PT72], in a manifold that admits an Anosov
flow the fundamental group grows exponentially and this of course not the case
for $T^{3}$.

######  Definition 2.3 (Mitsumatsu [Mit95]).

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

$\frac{\lVert D\phi^{t}(v^{u})\rVert}{\lVert D\phi^{t}(v^{s})\rVert}\geq
Ae^{Bt}\frac{\lVert v^{u}\rVert}{\lVert v^{s}\rVert}\;\;\;\;\;\;\;\text{for
any}\;v^{u}\in\mathcal{E}^{u}\;\text{and}\;v^{s}\in\mathcal{E}^{s}$

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 [ET98]). However, when they are smooth they also are uniquely
integrable. We call these flows regular projectively Anosov (see [Nod04],
[NT02] and [Asa04] for a complete classification). The following result gives
a geometric characterization of the family of projectively Anosov flows

######  Proposition 2.4 (Mitsumatsu [Mit95]).

Let $X$ be a $C^{1}$ vector field on $M$. Then $X$ is projectively Anosov if
and only if it is defined by a bi-contact structure.

######  Remark 2.5.

We assume the reader familiar with contact geometry. We refer to [Hoz20] and
[Hoz21] for the basic notions in bi-contact geometry and projectively Anosov
flows and for a precise discussion on the regularity of the plane fields,
bundles and foliations involved. The bi-contact structures in the present work
are defined by $C^{1}$ 1-forms. In particular a bi-contact structure
$(\xi_{-},\xi_{+})$ is a pair of transverse plane fields
$(\xi_{-}=\ker\alpha_{-},\xi_{+}=\ker\alpha_{+})$ for a pair of $C^{1}$
1-forms $(\alpha_{-},\alpha_{+})$ such that $\alpha_{+}\wedge d\alpha_{+}>0$
is a positive $C^{0}$ volume form. Note that the Reeb vector fields
$R_{\alpha_{-}}$ and $R_{\alpha_{+}}$ are a priori just $C^{0}$. A
$C^{\infty}$ vector field $X$ defines a contact Anosov flow if $X$ is Anosov
and is the Reeb vector field of a $C^{\infty}$ contact form $\beta$.

###  2.1. Reeb dynamics of a bi-contact structure defining an Anosov flows

We now present a characterization of Anosov flows that uses the Reeb dynamics
of the underling bi-contact structure.

######  Definition 2.6 (Hozoori [Hoz20]).

Consider a bi-contact structure $(\xi_{-},\xi_{+})$. A vector field is
dynamically positive (negative) if at every $p\in M$ it lies in the interior
of the first or third (second or forth) region in Figure 10.

######  Theorem 2.7 (Hozoori [Hoz20]).

Let $\phi^{t}$ be a projectively Anosov flow on $M$. The following are
equivalent.

  1. (1)

The flow $\phi^{t}$ is Anosov,

  2. (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. (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.

A straightforward consequence of Theorem 2.7 is the following useful
sufficient condition of Anosovity.

######  Corollary 2.8.

Let $X$ be a projectively Anosov flow and let
$(\ker\alpha_{-}=\xi_{-},\ker\alpha_{+}=\xi_{+})$ be a bi-contact structure
defining $X$ such that the Reeb vector field $R_{\alpha_{-}}$ is contained in
$\xi_{+}$. Then $X$ is an Anosov vector field.

Hozoori [Hoz21] shows that it is always possible to ensure the condition of
Corollary 2.8 in a neighborhood of a closed orbit.

######  Theorem 2.9 (Hozoori [Hoz21]).

Let $\phi^{t}$ be a $C^{1}$-Hölder Anosov flow with orientable weak invariant
foliations and $\gamma$ 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 neighborhood $N$ of
$\gamma$, the Reeb vector field $R_{\alpha_{-}}$ of $\alpha_{+}$ satisfies the
condition $\alpha_{+}(R_{\alpha_{-}})=0$.

We will systematically use the following consequence of Corollary 2.8.

######  Lemma 2.10.

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
$\xi_{+}$ in a open set $N\subset M$ and such that $R_{\alpha_{-}}$ is
dynamically positive in $M$. An isotopy of $\xi_{+}$ with support in $N$ along
the flowlines of $R_{\alpha_{-}}$ defines a path of orbit equivalent Anosov
flows.

###### Proof.

Assume that $R_{\alpha_{-}}$ is positive in $M$. An isotopy of
$\xi_{+}=\ker\alpha_{+}$ as above defines a path of bi-contact structures
$(\xi_{-},\ker(\alpha_{+})_{t})_{t\in[0,1]}$ such that
$R_{\alpha_{-}}\subset(\xi_{+})_{t}=\ker(\alpha_{+})_{t}$ in $N$. The
statement follows from Corollary 2.8 and $C^{1}$-structural stability of
Anosov flows. ∎

A volume preserving Anosov flow is an Anosov flow preserving a continuous
volume form. It is known that if the flow is $C^{k}$ such a volume form is
automatically $C^{k}$ (see [Hoz21] more references).

Hozoori shows [Hoz21] the following characterization of bi-contact structures
defining volume preserving Anosov flows.

######  Theorem 2.11 (Hozoori [Hoz21]).

Let $\phi^{t}$ be a projectively Anosov flow on $M$. $\phi^{t}$ is a volume
preserving Anosov flow if and only if there are pairs of positive and negative
contact forms defining $\phi^{t}$ such that the Reeb vector field of
$\alpha_{-}\subset\ker\alpha_{+}$ and $\alpha_{+}\subset\ker\alpha_{-}$.

##  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 [Ano63]). This
object is particularly important in our context since it is the ambient flow
where several surgery constructions (i.e. the ones introduced by Handel-
Thurston and Foulon-Hasselblatt) are defined.

###  3.1. Geometric structures on UTS

$L$ at 260 152 UT$\gamma$ at 448 155 $\gamma^{-}$ at 268 38 $\gamma^{+}$ at
268 275 $X$ at 185 174

Figure 2. On the top right, a Birkhoff torus associated to simple closed
geodesic $\gamma$. On the right side, the contact structures described in 3.2.
In grey, the contact structure $\eta_{+}=\ker\beta_{+}$ preserved by the
geodesic flow. The closed orbits $\gamma^{+}$ and $\mathfrak{\gamma^{-}}$ are
Legendrian knots for the bi-contact structure $(\xi_{-},\xi_{+})$ that
generates the geodesic flow. The contact structure $\xi_{+}=\ker\alpha_{+}$ is
represented in red while the $\xi_{-}=\ker\alpha_{-}$, is represented in blue.
The special knot $L$ is represented by the dashed line.

The geodesic flow on the unit tangent bundle of a hyperbolic surface $S$
carries some remarkable geometric structure that can be interpreted in the
context of bi-contact geometry. We follow the discussion of [FHV21]. Using the
identification of 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 $X$ that generates the flow,
the periodic vector field $V$ pointing in the fiber direction, and the vector
field defined by $H:=[V,X]$. This frame satisfies the following relations

(1) $[V,X]=H,\;\;\;\;[H,X]=V,\;\;\;\;[H,V]=X.$

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 also that (1) implies the existence of three 1-forms
$\alpha_{-},\alpha_{+}$ and $\beta_{+}$ defining mutually transverse contact
structures (see Figure 2). They are defined by the following

$\beta_{+}(V)=0=\beta_{+}(H),\;\;\;\;\alpha_{+}(X)=0=\alpha_{+}(V),\;\;\;\;\alpha_{-}(X)=0=\alpha_{-}(H),$
(2)
$\beta_{+}(X)=1,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\alpha_{+}(H)=1,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\alpha_{-}(V)=1.$

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 in the unit
tangent bundle of a hyperbolic surface is contact. Finally we have the
relations $\alpha_{+}=d\beta_{+}(V,\cdot)$ and
$\alpha_{-}=-d\beta_{+}(H,\cdot)$.

######  Theorem 3.1 (Anosov).

The geodesic flow on the unit tangent bundle of a hyperbolic surface is a
(contact) Anosov flow.

Anosovity has been proven by Anosov [Ano63] (see also [FHV21]). We remark that
Anosovity can be derived easily from Hozoori’s criterion. In fact, by the
discussion above there is a bi-contact structure
$(\ker\alpha_{-},\ker\alpha_{+})$ such that the Reeb vector fields of the
contact forms are dynamically positive and negative.

$K$ at 208 135 $\gamma^{+}$ at 210 12 $\gamma^{+}$ at 210 175 $\gamma^{-}$ at
210 95

$\gamma^{+}$ at 465 12 $\gamma^{+}$ at 465 175 $\gamma^{+}$ at 465 95 $K$ at
463 133

Figure 3. On the left, the foliation on the Birkhoff torus induced by the
stable and unstable weak foliations. On the right, the characteristic
foliations induced by the bi-contact structure
$(\ker\alpha_{+},\ker\alpha_{-})$.

###  3.2. Structures on a Birkhoff torus and Legendrian knots

Let S$\;=\mathbb{H}^{2}/\Gamma$ be a hyperbolic surface and choose a closed
geodesic $\gamma$. The lift 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 is defined by the angle
$\theta=\frac{\pi}{2}$ on each fiber along $\gamma$ and has the following
remarkable properties

  1. (1)

$L$ is a Legendrian knot for the contact structure $\eta_{+}$ preserved by the
flow,

  2. (2)

$L$ is transverse to the weak stable and unstable foliations,

  3. (3)

$L$ is a Legendrian-transverse knot with respect to the bi-contact structure
$(\ker\alpha_{-},\ker\alpha_{+})$.

##  4\. Contact Anosov flows and Foulon–Hasselblatt contact surgery

Given a $C^{k}$ contact form $\alpha$, its Reeb vector field $R_{\alpha}$ and
a Legendrian knot $L$ for $\xi=\ker\alpha$ Foulon and Hasselblatt describe in
[FH13] a family of contact surgeries of the Dehn type that generates a new
$C^{k}$ contact form $\tilde{\alpha}$ in the new manifold $\tilde{M}$ and a
$C^{k-1}$ vector field $R_{\tilde{\alpha}}$ such that

  1. (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 neighborhood of an annulus $C$ transverse to
the flow of $R_{\alpha}$.

  2. (2)

$R_{\tilde{\alpha}}$ is the Reeb vector field of $\tilde{\alpha}$. Moreover
$R_{\tilde{\alpha}}$ is collinear to $R_{\alpha}$ in $M\setminus C$ and is a
reparametrization of $R_{\alpha}$ in a neighborhood of $C$.

In other words 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

$K$ at 475 83

Figure 4. The surgery annulus in Foulon-Hasselblatt construction.

Given a 3-manifold with a contact structure $\xi=\ker\alpha$ and a Legendrian
knot $L$ there is a coordinate system

$(s,t,w)\in N=(-\delta,\delta)\times S^{1}\times(-\epsilon,\epsilon),$

where the parameters $(s,w)$ are defined on the surgery annulus
$C=\\{0\\}\times S^{1}\times(-\epsilon,\epsilon)$ (see Figure 4). 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 $R_{\alpha}=\frac{\partial}{\partial
t}$, 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 $\gamma=dt+w\>ds$.

The surgery can be thought 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 the one described by coordinates $(s+g(w),w)$ on the
other side. Here $g:[-\epsilon,\epsilon]\rightarrow S^{1}$ is a non decreasing
function such that $g(-\epsilon)=0$ and $g(\epsilon)=2\pi q$ and satisfying a
number of additional requirements. The transition map $G:C\rightarrow
C,\;(s,w)\rightarrow(s+g(w),w)$ is often called shear. Since

$G_{*}\frac{\partial}{\partial t}=\frac{\partial}{\partial t}$

the shear define 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.

Let us now 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

$G^{*}(\alpha)^{+}=(\alpha)^{-}+wg^{\prime}(w)\>dw$

therefore the shear defines a smooth vector field from $R_{\alpha}$ but not 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 $C^{1}$ and $dh$ is the
differential of the following function

$h(t,w)=\lambda(t)\int_{-\epsilon}^{w}xg^{\prime}(x)\;dx.$

Here $\lambda:\mathbb{R}\rightarrow[0,1]$ is $C^{1}$ bump function with
support in some interval $(0,\delta)$ with $\delta>0$. With this choice we
have $G^{*}(\alpha-dh)^{+}=(\alpha)^{-}$. Therefore $\tilde{\alpha}$ is a well
defined 1-form of class $C^{1}$ in $\tilde{M}$. Note that the deformation
depends on the shear and it is not immediately clear if after deformation the
plane field distribution $\ker\tilde{\alpha}$ still defines a contact
structure. The authors show that if we choose $0<\epsilon<\frac{\delta}{2\pi
q}$ this is in fact the case. Moreover the Reeb vector field of
$\tilde{\alpha}$ takes the form

$R_{\tilde{\alpha}}=\frac{R_{\alpha}}{1+dh(R_{\alpha})}.$

###  4.2. Application to contact Anosov flows

Foulon–Hasselblatt construction is purely contact geometric, it does not
require the Reeb vector field of $\gamma$ to be Anosov and it can be performed
in a neighborhood of any Legendrian knot.

If applied to a geodesic flow on the unit tangent bundle of an hyperbolic
surface we have the following.

######  Theorem 4.1 (Foulon–Hasselblatt–Vaugon [FHV21]).

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
$\theta=\frac{\pi}{2}$ on each fiber along $\gamma$. Let $2\epsilon$ be the
width of the surgery annulus.

  1. (1)

The $(1,q)$-Dehn surgery along $L$ defined in Subsection 4.1 does produce an
Anosov flow if $q>0$ for every $\epsilon>0$.

  2. (2)

It does not produce an Anosov if $-q/\epsilon$ is large enough, i.e., if
either $q<0$ is fixed and $\epsilon$ is small enough or if $\epsilon>0$ is
fixed and $q<0$ with $\left|q\right|$ 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 [FH13]).

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 chose $\epsilon>0$ small enough
to ensure that the surgery annulus $C_{\epsilon}$ is embedded in $M$.

##  5\. Legendrian-transverse surgery on Anosov flows

We introduce a new type of Dehn surgery on an Anosov flow with weak orientable
invariant foliations. Under this assumption the flow is defined by a bi-
contact structure $(\xi_{-},\xi_{+})$. Our construction is defined in a
neighborhood $N$ 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 $\alpha_{-}$ is contained in $\xi_{+}=\ker\alpha_{+}$. As
Hozoori shows in [Hoz21] (see 1), if the flow is Anosov and volume preserving
there is a bi-contact structure
$(\ker\alpha_{-}=\xi_{-},\ker\alpha_{+}=\xi_{+})$ such that
$R_{\alpha_{-}}\in\ker\alpha_{+}=\xi_{+}$ and
$R_{\alpha_{+}}\in\ker\alpha_{-}=\xi_{-}$, therefore the above condition is
verified for every Legendrian-transverse knot $K$. On the other hand, if the
flow is just Anosov Hozoori [Hoz21] shows that the condition
$R_{\alpha_{-}}\in\ker\alpha_{+}=\xi_{+}$ can be always achieved in a
neighborhood of a closed orbit. Moreover we have the following.

######  Lemma 5.1.

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 neighborhood
$N$ of $\gamma$ the Reeb vector field $R_{\alpha_{-}}$ belongs to
$\ker\alpha_{+}=\xi_{+}$. The knot $K=\phi^{t}(\gamma)$ is a Legendrian-
transverse knot for $0<t<1$ sufficiently small.

######  Definition 5.2.

Consider a closed orbit $\gamma$ and a pair of contact forms
$(\alpha_{-},\alpha_{+})$ such that in a neighborhood $N$ 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
$\xi_{+}=\ker\alpha_{+}$ obtained by translating $\gamma$ using the flow of
$R_{\alpha_{-}}$.

Figure 5. Legendrian-transverse push-off of a closed orbit. In blue
$R_{\alpha_{-}}$. In black, the Anosov flow.

In the following we sketch the construction of the Lgendrian-trnaverse
surgery. Let $K$ be a Legendrian-transverse knot. 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 Anosov flow $\phi^{t}$. Note
that $A_{0}$ 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
$\ker\tilde{\alpha}_{+}=\tilde{\xi}_{+}$. Therefore the new bi-contact
structure $(\ker\tilde{\alpha}_{-},\ker\tilde{\alpha}_{+})$ satisfies the
condition in Corollary 2.8 and the supported vector field $\tilde{X}$ is
Anosov.

###  5.1. Definition of the Legendrian-transverse surgery

We first construct a special coordinate system in a neighborhood $N$ of the
Legendrian-transverse knot $K$ where two supporting contact forms
$(\alpha_{-},\alpha_{+})$ take a very simple form.

We remark again that the condition on the Reeb vector fields in the following
lemma can be easily achieved in a volume preserving Anosov flow or near a
closed orbit of a general Anosov flow.

Figure 6. The surgery annulus $A_{0}$ spanned by the flow. The dashed line is
the Legendrian-transverse knot $K$. In blue $R_{\alpha_{-}}$. In black, the
Anosov flow.

######  Lemma 5.3.

Suppose $M$ is a 3-manifold endowed with a projectively Anososv flow $X$
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 pair of contact forms $(\alpha_{-},\alpha_{+})$
defining $(\xi_{-},\xi_{+})$ such that the Reeb vector field of $\alpha_{-}$
is contained in $\ker\alpha_{+}=\xi_{+}$ in a neighborhood $N$ of $K$. We can
chose $N=A_{0}\times(-\epsilon,\epsilon)$ equipped with coordinates $(s,v,w)$
such that a pair of contact forms $(\alpha_{-},\alpha_{+})$ supporting
$(\xi_{-},\xi_{+})$ can be written in $N$ as

$\alpha_{-}=dw+v\>ds$
$\alpha_{+}=ds-b(s,v,w)\>dv\;\;\;with\;\;\alpha_{+}=ds\;\;on\;\;A_{0}.$

Here $s$ and $v$ are coordinates on $A_{0}$ such that $s\in S^{1}$ is the
parameter for $K$ and $v\in(-\delta,\delta)$ with $\delta\in\mathbb{R}^{+}$
while $w\in(-\epsilon,\epsilon)$ is a transverse parameter to $A_{0}$.

###### 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 set the
tangent annulus

$A_{0}=\bigcup_{t\in[-\tau,\tau]}\phi^{t}(K)$

spanned by the flow $\phi^{t}$ for $t\in[-\tau,\tau]$ and some $\tau>0$. Let
$\alpha_{-}$ be a 1-form such that $\ker\alpha_{-}=\xi_{-}$. Since $K$ is
Legendrian we can choose $\tau$ such that the Reeb vector field
$R_{\alpha_{-}}$ is transverse to $A_{0}$.

Let $\phi^{w}$ the flow of the Reeb vector field $R_{\alpha_{-}}$ of
$\alpha_{-}$. There is an auxiliary coordinate system $(s,u,w)$ in a
neighborhood of $A$ such that

$\alpha_{-}=dw+a(s,u)\>ds.$

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

$(s,u)\rightarrow(s,a(s,u))=:(s,v)$

is non singular and therefore we have (see [FH13])

$\alpha_{-}=dw+v\>ds.$

Since the flow lines of $R_{\alpha_{-}}$ are Legendrian for $\ker\alpha_{+}$
by hypothesis and the $v$-curves are transverse to $\ker\alpha_{+}$ in a
neighborhood $N$ of $K$, we can write

$\alpha_{+}=ds-b(s,v,w)\>dv.$

The $v$-curves do not describe the same codimension two foliation described by
the flowlines of $X$. However, the foliations coincide on the surgery annulus.
Therefore $b(s,v,0)=0$ and $\alpha_{+}=ds$.

∎

######  Remark 5.4.

In the hypothesis of Lemma 5.3 the contact form
$\alpha_{+}^{0}=ds-b(s,v,w)\>dv$ can be isotoped along the flow lines of
$R_{\alpha_{-}}$ to contact form independent on the $s$-coordinate.

$\alpha_{+}^{1}=ds-b(v,w)\>dv.$

If the initial flow $\phi^{t}$ defined by $(\alpha_{-},\alpha_{+}^{0})$ is
Anosov, there is a path of Anosov flows connecting $\phi^{t}$ to the flow
$\psi^{t}$ defined by $(\alpha_{-},\alpha_{+}^{1})$. In particular $\phi^{t}$
and $\psi^{t}$ are orbit equivalent (see Lemma 2.10).

Suppose that we are in the hypothesis of Lemma 5.3 and Remark 5.4.

####  5.1.1. Deformations and gluings

Our procedure can be thought 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 $A_{0}^{-}$ where $w\leq 0$ to points $(s+f(v),v)$ on
$A_{0}^{+}$, the side of the cut that bounds the region where $w\geq 0$. We
define

$f:[-\delta,\delta]\rightarrow S^{1},\;\;\;v\rightarrow f(v),$

where $q\in\mathbb{Z}$ and $f:\mathbb{R}\rightarrow[0,2\pi]$ non increasing in
$(-\delta,\delta)$, smooth, and such that $f((-\infty,-\delta))=0$,
$f([\delta,\infty))=-2q\pi$.

######  Remark 5.5.

Note that the function $f$ used here to define the shear is non-increasing
while the function used in Section 4 to define Foulon-Hasselblatt construction
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 a 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 Subsection 4.1 the shear

$F:A_{0}\rightarrow A_{0},\;(s,v)\rightarrow(s+f(v),v)$

defines a smooth new $3$-manifold $\tilde{M}$ but it does not preserve the
contact form $\alpha_{-}$ since on the surgery annulus $A_{0}$

$F^{*}(\alpha_{-})^{+}=dw+v\>d(s+f(v))=(\alpha_{-})^{-}+vf^{\prime}(v)\>ds.$

An analogous calculation shows that $\alpha_{+}$ is not preserved either since

$F^{*}(\alpha_{+})^{+}=F^{*}ds=d(s+f(v))=ds+f^{\prime}(v)\>dv=(\alpha_{+})^{-}+f^{\prime}(v)\>dv.$

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.

The deformation that we apply to $\alpha_{-}$ is the same one used by Foulon
and Hasselblatt to define their contact surgery.

We set $(\tilde{\alpha}_{-})^{+}=(\alpha_{-})^{+}-dh$ where

$h(v,w)=\lambda_{1}(w)\int_{-\delta}^{v}xf^{\prime}(x)\;dx.$

Here $\lambda_{1}:\mathbb{R}:\rightarrow[0,1]$ is a smooth bump function
supported in $(0,\epsilon)$ such that $\lambda_{1}(0)=1$ and
$\left|\lambda_{1}^{\prime}\right|<\frac{1}{\epsilon}+\iota$ with $\iota$
arbitrarily small. Note that in a neighborhood of $A$ we have [FHV21]

$\tilde{\alpha}_{-}=dw+v\>ds-f^{\prime}(v)v\>dv,$

therefore

$F^{*}(\alpha_{-}-dh)^{+}=(\alpha_{-})^{-}$

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
$\tilde{\alpha}_{+}=\alpha_{+}-\sigma$ and consider the 1-form

(3) $\sigma=\lambda_{2}(w)\>f^{\prime}(v)\>dv$

where $\lambda_{2}:\mathbb{R}\rightarrow[0,1]$ is a smooth bump function, with
support in $(0,\epsilon)$. More explicitly

$\tilde{\alpha}_{+}=ds-(b(v,w)+\lambda_{2}(w)\>f^{\prime}(v))\;dv.$

Note that since $f^{\prime}(v)$ has support in $A_{0}$ the 1-form $\sigma$ is
smooth with support in the interior of $N=A_{0}\times[-\epsilon,\epsilon]$.

We check that on the surgery annulus $A_{0}$ we have
$F^{*}(\alpha_{+}-\sigma)^{+}=(\alpha_{+})^{-}$. A direct computation shows
that

$F^{*}(\tilde{\alpha}_{+})^{+}=F^{*}(ds-b\;dv-\lambda_{2}(w)\>f^{\prime}(v)\;dv)=ds+f^{\prime}(v)\;dv-b\;dv-f^{\prime}(v)\;dv=(\alpha_{+})^{-}$

therefore $\tilde{\alpha}_{+}$ is a well defined $1$-form on $\tilde{M}.$

The plane field distribution $\tilde{\xi}=ker(\tilde{\alpha})$ and the plane
field distribution $\tilde{\eta}=ker(\tilde{\beta})$ are transverse.

######  Theorem 5.6.

Let $(\xi_{-}=\ker\alpha_{-},\xi_{+}=\ker\alpha_{+})$ a bi-contact structure
such that in a neighborhood of a knot $K$ that is simultaneously Legendrian
for $\ker\alpha_{-}$ and transverse for $\ker\alpha_{+}$ we have
$R_{\alpha_{-}}\subset\ker\alpha_{+}$. If $q>0$ a Legendrian-transverse
$(1,q)$-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 [FH13] interpreting the surgery
annulus $A$ as transverse annulus to the Reeb vector field of $\alpha$. We use
a slightly different proof from the one 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 [FH13] we have

$\tilde{\alpha}\wedge d\tilde{\alpha}=(-1+\frac{\partial h}{\partial w})\>dV,$

with $dV=ds\wedge dv\wedge dw$. Therefore the condition $\left|\frac{\partial
h}{\partial w}\right|<1$ ensures that $\tilde{\alpha}$ is contact.

$\left|\frac{\partial h}{\partial
w}\right|=\left|\lambda^{\prime}_{1}\int_{-\delta}^{v}xf^{\prime}(x)\;dx\right|,$

integrating by parts and since $f(-\delta)=0$ for every $q\in\mathbb{Z}$ we
get

$\left|\frac{\partial h}{\partial
w}\right|=\left|\lambda^{\prime}_{1}(vf(v)+\delta
f(-\delta)-\int_{-\delta}^{v}f(x)\;dx)\right|<\left|\lambda^{\prime}_{1}(vf(v)-\int_{-\delta}^{v}f(x)\;dx)\right|$

since $\left|\lambda^{\prime}_{1}\right|<\frac{1}{\epsilon}+\iota$ with
$\iota$ arbitrarily small, $v<\delta$, $\left|f(v)\right|<2\pi\left|q\right|$
and $\int_{-\delta}^{v}f(x)\;dx)<2\pi\delta$, we have

$\left|\frac{\partial h}{\partial w}\right|<\frac{1}{\epsilon}\delta
2\pi(\left|q\right|+1)<1.$

This shows that the 1-form $\tilde{\alpha}_{-}$ is contact if

(4) $0<\delta<\frac{\epsilon}{2\pi(\left|q\right|+1)}.$

######  Remark 5.7.

Note that condition (4) 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 [ET98]

(5) $\frac{\partial b}{\partial w}+\lambda_{2}^{\prime}(w)\>f^{\prime}(v)>0.$

Since $\frac{\partial b}{\partial w}>0$ by the contact condition and
$\lambda_{2}^{\prime}(w)\leq 0$ by the definition of the bump function
$\lambda_{2}$, the inequality is always satisfied if $f^{\prime}(v)\leq 0$.
Therefore, the family of $(1,q)$-Dehn surgeries produces bi-contact structures
for $q>0$.

∎

The proof of Theorem 5.6 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 [HT80], [FHV21] and [Bar05]) where there are choices of
the direction of the shear that weaken 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 Legendrian-transverse surgery and
contact geometry.

######  Theorem 9.

Let $M$ be a $3$-manifold endowed with an Anosov flow. 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.
∎

$A_{0}$ at 170 45 $A_{\epsilon}$ at 170 185 $K$ at 80 69 $C_{out}$ at 50 140
$C_{in}$ at 140 43 $\phi^{w}(K)$ at 107 174

Figure 7. On the left: the red contact structure rotates along the $w$ axis.
As usual, the Legendrian-transverse knot is denoted by a dashed line. On the
right: the annuli $A_{w}$ spanned by the flow.

##  6\. New Anosov flows for negative surgery coefficients

We now give a sufficient condition for a Legendrian-transverse $(1,q)$-surgery
to yield an Anosov flow when the operation is performed in the direction that
weaken the hyperbolicity. The condition is purely contact geometric and
involves the behaviour of the the bi-contact structure along the boundary of a
neighborhood $N$ of a Legendrian-transverse knot $K$. We remark that the
condition in the hypothesis of the next theorem can be always achieved in a
volume preserving Anosov flow or near a closed orbit of a general Anosov flow.

######  Theorem 6.1.

Let $K$ be a Legendrian-transverse knot $K$ in a bi-contact structure
$(\xi_{-}=\ker\alpha_{-},\xi_{+}=\ker\alpha_{+})$ defining an Anosov flow,
such that $R_{\alpha_{-}}\subset\xi_{+}$. There is a neighborhood $N$ of $K$
such that if $k$ is the slope of the characteristic foliation induced by
$\xi_{+}$ on $\partial N$ a $(1,q)$-surgery with $q>-k$ yields an Anosov flow.

###### Proof.

Consider a flow-box neighborhood $N$ of $K$ bounded by the surgery annulus
$A=A_{0}$, the transverse annuli $C_{in}$ and $C_{out}$ constructed by flowing
the boundary components of $A_{0}$ using the flow of $R_{\alpha_{-}}$, and on
the top, by an annulus

$A_{\epsilon}=\bigcup_{t\in[-\tau,\tau]}\phi^{t}(\phi^{\epsilon}(K)).$

Without loss of generality we consider values of $\epsilon$ such that the
characteristic foliation induced on $\partial N$ is composed by closed curves
homotopic to a curve $r\mu+sK$ where $\mu$ is a meridian of $\partial N$. We
define the slope of the characteristic foliation $k=\frac{s}{r}>0$. Let $q$ be
the integral part of $-k$ if $-k$ is not an integer and set $q=-k+1$ if $-k$
is an integer. Since $q>-k$ there is a family of curves

$f_{s}(v):[-\delta,\delta]\rightarrow S^{1}$

and a shear

$F:A_{0}\rightarrow A_{0},\;\;\;(s,v)\rightarrow(s+f_{s}(v),v).$

After gluing, we obtain a new neighborhood $N$ with coordinates $(s,v,w)$ such
that the projection along the $w$-curves of the characteristic foliation
induced on $A_{\epsilon}$ by $\xi_{+}$ positively intesects the characteristic
foliation induced by $\xi_{+}$ on $A_{0}$. Therefore we can extend the
$C^{1}$-contact structure $\xi_{+}$ defined on $M\setminus N$ to a
$C^{1}$-contact structure $\tilde{\xi}_{+}$ on $M$ whose planes positively
rotate along the $w$-curves. Since the $w$-curves are flow lines of
$R_{\alpha_{-}}$, Anosovity follows by Hozoori’s criterion.

∎

######  Corollary 6.2.

In a neighborhood of a closed orbit $\gamma$ of an Anosov flow with orientable
weak invariant foliations there is a fixed tangent annulus $A_{0}$ such that
for every $q\in\mathbb{Z}$ a Legendrian-transverse $(1,q)$-surgery yields an
Anosov flow for every $q\in\mathbb{Z}$.

###### Proof.

We have to prove the statement just for $q<0$. By Theorem 2.9 there is a bi-
contact structure $(\ker\alpha_{-},\ker\alpha_{+})$ such that
$R_{\alpha_{-}}\subset\ker\alpha_{+}$ in a neighborhood $N$ of $\gamma$.
Therefore there is a knot $K$ that is a Legendrian-transverse push-off of
$\gamma$. Let $\epsilon_{\gamma}$ be the value of the parameter $w$ defined by
the flow of $R_{\alpha}$ along $\gamma$. Construct a flow-box neighborhood as
in Theorem 6.1. For $\epsilon$ approaching the value $\epsilon_{\gamma}$ we
have $k\rightarrow\infty$ for a fixed $A_{0}$. Therefore for every $k>0$ there
is a flow-box neigborhood $N_{k}$ as in Theorem 6.1 bounded below by the
annulus $A_{0}$. ∎

######  Remark 6.3.

In a neighborhood of a closed orbit $\gamma$ there is a sequence of nested
neighborhoods of the Legendrian-transverse knot $N_{-1}\subset
N_{-2}\subset\cdots\subset N_{q}\subset\cdots\subset N_{-\infty}$ where

$N_{q}=\bigcup_{w\in[0,\epsilon_{q}]}A_{w},$

where

$A_{w}=\bigcup_{t\in[-\tau,\tau]}\phi^{t}(\phi^{w}(K))$

where $\epsilon_{q}$ is a sufficiently close to $\epsilon_{\gamma}$. Moreover

$N_{-\infty}=\overline{\bigcup_{w\in[0,\epsilon_{\gamma})}A_{w}}.$

All the neighborhood are bounded on one side by $A_{0}$. For
$q\in\mathbb{Z}^{-}$ the neighborhood $N_{q}$ is a flow-box.

##  7\. Relations with Goodman surgery

In this chapter we study the relationship between the Legendrian-transverse
surgery and the surgery introduced by Goodman [Goo83].

At a first glance the two operations look really different. First of all the
surgery annulus used in the Legendrian-transverse surgery is tangent to the
generating vector field $X$. In Goodman’s construction we use an annulus
transverse to the flow. Secondly, in the Legendrian-transverse surgery we
introduce a deformation of the bi-contact structure induces a $3$-dimensional
perturbation of the flow in a neighborhood of the surgery annulus. In Goodman
construction the flow lines of the new flow coincide to the old ones outside
the (transverse) surgery annulus. Despite these differences there are always
choices such that the flows generated by the Legendrian-transverse surgery and
the ones generated by Goodman’s construction are orbit equivalent.

$A_{0}$ at 230 25 $A_{w}$ at 230 80 $v$ at 200 61 $w$ at 173 110 $C_{out}$ at
330 50 $C_{in}$ at 133 70 $s$ at 330 10

Figure 8. The flow-box $N$. A flow constructed by $(1,q)$-Legendrian-
transverse surgery along $A_{0}$ is orbit equivalent to a flow geberated by
$(1,q)$-Goodman surgery along $C_{out}$. A negative Dehn twist on $A_{0}$ (in
blue) corresponds to a positive Dehn twist on $C_{out}$ (in green).

######  Theorem 7.1.

Suppose that $K$ is a Legendrian-transverse knot in a bi-contact structure
$(\xi_{-},\xi_{+})$ defined by two contact forms $(\alpha_{-},\alpha_{+})$
such that in a neighborhood $N$ of $K$ the Reeb vector field $R_{\alpha_{-}}$
is contained in $\xi_{+}$. A flow generated by a Legendrian-transverse
$(1,q)$-surgery along the tangent annulus $A_{0}$ is orbit equivalent to a
flow generated by a $(1,q)$-Goodman surgery along a transverse annulus $C$.

###### Proof.

Let $N$ be the flow-box bounded by the tangent surgery annulus $A_{0}$, the
annulus

$A_{\epsilon}=\bigcup_{t\in[-\tau,\tau]}\phi^{t}(\phi^{\epsilon}(K))$

where $\epsilon>0$ and the transverse annuli $C_{in}$ and $C_{out}$
constructed by flowing the boundary components of $A_{0}$ using the flow of
$R_{\alpha_{-}}$.

Set $C=C_{out}$ and let $\mathcal{F}$ be the foliation obtained by projecting
on $C=C_{out}$ the $w$-curves (the flow-lines of $R_{\alpha_{-}}$) defined on
$C_{in}$ using the flow of $X$. Consider a Legendrian-transverse
$(1,q)$-surgery along the tangent annulus $A_{0}$ with shear

$F:A_{0}\rightarrow A_{0},\;\;\;(v,s)\rightarrow(v,s+f(v))$

with $f(-\delta)=0$ and $f(\delta)=-2\pi q$. Let $\tilde{X}$ be the new vector
field. On $A_{0}$ the vector field $X$ induces a foliation directed by the
vector $\frac{\partial}{\partial v}$ while $\tilde{X}$ induces a foliation
directed by $f^{\prime}(v)\>\frac{\partial}{\partial
s}+\frac{\partial}{\partial v}$. The new foliation on $A_{0}$ is obtained from
the one induced by $X$ by adding a full $(1,q)$-Dehn twist to the curves of
the original foliation. Consequently the curves of the foliation
$\mathcal{\tilde{F}}$ of $C_{out}$ constructed by projecting the $w$-curves
defined on $C_{in}$ using the flow of $\tilde{X}$ can be obtained from the
curves of $\mathcal{F}$ by adding a full $(1,-q)$-Dehn twist. An orbit
equivalent flow is constructed by cutting $M$ along $C=C_{out}$ and gluing
back $C^{-}$ to $C^{+}$ using some shear $G$ defined by

$G:C^{-}\rightarrow C^{+},\;\;\;(w,s)\rightarrow(w,s+g_{s}(w))$

with $g_{s}(0)=0$ and $g_{s}(\epsilon)=2\pi q$. In general
$g_{s}:[0,\epsilon]\rightarrow S^{1}$ depends on the coordinate $s$. ∎

######  Corollary 7.2.

Suppose that $K$ is a Legendrian-transverse knot in a bi-contact structure
$(\xi_{-}=\ker\alpha_{-},\xi_{+}=\ker\alpha_{+})$ defined by two contact forms
such that $R_{\alpha_{-}}\subset\xi_{+}$ in a neighborhood $N$ of $K$. There
is a transverse embedded annulus $C$ such that a $(1,q)$-Goodman surgery on
$C$ with $q>-k$ yields an Anosov flow.

######  Corollary 7.3.

In a neighborhood of a closed orbit $\gamma$ there is a sequence of nested
annuli $C_{-1}\subset C_{-2}\subset\cdots\subset C_{q}\subset\cdots\subset
C_{-\infty}$ bounded on one side by the the Legendrian-transverse knot $K$.
Every $C_{q}$ is the is the core of $N_{q}$ and it is transverse to the flow
if $q\in\mathbb{Z}$. The annulus $C_{-\infty}$ quasi-transverse and it is
bounded by $K$ on one side and by the closed orbit $\gamma$ on the other side.

Figure 9. If a $(1,q)$-Goodman surgery along an annulus $C_{q}$ produces an
Anosov flow for a fixed $q<0$, a $(1,q-1)$-Goodman produces an Anosov flow on
an annulus $C_{q-1}$ obtaine expanding $C_{q}$ towards the closed orbit.

##  8\. Surgery along simple closed geodesics in a geodesic flow

Let $X$ be the geodesic flow on the unit tangent bundle of a hyperbolic
surface $S$ and let $\beta_{+}$ 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
$\theta=\frac{\pi}{2}$ on each fiber along $\gamma$. $L$ is a Legendrian knot
for $\ker\gamma$. Let $C_{\epsilon}=S^{1}\times[-\epsilon,\epsilon]\subset$
UT$\gamma$ be a transverse annulus with $\epsilon$ be small enough to ensure
that $C_{\epsilon}$ is not self intersecting. As shown by Foulon and
Hasselblatt for any $q>0$ their construction ensure 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 shows that Goodman surgery produces a 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.

To this hand we prove the following lemma showing that while Foulon and
Hasselblatt construction requires an arbitarily thin annulus in order 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 field of preserving a positive
contact form $\beta_{+}$. For $q<0$ Foulon and Hasselblatt surgery along a
transverse embedded annulus produces a Reeb vector field 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

$G:C_{out}\rightarrow C_{out},\;\;\;(s,w)\rightarrow(w,s+g(w))$

along $C_{out}$ preserves a positive contact form

$\tilde{\beta}_{+}=dt+w\>ds-dh$

with

$h(t,w)=\lambda(t)\int_{-\epsilon}^{w}xg^{\prime}(x)\;dx,$

hence

$\tilde{\beta}_{+}=dt+w\>ds-\lambda^{\prime}(t)\int_{-\epsilon}^{w}xg^{\prime}(x)\;dx\;dt-\lambda(t)wg^{\prime}(w)\;dw.$

They indeed show that the contact condition

(6) $\tilde{\beta}_{+}\wedge d\tilde{\beta}_{+}=(1-\frac{\partial h}{\partial
t})\>dV>0$

is satisfied regardless of the sign of the Dehn twist if the surgery annulus
is sufficiently thin. Note that if

(7) $\frac{\partial h}{\partial
t}=\lambda^{\prime}(t)\int_{-\epsilon}^{w}xg^{\prime}(x)\;dx\;dt\leq 0$

the contact condition (6) is satisfied independently of the thickness of the
surgery annulus. If $g^{\prime}(w)<0$, we have
$\int_{-\epsilon}^{w}xg^{\prime}(x)\;dx\geq 0$ for $w\in(-\epsilon,\epsilon)$
and since since $\lambda^{\prime}(t)\leq 0$ for $t>0$ inequality (7) is
satisfied. This corresponds to $(1,q)$-Foulon-Hasselblatt surgery with $q<0$.

∎

######  Lemma 8.2.

In a neighborhood $\Lambda$ of a Birkhoff torus 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:

$\alpha_{-}=dw+v\>ds$ $\alpha_{+}=e^{\frac{1}{2}v^{2}}(\cos w\>ds-\sin w\>dv)$
$\beta_{+}=e^{\frac{1}{2}v^{2}}(-\sin w\>ds-\cos w\>dv).$

###### Proof.

An elementary calculation gives the following expressions for the Reeb vector
fields in $N$:

$V=R_{\alpha_{-}}=\frac{\partial}{\partial w}$
$H=R_{\alpha_{+}}=e^{-\frac{1}{2}v^{2}}(\cos w\>\frac{\partial}{\partial
s}-\sin w\>\frac{\partial}{\partial v}-v\cos w\>\frac{\partial}{\partial w})$

$X=R_{\beta_{+}}=e^{-\frac{1}{2}v^{2}}(-\sin w\>\frac{\partial}{\partial
s}-\cos w\>\frac{\partial}{\partial v}+v\sin w\>\frac{\partial}{\partial w}).$

The vector field $V$ is periodic in $\Lambda$ and

$\alpha_{+}(V)=\alpha_{-}(H)=\beta_{+}(V)=\beta_{+}(H)=0.$

∎

$L$ at 55 174 $C_{out}$ at 550 100 $A_{0}$ at 50 144 $w$ at 520 150
$A_{\epsilon}$ at 520 194 $A_{-\epsilon}$ at 525 40

Figure 10. On the left, a neighborhood of a Legendrian knot $L$ associated to
a simple closed geodesic on a geodesic flow. The blue curve is the Dehn twist
corrsponding to the Legendrian-transverse surgery. On the left: the
characteristic foliations induced by $\xi_{+}$ on $A_{\epsilon}$ and
$A_{-\epsilon}$. We perform a Legendrian transverse surgery on $A_{0}$
"splitting the difference" as in [FH13]. This method consists in 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 $w>0$. The green curve
on $C_{out}$ is the Dehn twist corresonding to the induced Goodman surgery.

######  Theorem 8.3.

Let $C\subset$ UT$\gamma$ be a quasi-transverse annulus associated to a simple
closed geodesic. For every $q<0$ there is an embedded transverse annulus
$C_{q}\subset C$ centered in $L$ such that a $(1,q)$-Goodman surgery produces
a contact Anosov flow.

###### Proof.

We first show that, for every $q\in\mathbb{Z}$, there are choices such that
the Legendrian-transverse surgery along a tangent annulus $A_{0}$ containing
$L$ produces an Anosov flow. Then we show that this flows are orbit equivalent
to flows constructed by $(1,q)$-Foulon-Hasselblatt surgery.

Consider the $(s,v,w)$-coordinate system in a neighborhood $\Lambda$ of
UT$\gamma$ as in Lemma 8.2. Let $N$ be a flow-box neighborhood of $L$ bounded
by the tangent surgery annulus $A=A_{0}$, the annulus

$A_{\epsilon}=\bigcup_{t\in[-\tau,\tau]}\phi^{t}(\phi^{\epsilon}(K))$

where $\epsilon>0$ and the transverse annuli $C_{in}$ and $C_{out}$
constructed by flowing the boundary components of $A_{0}$ using the flow of
$R_{\alpha_{-}}$. Since by Lemma 8.2 $\xi_{+}$ does not depend on the
$s$-coordinate, the characteristic foliation induced by $\xi_{+}$ on $\partial
N$ does not depend on the $s$-coordinate. By Remark 6.3 for every
$q\in\mathbb{Z}$ there is a neighborhood $N_{q}$ and a new pair of contact
structures $(\tilde{\xi}_{-},\tilde{\xi}_{+})_{q}$ on the new manifold
$\tilde{M}_{q}$ defining an Anosov flow. Note that $N_{q}$ and the pair
$(\tilde{\alpha}_{-},\tilde{\alpha}_{+})_{q}$ can be chosen to be independent
on the $s$-coordinate. By Theorem 7.1 these flows are orbit equivalent to
flows obtained by $(1,q)$-Goodman surgery with shear

$G:C_{out}\rightarrow C_{out},\;\;\;(s,w)\rightarrow(w,s+g(w))$

where we chose $g^{\prime}(w)<0$. Since $L$ is Legendrian for $\ker\beta_{+}$
and $\beta_{+}$ is also not dependent on $s$ and rotates along the
$w$-coordinate, we can apply Foulon-Hasselblatt construction with surgery
annulus $C_{out}$ and coordinates $(s,w,t)$ where $(s,w)$ are defined on
$C_{out}$ and $t$ is the parameter given by the geodesic flow (the Reeb vector
field of $\beta_{+}$). By Lemma 8.1 the resulting flow is contact regardless
of the thickness of $C_{out}$. Therefore for every $q<0$ the resulting flow is
Anosov and contact.

∎

## References

  * [Ano63] D. V. Anosov.  Ergodic properties of geodesic flows on closed Riemannian manifolds of negative curvature.  Dokl. Akad. Nauk SSSR, 151:1250–1252, 1963.
  * [Ano67] D. V. Anosov.  Geodesic flows on closed Riemannian manifolds of negative curvature.  Trudy Mat. Inst. Steklov., 90:209, 1967.
  * [Asa04] Masayuki Asaoka.  A classification of three dimensional regular projectively Anosov flows.  Proc. Japan Acad. Ser. A Math. Sci., 80(10):194–197 (2005), 2004\.
  * [Bar05] Thierry Barbot.  De l’hyperbolique au globalment hyperbolique.  Habilitation à diriger des recherches, Université Claude Bernard de Lyon, 2005.
  * [Bar17] Thomas Barthelmé.  Anosov flows in dimension 3 preliminary verison.  Preprint, 2017.
  * [BF13] Thierry Barbot and Sérgio R. Fenley.  Pseudo-Anosov flows in toroidal manifolds.  Geom. Topol., 17(4):1877–1954, 2013.
  * [BF14] Thomas Barthelmé and Sergio R. Fenley.  Knot theory of $\mathbb{R}$-covered Anosov flows: homotopy versus isotopy of closed orbits.  J. Topol., 7(3):677–696, 2014.
  * [BF15] Thierry Barbot and Sérgio R. Fenley.  Classification and rigidity of totally periodic pseudo-Anosov flows in graph manifolds.  Ergodic Theory Dynam. Systems, 35(6):1681–1722, 2015.
  * [BF17] Thomas Barthelmé and Sergio R. Fenley.  Counting periodic orbits of Anosov flows in free homotopy classes.  Comment. Math. Helv., 92(4):641–714, 2017.
  * [BI20] Christian Bonatti and Ioannis Iakovoglou.  Anosov flows on 3-manifolds: the surgery and the foliations.  Preprint, 2020.
  * [ET98] Yakov M. Eliashberg and William P. Thurston.  Confoliations, volume 13 of University Lecture Series.  American Mathematical Society, Providence, RI, 1998.
  * [Fan09] Yong Fang.  Thermodynamic invariants of Anosov flows and rigidity.  Discrete Contin. Dyn. Syst., 24(4):1185–1204, 2009.
  * [FH13] Patrick Foulon and Boris Hasselblatt.  Contact Anosov flows on hyperbolic 3-manifolds.  Geom. Topol., 17(2):1225–1252, 2013.
  * [FHV21] Patrick Foulon, Boris Hasselblatt, and Anne Vaugon.  Orbit growth of contact structures after surgery.  Annales Henri Lebesgue, UFR de Mathématiques - IRMAR, In press. hal-03132118, 2021.
  * [Goo83] Sue Goodman.  Dehn surgery on Anosov flows.  In Geometric dynamics (Rio de Janeiro, 1981), volume 1007 of Lecture Notes in Math., pages 300–307. Springer, Berlin, 1983.
  * [Ham93] Ursula Hamenstädt.  Regularity of time-preserving conjugacies for contact Anosov flows with $C^{1}$-Anosov splitting.  Ergodic Theory Dynam. Systems, 13(1):65–72, 1993.
  * [Hoz20] Surena Hozoori.  Symplectic geometry of Anosov flows in dimension 3 and bi-contact topology.  Preprint, 2020.
  * [Hoz21] Surena Hozoori.  On Anosovity, divergence and bi-contact geometry.  Preprint, 2021.
  * [HT80] Michael Handel and William P. Thurston.  Anosov flows on new three manifolds.  Invent. Math., 59(2):95–103, 1980.
  * [KC22] Thomas Massoni August Moreno Kai Cielieback, Oleg Lazarev.  Floer theory of Anosov flows in dimension three.  2022\.
  * [Liv04] Carlangelo Liverani.  On contact Anosov flows.  Ann. of Math. (2), 159(3):1275–1312, 2004.
  * [Mar21] T. Marty.  Anosov flows and Birkhoff sections.  2021\.
  * [Mar23] T. Marty.  Skewed Anosov flows are orbit equivalent to Reeb Anosov flows in dimension 3.  2023\.
  * [Mit95] Yoshihiko Mitsumatsu.  Anosov flows and non-Stein symplectic manifolds.  Ann. Inst. Fourier (Grenoble), 45(5):1407–1421, 1995.
  * [Mit02] Yoshihiko Mitsumatsu.  Foliations and contact structures on 3-manifolds.  In Foliations: geometry and dynamics (Warsaw, 2000), pages 75–125. World Sci. Publ., River Edge, NJ, 2002.
  * [Nod04] Takeo Noda.  Regular projectively Anosov flows with compact leaves.  Ann. Inst. Fourier (Grenoble), 54(2):481–497, 2004.
  * [NT02] Takeo Noda and Takashi Tsuboi.  Regular projectively Anosov flows without compact leaves.  In Foliations: geometry and dynamics (Warsaw, 2000), pages 403–419. World Sci. Publ., River Edge, NJ, 2002.
  * [Pla71] Joseph Franklin Plante.  ANOSOV FLOWS.  ProQuest LLC, Ann Arbor, MI, 1971.  Thesis (Ph.D.)–University of California, Berkeley.
  * [PT72] J. F. Plante and W. P. Thurston.  Anosov flows and the fundamental group.  Topology, 11:147–150, 1972.
  * [Sha20] Mario Shannon.  Flot d’Anosov et hyperbolicité partiel en 3-varietes.  2020\.