# Anosov flows and Liouville pairs in dimension three

Thomas Massoni Department of Mathematics, Princeton University, USA. Email
address<EMAIL_ADDRESS>
###### Abstract

Building upon the work of Mitsumatsu and Hozoori, we establish a complete
homotopy correspondence between three-dimensional Anosov flows and certain
pairs of contact forms that we call _Anosov Liouville pairs_. We show a
similar correspondence between projectively Anosov flows and bi-contact
structures, extending the work of Mitsumatsu and Eliashberg–Thurston. As a
consequence, every Anosov flow on a closed oriented three-manifold $M$ gives
rise to a Liouville structure on $[-1,1]\times M$ which is well-defined up to
homotopy, and which only depends on the homotopy class of the Anosov flow. Our
results also provide a new perspective on the classification problem of Anosov
flows in dimension three.

###### Contents

  1. 1 Introduction
  2. 2 Anosov Liouville pairs
    1. 2.1 Preliminary definitions
    2. 2.2 Elementary properties of Anosov Liouville pairs
  3. 3 From Anosov flows to Anosov Liouville pairs and back
    1. 3.1 Anosov and projectively Anosov flows
    2. 3.2 From Anosov flows to Anosov Liouville pairs
    3. 3.3 From Anosov Liouville pairs to Anosov flows
    4. 3.4 Volume preserving Anosov flows
  4. 4 Spaces of Anosov Liouville pairs and bi-contact structures
    1. 4.1 Contractibility of fibers
    2. 4.2 Homotopy equivalences
    3. 4.3 The kernel map
  5. 5 Linear Liouville pairs
  6. A Smoothing lemmas
  7. B Almost volume preserving Anosov flows

##  1 Introduction

Throughout this article, $M$ denotes a closed, oriented, smooth manifold of
dimension three. Anosov and projectively Anosov flows on $M$ will always be
assumed to be _oriented_ , i.e., their stable and unstable foliations are
oriented. This can always be achieved by passing to a suitable double cover of
$M$. For simplicity, we will only consider smooth (i.e.,
$\mathcal{C}^{\infty}$) flows, as we are primarily interested in smooth
contact and symplectic structures. Our main results hold for (projectively)
Anosov flows generated by $\mathcal{C}^{1}$ vector fields with minor changes.
Moreover, the structural stability of $\mathcal{C}^{1}$ Anosov vector fields
[Rob75] ensures that any Anosov flow generated by a $\mathcal{C}^{1}$ vector
field is topologically equivalent to a smooth Anosov flow, and these two flows
are dynamically identical. The definitions and basic properties of Anosov and
projectively Anosov flows are recalled in Section 3.1.

The notion of Anosov flow, originally introduced by Anosov [Ano63, Ano67] as a
generalization of the geodesic flow on hyperbolic manifold, plays a central
role in the theory of smooth dynamical systems. The interplay between the
dynamical and topological properties of Anosov flows is particularly rich and
striking in dimension three. We refer to the nice survey [Bar17] for many
relevant results and references, and to the book [FH19] for a more complete
exposition. Eliashberg and Thurston [ET98], and independently Mitsumatsu
[Mit95], introduced the more general concept of a _conformally/projectively
Anosov flow_ on three-manifolds, and established a correspondence between such
flows and _bi-contact structures_ , i.e., transverse pairs of contact
structures with opposite orientations. Recently, Hozoori [Hoz22a] extended
this correspondence to Anosov flows, and showed that (oriented) Anosov flows
can be completely characterized in terms of bi-contact structures admitting a
pair of contact forms satisfying a natural symplectic condition. More
precisely, Hozoori showed the following

######  Theorem ([Hoz22a], Theorem 1.1.).

Let $\Phi$ be a non-singular flow on a closed oriented $3$-manifold $M$,
generated by a vector field $X$. $\Phi$ is Anosov if and only if there exist
transverse contact structures $\xi_{-}$ and $\xi_{+}$, negative and positive,
respectively, and contact forms $\alpha_{-}$ and $\alpha_{+}$ for $\xi_{-}$
and $\xi_{+}$, respectively, such that the $1$-forms

$(1-t)\alpha_{-}+(1+t)\alpha_{+}\qquad
and\qquad-(1-t)\alpha_{-}+(1+t)\alpha_{+}$

are positively oriented Liouville forms on $[-1,1]_{t}\times M$.

Recall that a Liouville form on a smooth manifold with boundary is a $1$-form
$\lambda$ such that $\omega=d\lambda$ is symplectic, i.e., non-degenerate, and
the Liouville vector field $Z$ defined by $\omega(Z,\cdot\,)=\lambda$ is
outward-pointing along the boundary. In particular, an Anosov flow on a
$3$-manifold $M$ (under some suitable orientability assumptions recalled in
Definition 3.1) gives rise to a Liouville structure on $[-1,1]\times M$ which
is _not Weinstein_ , since the latter manifold has a non-trivial third
homology group and disconnected boundary. It is natural to ask the following

######  Questions.

How do the Liouville structures constructed by Hozoori depend on the
underlying Anosov flow? More precisely,

  1. (1)

For a given Anosov flow $\Phi$, is the space of pairs of contact forms
$(\alpha_{-},\alpha_{+})$ as in the previous Theorem path-connected?

  2. (2)

Does a path of Anosov flows induce a path of Liouville structures on
$[-1,1]\times M$?

  3. (3)

Does every bi-contact structure $(\xi_{-},\xi_{+})$ supporting an Anosov flow
admit a pair of contact forms $(\alpha_{-},\alpha_{+})$ as in the previous
Theorem?

Here, we say that a bi-contact structure $(\xi_{-},\xi_{+})$ supports a non-
singular flow generated by a vector field $X$ if $X\in\xi_{-}\cap\xi_{+}$ (in
the more precise Definition 2.1, we also add a condition on the orientations
of $\xi_{\pm}$).

In the present article, we give a complete answer to these questions and
upgrade Hozoori’s correspondence to a _homotopy equivalence_ between the space
of Anosov flows on $M$, and a space of suitable pairs of contact forms on $M$.
To that extent, we will consider a _different_ condition on the pair
$(\alpha_{-},\alpha_{+})$ than the one in Hozoori’s theorem, and we first show

######  Theorem 1.

Let $\Phi$ be a non-singular flow on a closed oriented $3$-manifold $M$,
generated by a vector field $X$. $\Phi$ is Anosov if and only if there exists
a pair of contact forms $(\alpha_{-},\alpha_{+})$ on $M$ such that
$X\in\ker\alpha_{-}\cap\ker\alpha_{+}$, and the $1$-forms

$e^{-s}\alpha_{-}+e^{s}\alpha_{+}\qquad
and\qquad-e^{-s}\alpha_{-}+e^{s}\alpha_{+}$

are positively oriented Liouville forms on $\mathbb{R}_{s}\times M$.

In the terminology of [MNW13, Definition 1], we say that a pair of contact
forms $(\alpha_{-},\alpha_{+})$ on a manifold $M$ is a Liouville pair if the
$1$-form

$e^{-s}\alpha_{-}+e^{s}\alpha_{+}$

is a positively oriented Liouville form on $\mathbb{R}_{s}\times M$.

######  Warning.

At first glance, Theorem 1 seems almost identical to Hozoori’s theorem.
However, we warn the reader that the condition on $(\alpha_{-},\alpha_{+})$
that we consider is different than Hozoori’s one. Indeed, there exist pairs of
contact forms $(\alpha_{-},\alpha_{+})$ which are Liouville pairs as defined
above, but such that

$(1-t)\alpha_{-}+(1+t)\alpha_{+}$

is not a Liouville form on $[-1,1]_{t}\times M$; see Lemma 5.2. It turns out
that our condition enjoys some nice symmetries which make it much easier to
work with.

Theorem 1 motivates the following

######  Definition 1.1.

An Anosov Liouville pair (AL pair for short) on an oriented $3$-manifold $M$
is a pair of contact forms $(\alpha_{-},\alpha_{+})$ such that both
$(\alpha_{-},\alpha_{+})$ and $(-\alpha_{-},\alpha_{+})$ are Liouville pairs.
We denote by
$\mathcal{AL}:=\mathcal{AL}(M)\subset\Omega^{1}(M)\times\Omega^{1}(M)$ the
space of Anosov Liouville pairs on $M$.

Notice that we do not assume that $\xi_{\pm}:=\ker\alpha_{\pm}$ are
transverse, since this is implied by the Liouville conditions; see Proposition
2.1. By Theorem 1, the intersection $\xi_{-}\cap\xi_{+}$ is spanned by an
Anosov vector field. A positive time reparametrization of an Anosov flow
remains Anosov, and we denote by $\mathcal{AF}:=\mathcal{AF}(M)$ the space of
smooth oriented Anosov flows on $M$ up to positive time reparametrization.
Alternatively, $\mathcal{AF}$ can be viewed as the space of smooth unit Anosov
vector fields on $M$ for an arbitrary Riemannian metric on $M$, or the space
of smooth $1$-dimensional oriented foliations spanned by Anosov vector fields
on $M$, together with some extra orientation data. Hence, there is a natural
continuous _intersection map_

$\begin{array}[]{rccc}\mathcal{I}:&\mathcal{AL}&\longrightarrow&\mathcal{AF}\\\
&(\alpha_{-},\alpha_{+})&\longmapsto&\ker\alpha_{-}\cap\ker\alpha_{+}\end{array}$

which sends an AL pair to the $1$-dimensional (oriented) distribution obtained
by intersecting the underlying contact structures. Here, we endow the spaces
$\mathcal{AL}$ and $\mathcal{AF}$ with the $\mathcal{C}^{\infty}$ topology.
Denoting by $\mathcal{BC}$ the space of smooth bi-contact structures on $M$
and by $\mathbb{P}\mathcal{AF}$ the space of smooth projectively Anosov flows
on $M$ up to positive time reparametrization, we have a similar intersection
map

$\begin{array}[]{rccc}\mathbb{P}\mathcal{I}:&\mathcal{BC}&\longrightarrow&\mathbb{P}\mathcal{AF}\\\
&(\xi_{-},\xi_{+})&\longmapsto&\xi_{-}\cap\xi_{+}\end{array}$

as well as a _kernel map_

$\begin{array}[]{rccc}\underline{\mathrm{ker}}:&\mathcal{AL}&\longrightarrow&\mathcal{BC}\\\
&(\alpha_{-},\alpha_{+})&\longmapsto&(\ker\alpha_{-},\ker\alpha_{+})\end{array}$

The main results of this paper, answering the Questions (1), (2) and (3)
above, can be summarized as follows.

######  Theorem 2.

The maps in the commutative diagram

${\mathcal{AL}}$${\mathcal{BC}}$${\mathcal{AF}}$${\mathbb{P}\mathcal{AF}}$$\scriptstyle{\underline{\ker}}$$\scriptstyle{\mathcal{I}}$$\scriptstyle{\mathbb{P}\mathcal{I}}$

satisfy the following properties.

  * •

$\mathcal{I}$ and $\mathbb{P}\mathcal{I}$ are acyclic Serre fibrations
(Theorem 4.4 and Theorem 4.5).

  * •

$\underline{\ker}$ is an acyclic Serre fibration onto its image (Theorem 4.6).

  * •

The inclusion
$\underline{\mathrm{ker}}(\mathcal{AL})\subset\mathbb{P}\mathcal{I}^{-1}\big{(}\mathcal{AF}\big{)}$
is strict in general (Theorem 3.1), but it is a homotopy equivalence (Theorem
4.7).

Recall that an acyclic Serre fibration is a Serre fibration which is also a
weak homotopy equivalence, or equivalently, whose fibers are weakly
contractible. All the topological spaces under consideration have the homotopy
type of a CW complex (see the beginning of Section 4), so these acyclic Serre
fibrations are homotopy equivalences by the Whitehead theorem. Unpacking the
notations,

  * •

$\underline{\mathrm{ker}}(\mathcal{AL})$ is the space of bi-contact structures
$(\xi_{-},\xi_{+})$ admitting contact forms $\alpha_{-},\alpha_{+}$ such that
$(\alpha_{-},\alpha_{+})$ is an AL pair,

  * •

$\mathbb{P}\mathcal{I}^{-1}\big{(}\mathcal{AF}\big{)}$ is the space of bi-
contact structures supporting an Anosov flow.

We emphasize that the top row in the diagram of Theorem 2 _only involves
concepts from contact and symplectic geometry._ This enables us to identify
projectively Anosov flows with bi-contact structures, and Anosov flows with
bi-contact structures satisfying a quantitative constraint, coming from the
existence of a suitable pair of contact forms. Moreover, the space of AL pairs
for a fixed underlying bi-contact structure is (weakly) contractible if non-
empty. Hence, AL pairs can be thought of as _auxiliary data attached to bi-
contact structures_.

Our results can be summarized by the following slogan:

The topological properties of the spaces $\mathcal{AF}$,
$\mathbb{P}\mathcal{AF}$ and the inclusion
$\mathcal{AF}\subset\mathbb{P}\mathcal{AF}$ can be translated into topological
properties of the spaces $\mathcal{AL}$, $\mathcal{BC}$, and the map
$\underline{\mathrm{ker}}:\mathcal{AL}\rightarrow\mathcal{BC}$, and vice
versa.

One important missing piece in this correspondence between Anosov dynamics and
contact topology is the mirror notion of _topological_ or _orbit equivalence_
of flows in the contact world.

######  Definition 1.2.

Two Anosov flows $\Phi=\\{\phi^{t}\\}$ and $\Psi=\\{\psi^{t}\\}$ on $M$ are
topologically equivalent, or orbit equivalent, if there exist a homeomorphism
$h:M\rightarrow M$ and a continuous map $\tau:\mathbb{R}\times
M\rightarrow\mathbb{R}$ such that $\tau(t,x)\geq 0$ for $t\geq 0$, and

$\psi^{\tau(t,x)}=h\circ\phi^{t}\circ h^{-1}(x)$

for every $t\in\mathbb{R}$ and $x\in M$.

In other words, the topological equivalence $h$ sends the oriented
trajectories of $\phi$ onto the oriented trajectories of $\psi$, but does not
necessarily preserves the parametrization. The structural stability of Anosov
flows with smooth dependence on parameters [LMM86, Theorem A.1.] implies that
two smooth Anosov flows which are homotopic through smooth Anosov flows are
topologically equivalent through a topological equivalence which is isotopic
to the identity. We do not know if the converse is true.

######  Question 1.

If two (smooth) Anosov flows on $M$ are topologically equivalent (via a
topological equivalence which is merely continuous), what can be said about
the spaces of AL pairs supporting them? How to characterize topological
equivalence in terms of AL pairs?

It is not clear to us how the (hyper)tight contact structure $\xi_{\pm}$
associated with a Anosov flow behave under topological equivalence. Solving
these questions could have a significant impact in the understanding of Anosov
flows from the perspective of contact geometry. For instance, it is known by
the work of Colin, Giroux and Honda [CGH09] that an atoroidal $3$-manifold
carries finitely many isotopy classes of _tight_ contact structures. This
naturally raises the following

######  Question 2.

On a closed atoroidal $3$-manifold, are there finitely many Anosov flows up to
homotopy? Up to topological equivalence?

A fundamental problem in $3$-dimensional Anosov dynamics is to determine if a
closed $3$-manifold always carry _finitely many_ topological equivalence
classes of Anosov flows [Bar17, Question 1.]. Although toroidal (and
irreducible) $3$-manifolds can carry infinitely many isotopy classes of tight
contact structures, all of them can be obtained from finitely many contact
structures by performing _Lutz twists_ along suitable tori; see [CGH09]. The
authors also show that there are finitely many tight contact structures for a
prescribed _Giroux torsion_ , up to isotopy and Dehn twists. We hope that this
_coarse classification_ of tight contact structures together with our homotopy
correspondence could lead to important results in the classification of Anosov
flows on $3$-manifolds. To this end, it is crucial to understand the following

######  Question 3.

Let $(\alpha_{-},\alpha_{+})$ be an AL pair on $M$. Fixing $\alpha_{+}$, what
can be said about the Anosov flow supported by an AL pair
$(\alpha^{\prime}_{-},\alpha_{+})$, where $\alpha^{\prime}_{-}$ is isotopic to
$\alpha_{-}$?

The main difficulty here is that a path $(\alpha^{t}_{-})_{t\in[0,1]}$ of
contact forms from $\alpha^{0}_{-}=\alpha_{-}$ to
$\alpha^{1}_{-}=\alpha^{\prime}_{-}$ might not induce a path of bi-contact
structures, as $\xi^{t}_{-}=\ker\alpha^{t}_{-}$ and $\xi_{+}=\ker\alpha_{+}$
might fail to be transverse for some $t\in(0,1)$. Even if transversality
holds, $(\alpha^{t}_{-},\alpha_{+})$ might fail to be an AL pair.
Nevertheless, one could try to analyze the failure of these properties for a
generic path $(\alpha^{t}_{-})_{t}$, and apply suitable modifications to it.
We wish to explore this direction in future work.

A closely related question, already raised by Hozoori [Hoz22a, Question 7.2.]
is the following.

######  Question 4.

Let $\Phi_{0}$ and $\Phi_{1}$ be two Anosov flows on $M$ and assume that they
are homotopic through _projectively_ Anosov flows. Equivalently, assume that
there exist two AL pairs $(\alpha_{-}^{0},\alpha_{+}^{0})$ and
$(\alpha_{-}^{1},\alpha_{+}^{1})$ supporting $\Phi_{0}$ and $\Phi_{1}$,
respectively, such that their underlying bi-contact structures are homotopic
(through bi-contact structures). Are $\Phi_{0}$ and $\Phi_{1}$ homotopic
through Anosov flows, i.e., are $(\alpha_{-}^{0},\alpha_{+}^{0})$ and
$(\alpha_{-}^{1},\alpha_{+}^{1})$ homotopic through AL pairs? Are $\Phi_{0}$
and $\Phi_{1}$ topologically equivalent?

From the point of view of Liouville geometry, it is natural to weaken the
definition of AL pairs as follows.

######  Definition 1.3.

A Liouville pair $(\alpha_{-},\alpha_{+})$ on $M$ is a weak Anosov Liouville
pair (wAL pair for short) if it satisfies the following two conditions.

  1. (1)

The contact plane fields $\xi_{\pm}:=\ker\alpha_{\pm}$ are everywhere
transverse,

  2. (2)

The intersection $\xi_{-}\cap\xi_{+}$ is spanned by an Anosov vector field.

An Anosov Liouville structure (AL structure for short) on
$V=\mathbb{R}_{s}\times M$ is a pair $(\omega,\lambda)$ where
$\omega=d\lambda$ is a symplectic form and

$\lambda=e^{-s}\alpha_{-}+e^{s}\alpha_{+}$

for a weak Anosov Liouville pair $(\alpha_{-},\alpha_{+})$. We call the triple
$(V,\omega,\lambda)$ an Anosov Liouville manifold.

An Anosov flow $\Phi$ is supported by the AL structure $(\omega,\lambda)$ if
the vector field $X$ generating $\Phi$ satisfies $X\in\xi_{-}\cap\xi_{+}$.

Note that the definition of wAL pairs _does_ make reference to the underlying
Anosov flow, as opposed to AL pairs. By Theorem 1, there is an inclusion
$\mathcal{AL}\subset\mathcal{AL}^{w}$, where $\mathcal{AL}^{w}$ denotes the
space of wAL pairs on $M$. This inclusion is strict in general. The map
$\mathcal{I}$ naturally extends to a map
$\mathcal{I}^{w}:\mathcal{AL}^{w}\rightarrow\mathcal{AF}$, and similarly to
the first bullet of Theorem 2, we have:

######  Theorem 3.

The map $\mathcal{I}^{w}:\mathcal{AL}^{w}\rightarrow\mathcal{AF}$ is an
acyclic Serre fibration, hence a homotopy equivalence.

######  Corollary 1.

Let $\Phi_{0}$ and $\Phi_{1}$ be two Anosov flows on $M$, supported by AL
structures $(\omega_{0},\lambda_{0})$ and $(\omega_{1},\lambda_{1})$,
respectively. If $\Phi_{0}$ and $\Phi_{1}$ are homotopic through Anosov flows,
then $(\omega_{0},\lambda_{0})$ and $(\omega_{1},\lambda_{1})$ are homotopic
through AL structures, and $(V,\omega_{0},\lambda_{0})$ and
$(V,\omega_{1},\lambda_{1})$ are exact symplectomorphic.

Here, an exact symplectomorphism
$\psi:(V,\omega_{0},\lambda_{0})\rightarrow(V,\omega_{1},\lambda_{1})$ is a
diffeomorphism such that $\psi^{*}\lambda_{1}=\lambda_{0}+df$ for some smooth
function $f:V\rightarrow\mathbb{R}$. In Corollary 1, we can further assume
that $df$ has compact support.

###### Proof of Corollary 1.

If $\Phi_{0}$ and $\Phi_{1}$ are homotopic through Anosov flows, Theorem 3
provides a _continuous_ path of smooth AL structures from
$(\omega_{0},\lambda_{0})$ to $(\omega_{1},\lambda_{1})$. This path can be
smoothen while ensuring the existence of some number $A>0$ such that the
corresponding Liouville vector fields are all transverse to $\\{\pm A\\}\times
M$. Then, [CE12, Proposition 11.8.] provides an exact symplectomorphism $\psi$
such that $\psi^{*}\lambda_{1}-\lambda_{0}$ is compactly supported. ∎

Anosov Liouville manifolds have numerous interesting invariants coming from
Floer theory, e.g., symplectic cohomology and wrapped Fukaya category. As an
important consequence of Corollary 1, these are _invariants of the underlying
Anosov flow_ , and only depend on its homotopy class in the space of Anosov
flows. Some of these invariants are studied in detail in [Cie+22]. To our
knowledge, this is the first thorough analysis of symplectic invariants of
non-Weinstein Liouville manifolds.

One can also consider Liouville pairs $(\alpha_{-},\alpha_{+})$ whose
underlying contact planes are everywhere transverse. We call such pairs
transverse Liouville pairs. They correspond to particular projectively Anosov
flow that we call semi-Anosov flows, see Remark 3.4. General Liouville pairs
(without the transversality assumption) are more complicated to understand,
but their underlying contact planes can only intersect _positively_ , see
Remark 2.2. In the terminology of [CF11], they constitute positive contact
pairs.

These geometric structures are summarized in the following diagram; the ones
in blue are the main protagonists of this article.

${{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\big{\\{}\text{AL
pairs}\big{\\}}}}$${{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\big{\\{}\text{Weak
AL pairs}\big{\\}}}}$${\big{\\{}\text{Transverse Liouville
pairs}\big{\\}}}$${\big{\\{}\text{Liouville
pairs}\big{\\}}}$${{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\big{\\{}\text{Bi-
contact structures}\big{\\}}}}$${\big{\\{}\text{Positive contact
pairs}\big{\\}}}$$\subset$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,1}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\underline{\ker}}$$\subset$$\subset$$\scriptstyle{\underline{\ker}}$$\scriptstyle{\underline{\ker}}$$\subset$

#### Acknowledgments.

I am grateful to my PhD advisor John Pardon for his constant support and
encouragement. I would like to thank Sergio Fenley, Jonathan Zung and Malo
Jézéquel for insightful conversations about Anosov flows, and Surena Hozoori
for multiple discussions about his work and fruitful exchanges.

##  2 Anosov Liouville pairs

###  2.1 Preliminary definitions

If $X$ is a non-singular vector field on $M$, we write

$N_{X}:=TM/\langle X\rangle.$

An orientation on $M$ naturally determines an orientation on the plane bundle
$N_{X}\rightarrow M$. We denote by $\pi:TM\rightarrow N_{X}$ the quotient map.
There is a correspondence between $n$-forms $\alpha$ on $M$ satisfying
$\iota_{X}\alpha=0$ and $n$-forms $\overline{\alpha}$ on $N_{X}$. Moreover, a
vector field $Y$ on $M$ induces a section $\overline{Y}:=\pi(Y)$ on $N_{X}$.
The operator $\mathcal{L}_{X}$, the Lie derivative along $X$, naturally
induces an operator, still denoted by $\mathcal{L}_{X}$, on sections of
$N_{X}$ and on $n$-forms on $N_{X}$.

######  Definition 2.1.

A bi-contact structure on an oriented $3$-manifold $M$ is a pair of co-
oriented contact structures $(\xi_{-},\xi_{+})$ such that $\xi_{-}$ is
negative, $\xi_{+}$ is positive and $\xi_{-}$ and $\xi_{+}$ are transverse
everywhere.

A non-singular flow $\Phi$ on $M$ generated by a vector field $X$ is supported
by a bi-contact structure $(\xi_{-},\xi_{+})$ if $X\in\xi_{-}\cap\xi_{+}$, and
the following orientation compatibility condition holds. Let
$\overline{\xi}_{\pm}\subseteq N_{X}$ be the image of $\xi_{\pm}$ under the
quotient map $\pi:TM\rightarrow N_{X}$. The orientations on $M$, $\xi_{\pm}$
and $X$ induce natural orientations on $N_{X}$ and $\overline{\xi}_{\pm}$. We
require that the orientation on $N_{X}$ coincides with the one on
$\overline{\xi}_{-}\oplus\overline{\xi}_{+}$ (see Figure 1 and Figure 2).

Similarly, $\Phi$ is supported by a (weak) Anosov Liouville pair
$(\alpha_{-},\alpha_{+})$ if it is supported by the bi-contact structure
$(\xi_{-},\xi_{+})=(\ker\alpha_{-},\ker\alpha_{+})$.

Note that the definitions of bi-contact structures and (weak) Anosov Liouville
pairs still make sense if $\xi_{\pm}$, or $\alpha_{\pm}$, are merely
$\mathcal{C}^{1}$. We will always assume that bi-contact structures and (weak)
Anosov Liouville pairs are smooth unless stated otherwise. Bi-contact
structures and a (weak) Anosov Liouville pairs obviously constitute open
subsets for the $\mathcal{C}^{1}$ topology of the space of pairs of $2$-plane
fields on $M$ and the space of pairs of $1$-forms on $M$, respectively.

Figure 1: Co-orientation convention for bi-contact structures supporting a
vector field or a flow.

If $(\xi_{-},\xi_{+})$ is a bi-contact structure supporting a non-singular
flow $\Phi=\\{\phi^{t}\\}$, then the bi-contact structure obtained from
$(\xi_{-},\xi_{+})$ by reversing the co-orientations of both $\xi_{-}$ and
$\xi_{+}$ supports $\Phi$ as well. Reversing the co-orientation of $\xi_{-}$
or $\xi_{+}$ only yields a bi-contact structure supporting the reversed flow
$\Phi^{-1}=\\{\phi^{-t}\\}$.

It is easy to deduce from Theorem 1 the very well-known

######  Corollary 2.1.

The space of (smooth, $\mathcal{C}^{1}$) Anosov vector fields on $M$ is open
in the $\mathcal{C}^{1}$ topology.

###### Proof.

Let $X$ be an Anosov vector field on $M$ and $(\alpha_{-},\alpha_{+})$ be an
AL pair supporting $X$. We choose a $1$-form $\theta$ such that
$\theta(X)\equiv 1$. If $X^{\prime}$ is another vector field which is
sufficiently $\mathcal{C}^{1}$-close to $X$, the pair
$(\alpha^{\prime}_{-},\alpha^{\prime}_{+})$ defined by

$\displaystyle\alpha^{\prime}_{\pm}:=\alpha_{\pm}-\frac{\alpha_{\pm}(X^{\prime})}{\theta(X^{\prime})}\theta$

is an AL pair supporting $X^{\prime}$ and by Theorem 1, $X^{\prime}$ is
Anosov. ∎

If $(\alpha_{-},\alpha_{+})$ is an AL pair on $M$ and
$\sigma:M\rightarrow\mathbb{R}$ is a smooth function, one easily checks that

$\sigma\cdot(\alpha_{-},\alpha_{+}):=\big{(}e^{-\sigma}\alpha_{-},e^{\sigma}\alpha_{+}\big{)}$

is also an AL pair that defines the same bi-contact structure as
$(\alpha_{-},\alpha_{+})$. These two AL pairs will be called equivalent. This
defines an action of $\mathcal{C}^{\infty}(M,\mathbb{R})$ on the space of AL
pairs.

######  Definition 2.2.

A pair of contact forms $(\alpha_{-},\alpha_{+})$ on M, negative and positive,
respectively, is balanced if

$\alpha_{+}\wedge d\alpha_{+}=-\alpha_{-}\wedge d\alpha_{-}.$

In other words, $(\alpha_{-},\alpha_{+})$ is balanced if $\alpha_{\pm}$ define
opposite volume forms on $M$.

######  Lemma 2.1.

Two equivalent AL pairs on $M$ define Liouville isomorphic Liouville
structures on $\mathbb{R}\times M$. Any AL pair on $M$ is equivalent to a
(unique) balanced one.

###### Proof.

Let $(\alpha_{-},\alpha_{+})$ be an AL pair on $M$ and
$\lambda:=e^{-s}\alpha_{-}+e^{s}\alpha_{+}$ be the corresponding Liouville
form. If $\sigma\in\mathcal{C}^{\infty}(M,\mathbb{R})$ and
$\lambda^{\prime}:=e^{-(s+\sigma)}\alpha_{-}+e^{s+\sigma}\alpha_{+}$, the
diffeomorphism

$\begin{array}[]{rccc}\Psi:&\mathbb{R}\times
M&\longrightarrow&\mathbb{R}\times M\\\
&(s,x)&\longmapsto&(s-\sigma(x),x)\end{array}$

satisfies $\Psi^{*}\lambda^{\prime}=\lambda$. Moreover, if
$f:M\rightarrow\mathbb{R}_{>0}$ is such that

$\alpha_{-}\wedge d\alpha_{-}=-f\,\alpha_{+}\wedge d\alpha_{+},$

and $\sigma:=\frac{1}{4}\ln f$, then $\sigma\cdot(\alpha_{-},\alpha_{+})$ is
balanced. ∎

As a straightforward application of Gray’s stability theorem and the above
lemma, we have the following

######  Lemma 2.2.

Let $(\alpha_{-},\alpha_{+})$ be an AL pair and let $\xi_{+}:=\ker\alpha_{+}$.
If $\xi^{\prime}_{+}=\ker\alpha^{\prime}_{+}$ is a contact structure homotopic
to $\xi_{+}$, then there exists a path of AL pairs
$(\alpha^{t}_{-},\alpha^{t}_{+})$, $t\in[0,1]$, such that
$(\alpha^{0}_{-},\alpha^{0}_{+})=(\alpha_{-},\alpha_{+})$ and
$\alpha^{1}_{+}=\alpha^{\prime}_{+}$.

######  Definition 2.3.

A pair of contact forms $(\alpha_{-},\alpha_{+})$ on M, negative and positive,
respectively, is closed if $\alpha_{-}\wedge\alpha_{+}$ is a closed $2$-form.

It is straightforward to check that a closed pair $(\alpha_{-},\alpha_{+})$ is
an AL pair (see also Lemma 2.3 below). As we will see in Proposition 3.1,
closed AL pairs are in correspondence with volume preserving Anosov flows.

###  2.2 Elementary properties of Anosov Liouville pairs

The notion of Anosov Liouville pair can be conveniently characterized in the
following way, which only involves the forms and their exterior differentials.

######  Lemma 2.3.

Let $(\alpha_{-},\alpha_{+})$ be a pair of $1$-forms on $M$. We write

$\displaystyle\alpha_{+}\wedge d\alpha_{+}$
$\displaystyle=f_{+}\,\mathrm{dvol},$ $\displaystyle\alpha_{-}\wedge
d\alpha_{-}$ $\displaystyle=-f_{-}\,\mathrm{dvol},$ $\displaystyle
d(\alpha_{-}\wedge\alpha_{+})$ $\displaystyle=f_{0}\ \mathrm{dvol},$

where $\mathrm{dvol}$ is any volume form on $M$ and
$f_{\pm},f_{0}:M\rightarrow\mathbb{R}$ are smooth functions. Then
$(\alpha_{-},\alpha_{+})$ is an AL pair if and only if $f_{\pm}>0$, and

$\displaystyle f_{0}^{2}<4f_{-}f_{+}.$ (2.1)

###### Proof.

Following [MNW13, Lemma 9.4], it is easy to see that $(\alpha_{-},\alpha_{+})$
is a Liouville pair if and only if for all constants $C_{-},C_{+}\geq 0$ with
$(C_{-},C_{+})\neq(0,0)$,

$(C_{+}\alpha_{+}-C_{-}\alpha_{-})\wedge(C_{+}d\alpha_{+}+C_{-}d\alpha_{-})>0,$

which is equivalent to

$C_{+}^{2}f_{+}+C_{-}C_{+}f_{0}+C_{-}^{2}f_{-}>0.$

Hence, $(\alpha_{-},\alpha_{+})$ is an AL pair if and only if $f_{\pm}>0$ and
for every $x\in\mathbb{R}$,

$x^{2}f_{+}+xf_{0}+f_{-}>0,$

which is equivalent to (2.1). ∎

######  Remark 2.1.

The proof also shows that $(\alpha_{-},\alpha_{+})$ is a Liouville pair if and
only if $f_{\pm}>0$ and $-f_{0}<2\sqrt{f_{-}f_{+}}$.

We now use this criterion to show some natural geometric properties of Anosov
Liouville pairs.

######  Proposition 2.1.

Let $(\alpha_{-},\alpha_{+})$ be an Anosov Liouville pair. Then it defines a
bi-contact structure $(\xi_{-},\xi_{+})=(\ker\alpha_{-},\ker\alpha_{+})$.
Moreover, if $X\in\xi_{-}\cap\xi_{+}$ is a nowhere vanishing vector field and
$R_{\pm}$ is the Reeb vector field of $\alpha_{\pm}$, then
$\\{X,R_{-},R_{+}\\}$ is a basis at every point of $M$.

###### Proof.

We first show that $\xi_{-}$ and $\xi_{+}$ intersect transversally everywhere.
Assume by contradiction that there exist a point $x\in M$ and two linearly
independent vectors $X,Y\in T_{x}M$ such that $\alpha_{\pm}(Y)=0$. In what
follows, all the quantities will be implicitly evaluated at this point $x$. We
can assume without loss of generality that $d\alpha_{+}(X,Y)>0$ and
$\mathrm{dvol}(X,Y,R_{+})=1$. We compute

$\displaystyle\alpha_{+}\wedge d\alpha_{+}(X,Y,R_{+})$
$\displaystyle=d\alpha_{+}(X,Y)=f_{+},$ $\displaystyle\alpha_{-}\wedge
d\alpha_{-}(X,Y,R_{+})$
$\displaystyle=\alpha_{-}(R_{+})d\alpha_{-}(X,Y)=-f_{-},$
$\displaystyle\alpha_{-}\wedge d\alpha_{+}(X,Y,R_{+})$
$\displaystyle=\alpha_{-}(R_{+})d\alpha_{+}(X,Y),$
$\displaystyle\alpha_{+}\wedge d\alpha_{-}(X,Y,R_{+})$
$\displaystyle=d\alpha_{-}(X,Y),$

hence

$\displaystyle f_{0}^{2}-4f_{-}f_{+}$
$\displaystyle=\begin{multlined}(d\alpha_{-}(X,Y))^{2}-2\alpha_{-}(R_{+})d\alpha_{-}(X,Y)d\alpha_{+}(X,Y)\\\
+\alpha_{-}(R_{+})^{2}(d\alpha_{+}(X,Y))^{2}+4\alpha_{-}(R_{+})d\alpha_{-}(X,Y)d\alpha_{+}(X,Y)\end{multlined}(d\alpha_{-}(X,Y))^{2}-2\alpha_{-}(R_{+})d\alpha_{-}(X,Y)d\alpha_{+}(X,Y)\\\
+\alpha_{-}(R_{+})^{2}(d\alpha_{+}(X,Y))^{2}+4\alpha_{-}(R_{+})d\alpha_{-}(X,Y)d\alpha_{+}(X,Y)$
$\displaystyle=\left(d\alpha_{-}(X,Y)+\alpha_{-}(R_{+})d\alpha_{+}(X,Y)\right)^{2}$
$\displaystyle\geq 0,$

contradicting (2.1).

For the second part, we write

$\displaystyle\alpha_{-}\wedge d\alpha_{+}=g_{+}\,\mathrm{dvol},$
$\displaystyle\alpha_{+}\wedge d\alpha_{-}=g_{-}\,\mathrm{dvol},$

where $g_{\pm}:M\rightarrow\mathbb{R}$ are smooth functions (note that
$f_{0}=g_{-}-g_{+}$) and we compute

$\displaystyle\alpha_{+}\wedge d\alpha_{+}(X,R_{+},\cdot\,)$
$\displaystyle=-d\alpha_{+}(X,\cdot\,)=f_{+}\,\mathrm{dvol}(X,R_{+},\cdot\,),$
$\displaystyle\alpha_{-}\wedge d\alpha_{-}(X,R_{+},\cdot\,)$
$\displaystyle=-\alpha_{-}(R_{+})d\alpha_{-}(X,\cdot\,)+d\alpha_{-}(X,R_{+})\alpha_{-}=-f_{-}\,\mathrm{dvol}(X,R_{+},\cdot\,),$
$\displaystyle\alpha_{-}\wedge d\alpha_{+}(X,R_{+},\cdot\,)$
$\displaystyle=-\alpha_{-}(R_{+})d\alpha_{+}(X,\cdot\,)=g_{+}\,\mathrm{dvol}(X,R_{+},\cdot\,),$
$\displaystyle\alpha_{+}\wedge d\alpha_{-}(X,R_{+},\cdot\,)$
$\displaystyle=-d\alpha_{-}(X,\cdot\,)+d\alpha_{-}(X,R_{+})\alpha_{+}=g_{-}\,\mathrm{dvol}(X,R_{+},\cdot\,).$

Let us assume that $\mathrm{dvol}(X,R_{-},R_{+})=0$ at a point $x\in M$. In
what follows, all the quantities will be implicitly evaluated at this point
$x$. Plugging in $R_{-}$ yields

$d\alpha_{-}(X,R_{+})=d\alpha_{+}(X,R_{-})=0.$

Note that $X$ and $R_{+}$ are not colinear since $\alpha_{+}(X)=0$ and
$\alpha_{+}(R_{+})=1$. We obtain that $\alpha_{-}(R_{+})\neq 0$ and

$\displaystyle f_{+}$ $\displaystyle=\frac{1}{\alpha_{-}(R_{+})}g_{+},$
$\displaystyle f_{-}$ $\displaystyle=-\alpha_{-}(R_{+})g_{-}.$

Finally,

$\displaystyle f_{0}^{2}-4f_{-}f_{+}$
$\displaystyle=\big{(}g_{-}-g_{+}\big{)}^{2}+4g_{-}g_{+}$
$\displaystyle=\big{(}g_{-}+g_{+}\big{)}^{2}$ $\displaystyle\geq 0,$

contradicting (2.1). ∎

######  Remark 2.2.

A (non-Anosov) Liouville pair may not define a bi-contact structure, namely
$\xi_{-}=\ker\alpha_{-}$ and $\xi_{+}=\ker\alpha_{+}$ may not be transverse
everywhere. Nevertheless, the first part of the proof can easily be adapted to
show that at a point where $\xi_{-}$ and $\xi_{+}$ coincide, their
orientations coincide (and their co-orientations are opposite). In the
terminology of [CF11], $(\xi_{-},\xi_{+})$ is a _positive pair_ of contact
structures. After a generic perturbation of $\alpha_{-}$ and/or $\alpha_{+}$,
the singular set $\Delta:=\\{x\in M:\xi_{-}(x)=\xi_{+}(x)\\}$ is a smoothly
embedded link in $M$. Moreover, $f_{0}>0$ along $\Delta$ so the Liouville
condition of Remark 2.1 is largely satisfied.

For any AL pair $(\alpha_{-},\alpha_{+})$, if $X$ (or $\mathrm{dvol}$) is
chosen so that $\mathrm{dvol}(X,R_{-},R_{+})=1$, then

$\displaystyle f_{+}$
$\displaystyle=d\alpha_{+}(X,R_{-})=\mathcal{L}_{X}\alpha_{+}(R_{-}),$
$\displaystyle f_{-}$
$\displaystyle=d\alpha_{-}(X,R_{+})=\mathcal{L}_{X}\alpha_{-}(R_{+}),$
$\displaystyle g_{+}$ $\displaystyle=\alpha_{-}(R_{+})f_{+},$ $\displaystyle
g_{-}$ $\displaystyle=-\alpha_{+}(R_{-})f_{-}.$

Moreover, if $(\alpha_{-},\alpha_{+})$ is balanced, i.e., if $f_{+}=f_{-}$,
the condition (2.1) becomes

$\displaystyle|\alpha_{-}(R_{+})+\alpha_{+}(R_{-})|<2.$ (2.2)

In fact, (balanced) AL pairs can be completely characterized by their Reeb
vector fields.

######  Proposition 2.2.

Let $(\alpha_{-},\alpha_{+})$ be a pair of contact forms on $M$, negative and
positive, respectively, and assume that it is balanced. Then it is an AL pair
if and only if (2.2) is satisfied.

###### Proof.

We only have to show that under these hypothesis, the conclusions of
Proposition 2.1 are satisfied, since these imply that
$g_{+}=\alpha_{-}(R_{+})f_{+}$ and $g_{-}=-\alpha_{+}(R_{-})f_{-}$ and Lemma
2.3 concludes the proof.

Assume first that $\xi_{-}$ and $\xi_{+}$ are not transverse at a point $x\in
M$. With the same notations as in the proof of Proposition 2.1, similar
computations show that at this point,

$\displaystyle\alpha_{+}\wedge d\alpha_{+}(X,Y,R_{+})$
$\displaystyle=d\alpha_{+}(X,Y)=f_{+},$ $\displaystyle\alpha_{-}\wedge
d\alpha_{-}(X,Y,R_{+})$
$\displaystyle=\alpha_{-}(R_{+})d\alpha_{-}(X,Y)=-f_{-},$
$\displaystyle\alpha_{+}\wedge d\alpha_{+}(X,Y,R_{-})$
$\displaystyle=\alpha_{+}(R_{-})d\alpha_{+}(X,Y)=f_{+}\,\mathrm{dvol}(X,Y,R_{-}),$
$\displaystyle\alpha_{-}\wedge d\alpha_{-}(X,Y,R_{-})$
$\displaystyle=d\alpha_{-}(X,Y)=-f_{-}\,\mathrm{dvol}(X,Y,R_{-}),$

hence

$\mathrm{dvol}(X,Y,R_{-})=\alpha_{+}(R_{-})$

by the first and third equalities, and

$\alpha_{-}(R_{+})\alpha_{+}(R_{-})=1$

by the second and fourth equalities, contradicting (2.2) by the inequality of
arithmetic and geometric means.

Assuming now that $\mathrm{dvol}(X,R_{-},R_{+})=0$ at a point $x\in M$, the
proof of Proposition 2.1 showed that at this point,

$d\alpha_{-}(X,R_{+})=d\alpha_{+}(X,R_{-})=0,$

and

$g_{+}=\alpha_{-}(R_{+})f_{+}.$

Similarly,

$\displaystyle\alpha_{+}\wedge d\alpha_{+}(X,R_{-},\cdot\,)$
$\displaystyle=-\alpha_{+}(R_{-})d\alpha_{+}(X,\cdot\,)=f_{+}\,\mathrm{dvol}(X,R_{-},\cdot\,),$
$\displaystyle\alpha_{-}\wedge d\alpha_{+}(X,R_{-},\cdot\,)$
$\displaystyle=-d\alpha_{+}(X,\cdot\,)=g_{+}\,\mathrm{dvol}(X,R_{-},\cdot\,),$

hence

$f_{+}=\alpha_{+}(R_{-})g_{+}.$

Once again, we obtain that

$\alpha_{-}(R_{+})\alpha_{+}(R_{-})=1,$

contradicting (2.2). ∎

##  3 From Anosov flows to Anosov Liouville pairs and back

In this section, we adapt the proof of [Hoz22a, Theorem 1.1] to the setting of
Anosov Liouville as defined in the Introduction.

###  3.1 Anosov and projectively Anosov flows

We recall the definitions of Anosov and projectively Anosov flows with an
emphasis on our orientation conventions, and recast them in terms of the
existence of suitable $1$-forms.

######  Definition 3.1.

Let $\Phi=\\{\phi^{t}\\}_{t\in\mathbb{R}}$ be a flow on $M$ generated by a
non-singular $\mathcal{C}^{1}$ vector field $X$.

  * •

$\Phi$ is Anosov if there exists a continuous invariant hyperbolic splitting

$\displaystyle TM=\langle X\rangle\oplus E^{s}\oplus E^{u}$ (3.1)

where $E^{s},E^{u}$ are $1$-dimensional bundles such that for some (any)
Riemannian metric $g$ on $M$, there exist constants $C,a>0$ such that for all
$v\in E^{s}$ and $t\in\mathbb{R}$,

$\|d\phi^{t}(v)\|\leq Ce^{-at}\|v\|,$

and for all $v\in E^{u}$ and $t\in\mathbb{R}$,

$\|d\phi^{t}(v)\|\geq Ce^{at}\|v\|.$

$E^{s}$ and $E^{u}$ are called the (strong) stable and unstable bundles of
$\Phi$, respectively.

  * •

$\Phi$ is projectively Anosov if there exists a continuous invariant dominated
splitting

$\displaystyle TM/\langle
X\rangle=N_{X}=\overline{E}^{s}\oplus\overline{E}^{u}$ (3.2)

where $\overline{E}^{s},\overline{E}^{u}$ are $1$-dimensional bundles such
that for some (any) Riemannian metric $\overline{g}$ on $N_{X}$, there exist
constants $C,a>0$ such that for all unit vectors
$v_{s}\in\overline{E}^{s},v_{u}\in\overline{E}^{u}$, and $t\in\mathbb{R}$,

$\|d\phi^{t}(v_{u})\|\geq Ce^{at}\,\|d\phi^{t}(v_{s})\|.$

We denote by $E^{ws}:=\pi^{-1}\big{(}\overline{E}^{s}\big{)}$ and
$E^{wu}:=\pi^{-1}\big{(}\overline{E}^{u}\big{)}$ the weak-stable and weak-
unstable bundles of $\Phi$, respectively.

  * •

In both cases, if the constant $C$ can be chosen to be $1$, the corresponding
metrics $g$ and $\overline{g}$ are called adapted to $\Phi$.

  * •

The Anosov (resp. projectively Anosov) flow $\Phi$ is oriented if $E^{s}$ and
$E^{u}$ are oriented (resp. $\overline{E}^{s}$ and $\overline{E}^{u}$ are
oriented) and their orientations are compatible with the splitting (3.1)
(resp. the splitting (3.2)).

Figure 2: Orientation convention for the Anosov splitting. There is a similar
picture for the dominated splitting. The vector field X points toward the
reader, watch out.

Anosov flows are projectively Anosov, with dominated splitting
$N_{X}=\pi\big{(}E^{s}\big{)}\oplus\pi\big{(}E^{u}\big{)}$. Every three
dimensional (projectively) Anosov flow admits a smooth adapted metric, see
[FH19, Proposition 5.1.5] (the proof is given for Anosov flows but it easily
generalizes to projectively Anosov flows in dimension three). Anosov famously
showed that the weak/strong stable/unstable bundles of an Anosov flow are
_uniquely integrable_. Moreover, $E^{ws}$ and $E^{wu}$ integrate into _taut
foliations_ $\mathcal{F}^{ws}$ and $\mathcal{F}^{wu}$, respectively. This is
not true for projectively Anosov flows; see [ET98, Example 2.2.9.]

The following definition appears in [Hoz22a, Definition 3.11.].

######  Definition 3.2.

Let $\Phi$ be a projectively Anosov on $M$ generated by a vector field $X$ and
$\overline{g}$ be a Riemannian metric on $N_{X}$. The expansion rates in the
stable and unstable directions for $\overline{g}$ are continuous functions
$r_{s},r_{u}:M\rightarrow\mathbb{R}$ defined by

$r_{s}:=\frac{\partial}{\partial
t}\Big{|}_{t=0}\ln\|d\phi^{t}(\overline{e}_{s})\|,\qquad
r_{u}:=\frac{\partial}{\partial
t}\Big{|}_{t=0}\ln\|d\phi^{t}(\overline{e}_{u})\|,$

where $\overline{e}_{s}$ and $\overline{e}_{u}$ are unit sections of
$\overline{E}^{s}$ and $\overline{E}^{u}$, respectively, which are continuous
and continuously differentiable along the flow $\Phi$. Moreover,

$\mathcal{L}_{X}\overline{e}_{s}=-r_{s}\overline{e}_{s},\qquad\mathcal{L}_{X}\overline{e}_{u}=-r_{u}\overline{e}_{u}.$

Many natural quantities (e.g., functions, vector fields, $1$-forms, Riemannian
metrics) defined for (projectively) Anosov flows are continuous and can be
upgraded to a quantities which are continuous _and_ continuously
differentiable along the flow by considering the averaging

$\frac{1}{T}\int_{0}^{T}\big{(}\phi^{t}\big{)}^{*}\\_\,dt$

for some $T>0$. Nevertheless, these quantities may not be $\mathcal{C}^{1}$.
It is therefore natural to consider the following spaces (only the cases
$k=0,1$ and $n=0,1$ will be relevant for us).

######  Definition 3.3.

Le $X$ be a smooth, non-singular vector field on $M$, and $k\geq 0$ be a non-
negative integer.

  * •

A $n$-form $\alpha$ on $M$ is of class $\mathcal{C}^{k}_{X}$ if $\alpha$ is
differentiable along $X$, and both $\alpha$ and $\mathcal{L}_{X}\alpha$ are of
class $\mathcal{C}^{k}$. We denote by $\Omega^{n}_{X,k}$ the space of
$n$-forms on $M$ of class $\mathcal{C}^{k}_{X}$ satisfying $\iota_{X}\alpha=0$
(which is vacuous for $n=0$, i.e., for functions). We also denote by
$\Omega^{n}_{X}=\Omega^{n}_{X,\infty}\subset\Omega^{n}$ the space of smooth
$n$-forms satisfying $\iota_{X}\alpha=0$.

  * •

On $\Omega^{n}_{X,k}$, there is a natural norm defined by

$|\alpha|_{\mathcal{C}_{X}^{k}}:=|\alpha|_{\mathcal{C}^{k}}+|\mathcal{L}_{X}\alpha|_{\mathcal{C}^{k}},$

making $\big{(}\Omega_{X,k}^{n},|\cdot|_{\mathcal{C}_{X}^{k}}\big{)}$ a Banach
space. $\Omega^{n}_{X}$ is naturally a Fréchet space as a closed subspace of
$\Omega^{n}$

These definitions naturally extend to sections of $N_{X}$ and $n$-forms on
$N_{X}$.

In Appendix A, we show some density results for these spaces which are
particularly useful when dealing with (projectively) Anosov flows generated by
$\mathcal{C}^{1}$ vector fields and can be used to bypass Hozoori’s delicate
approximation techniques in [Hoz22a, Section 4]. The results in Appendix A are
not needed (except in the proof of Theorem 4.2) if we restrict our attention
to smooth Anosov flows in view of Lemma 3.2 below.

In dimension three, the definitions of Anosov and projectively Anosov flows
can be rephrased in terms of the existence of certain $1$-forms. The following
lemma is essentially an adaptation of results of Mitsumatsu [Mit95] and
Hozoori [Hoz22a, Hoz22].

######  Lemma 3.1.

Let $\Phi$ be a smooth, non-singular flow on $M$ generated by a vector field
$X$, then

  1. (1)

$\Phi$ is projectively Anosov if and only if there exist
$(\alpha_{s},\alpha_{u})\in\Omega^{1}_{X,0}\times\Omega^{1}_{X,0}$ and
continuous functions $r_{u},r_{s}:M\rightarrow\mathbb{R}$ such that

$\displaystyle\overline{\alpha}_{s}\wedge\overline{\alpha}_{u}>0,$
$\displaystyle\mathcal{L}_{X}\alpha_{s}=r_{s}\,\alpha_{s},$
$\displaystyle\mathcal{L}_{X}\alpha_{u}=r_{u}\,\alpha_{u},$

and $r_{s}<r_{u}$. Here, $\overline{\alpha}_{s}$ and $\overline{\alpha}_{u}$
denote the $1$-forms on $N_{X}$ induced by $\alpha_{s}$ and $\alpha_{u}$,
respectively.

  2. (2)

$\Phi$ is Anosov if and only if there exist $\alpha_{s}$, $\alpha_{u}$,
$r_{s}$, $r_{u}$ as above such that $r_{s}<0<r_{u}$.

  3. (3)

$\Phi$ is volume preserving Anosov if and only if there exist $\alpha_{s}$,
$\alpha_{u}$, $r_{s}$, $r_{u}$ as above such that $r_{u}+r_{s}=0$.

Moreover, $\ker\alpha_{u}=E^{ws}$ and $\ker\alpha_{s}=E^{wu}$.

###### Proof.

(1) essentially follows from [Hoz22a, Proposition 3.15.]. We recall the main
arguments. If $\Phi$ is projectively Anosov, we can choose an adapted metric
and unit vector fields $\overline{e}_{s}$ and $\overline{e}_{u}\in N_{X}$ of
class $\mathcal{C}^{0}_{X}$ such that $\overline{e}_{s}$ spans
$\overline{E}^{s}$, $\overline{e}_{u}$ spans $\overline{E}^{u}$ and
$(\overline{e}_{s},\overline{e}_{u})$ is positively oriented. If
$(\overline{\alpha}_{s},\overline{\alpha}_{u})$ is the dual basis of
$(\overline{e}_{u},\overline{e}_{s})$, it induces a pair
$(\alpha_{s},\alpha_{u})\in\Omega^{1}_{X,0}\times\Omega^{1}_{X,0}$ and the
relations $\mathcal{L}_{X}\overline{e}_{s}=-r_{s}\overline{e}_{s}$ and
$\mathcal{L}_{X}\overline{e}_{u}=-r_{u}\overline{e}_{u}$, where $r_{s}<r_{u}$,
imply the desired relations for $\alpha_{s}$ and $\alpha_{u}$. Reciprocally,
if $(\alpha_{s},\alpha_{u})$ is such a pair, we define
$(\overline{e}_{s},\overline{e}_{u})$ as dual basis of
$(\overline{\alpha}_{s},\overline{\alpha}_{u})$ and we easily check that it
yields a projectively Anosov splitting of $N_{X}$. This is essentially because
for $\star\in\\{s,u\\}$,
$\phi^{T}_{*}\overline{e}_{\star}=\exp\left(\int^{T}_{0}r_{\star}\circ\phi^{t}\,dt\right)\overline{e}_{\star}$
and $r_{s}<r_{u}$, where $\Phi=\\{\phi^{t}\\}$. In particular, we have
$\ker\alpha_{s}=E^{wu}$ and $\ker\alpha_{u}=E^{ws}$ since these bundles are
uniquely determined by the flow.

(2) follows from (1) and [Hoz22a, Proposition 3.17.].

The forward direction of (3) follows from the proof of [Mit95, Theorem 3.].
Indeed, assuming that $\Phi$ is volume preserving Anosov, we can arrange that
$r_{u}+r_{s}=0$ in the following way. If $\mathrm{dvol}$ is a (smooth) volume
form preserved by $\Phi$, then $\tau:=\iota_{X}\mathrm{dvol}$ is a non-
degenerate $2$-form on $N_{X}$ invariant under $\Phi$. There exists an adapted
metric $g$ of class $\mathcal{C}^{0}_{X}$ for which the Anosov splitting is
orthogonal and the volume form for the induced metric $\overline{g}$ on
$N_{X}$ is precisely $\tau$. Hence, if $e_{s}$ and $e_{u}$ are
$\mathcal{C}^{0}_{X}$ unit vector fields spanning $E^{s}$ and $E^{u}$,
respectively, then $\tau(\overline{e}_{s},\overline{e}_{u})=1$.
Differentiating this equality along $X$ yields $r_{u}+r_{s}=0$ as desired, and
we obtain $(\alpha_{s},\alpha_{u})$ by dualization as before. For the reverse
direction, $r_{s}<0<r_{u}$ since $r_{s}<r_{u}$ and $r_{s}=-r_{u}$, so $\Phi$
is Anosov by (2). Moreover, if $\theta$ is a smooth $1$-form satisfying
$\theta(X)\equiv 1$, then
$\mathrm{dvol}:=\alpha_{s}\wedge\alpha_{u}\wedge\theta$ is a
$\mathcal{C}^{0}_{X}$ volume form preserved by $X$ and by [LMM86, Corollary
2.1.], this volume form is smooth. ∎

It is well-known that in dimension three, the regularity of the weak-stable
and weak-unstable bundles of a smooth (even $\mathcal{C}^{2}$) Anosov flow are
$\mathcal{C}^{1}$. We have the following

######  Lemma 3.2.

If $\Phi$ is Anosov and smooth, we can further assume that $\alpha_{s}$,
$\alpha_{u}$, $r_{s}$ and $r_{u}$ as in Lemma 3.1 are $\mathcal{C}^{1}$, i.e.,
$\alpha_{s}$ and $\alpha_{u}$ are $\mathcal{C}^{1}_{X}$.

###### Proof.

By [Has94, Corollary 1.8.], $E^{ws}$ and $E^{wu}$ are $\mathcal{C}^{1}$ and an
adapted metric can always be assumed to be smooth, so the construction in
Lemma 3.1 yields $\mathcal{C}^{1}$ $1$-forms $\alpha_{s}$ and $\alpha_{u}$.
The $\mathcal{C}^{1}$ regularity of $r_{s}$ and $r_{u}$ follows from a trick
of Simić [Sim97]. First, let us choose a $\mathcal{C}^{1}$ $1$-form
$\alpha_{u}$ such that $\ker\alpha_{u}=E^{ws}$ and
$\mathcal{L}_{X}\alpha_{u}=r_{u}\,\alpha_{u}$, where $r_{u}$ is continuous and
positive. Fix a smooth vector field $Z$ positively transverse to $E^{ws}$, so
that $f:=\alpha_{u}(Z)>0$. Here, $f$ is $\mathcal{C}^{1}$ and can be
approximated by a smooth function $\widetilde{f}>0$ so that
$h:=\frac{\widetilde{f}}{f}$ is $\mathcal{C}^{1}$-close to $1$. Setting
$\widetilde{\alpha}_{u}:=h\alpha_{u}$, $\widetilde{\alpha}_{u}$ is
$\mathcal{C}^{1}$ and satisfies
$\mathcal{L}_{X}\widetilde{\alpha}_{u}=\widetilde{r}_{u}\,\widetilde{\alpha}_{u}$,
where $\widetilde{r}_{u}$ is $\mathcal{C}^{0}$-close to $r_{u}$ and can be
assumed to be positive. We now show that $\widetilde{r}_{u}$ is
$\mathcal{C}^{1}$. Indeed, $\widetilde{\alpha}_{u}(Z)=\widetilde{f}$ and

$\displaystyle\widetilde{r}_{u}$
$\displaystyle=\left(\mathcal{L}_{X}\widetilde{\alpha}_{u}\right)(Z)$
$\displaystyle=\mathcal{L}_{X}\left(\widetilde{\alpha}_{u}(Z)\right)-\widetilde{\alpha}_{u}\left([X,Z]\right)$
$\displaystyle=X\cdot\widetilde{f}-\widetilde{\alpha}_{u}\left([X,Z]\right),$

and the last quantity is $\mathcal{C}^{1}$ since $X\cdot\widetilde{f}$ and
$[X,Z]$ are smooth and $\widetilde{\alpha}_{u}$ is $\mathcal{C}^{1}$. The same
argument applies to $\alpha_{s}$. ∎

######  Remark 3.1.

The proof actually shows more. Since
$\widetilde{r}_{u}=u+\widetilde{\alpha}_{u}(V)$ for some smooth function $u$
and some smooth vector field $V$, and
$\mathcal{L}_{X}\widetilde{\alpha}_{u}=\widetilde{r}_{u}\,\widetilde{\alpha}_{u}$,
an immediate induction argument shows that for every integer $n\geq 0$,
$\mathcal{L}_{X}^{n}\widetilde{\alpha}_{u}$ and
$\mathcal{L}_{X}^{n}\widetilde{r}_{u}$ exist and are $\mathcal{C}^{1}$, where
$\mathcal{L}_{X}^{n}:=\mathcal{L}_{X}\circ\dots\circ\mathcal{L}_{X}$ denotes
the Lie derivative along $X$ iterated $n$ times. In fact, it is well known
that in our setting, the individual leaves of the weak-(un)stable foliation
are smooth.

######  Remark 3.2.

The same argument works for smooth projectively Anosov flow whose weak-stable
and weak-unstable distributions are $\mathcal{C}^{1}$. There are known
examples of smooth projectively Anosov flows in dimension three whose weak
distributions are only $\mathcal{C}^{0}$; see [ET98, Example 2.2.9.].

We call a pair of $1$-forms $(\alpha_{s},\alpha_{u})$ as in $(1)$ (resp.
$(2)$, $(3)$) a defining pair for the projectively Anosov (resp. Anosov,
volume preserving Anosov) flow $\Phi$. We further require defining pairs for
(volume preserving) Anosov flows to be $\mathcal{C}^{1}$. We also impose the
following conditions on orientations:

  * •

The orientation on $E^{ws}$ coming from $\Phi$ agrees with the one induced by
$\alpha_{u}$,

  * •

The orientation on $E^{wu}$ coming from $\Phi$ agrees with the one induced by
$-\alpha_{s}$.111The somewhat strange minus sign is explained by the following
remark. In the Euclidean plane $\mathbb{R}^{2}$ with its standard oriented
basis $(e_{1},e_{2})$, the dual basis $(e^{*}_{1},e^{*}_{2})$ induces co-
orientations on the $x$ and $y$-axis corresponding to the natural orientation
on the $x$-axis and to the _opposite_ of the natural orientation on the
$y$-axis.

We denote by $\mathcal{D}_{\Phi}$ the space of defining pairs for $\Phi$
endowed with the $\mathcal{C}^{0}_{X}$ topology in the projectively Anosov
case, and with the $\mathcal{C}^{1}$ topology in the (volume preserving)
Anosov case.

######  Lemma 3.3.

The space $\mathcal{D}_{\Phi}$ of defining pairs for a projectively Anosov
(resp. Anosov, volume preserving Anosov) flow $\Phi$ with its corresponding
topology is contractible.

###### Proof.

Let us fix a defining pair $(\alpha_{s},\alpha_{u})\in\mathcal{D}_{\Phi}$ for
a projectively Anosov flow $\Phi$ generated by a vector field $X$. If
$(\alpha^{\prime}_{s},\alpha^{\prime}_{u})\in\mathcal{D}_{\Phi}$ is any other
defining pair, then $\ker\alpha_{u}=\ker\alpha^{\prime}_{u}$ and
$\ker\alpha_{s}=\ker\alpha^{\prime}_{s}$, and the orientations on these spaces
agree. Hence, there exist (unique) functions
$\rho_{s},\rho_{u}:M\rightarrow\mathbb{R}$ of class $\mathcal{C}^{0}_{X}$ such
that $\alpha^{\prime}_{u}=e^{\rho_{u}}\alpha_{u}$ and
$\alpha^{\prime}_{s}=e^{\rho_{s}}\alpha_{s}$, and they satisfy

$\displaystyle
r_{u}^{\prime}-r_{s}^{\prime}=X\cdot(\rho_{u}-\rho_{s})+r_{u}-r_{s}>0.$ (3.3)

Here, $r_{u}^{\prime}$ and $r_{s}^{\prime}$ are such that
$\mathcal{L}_{X}\alpha^{\prime}_{s}=r^{\prime}_{s}\,\alpha^{\prime}_{s}$ and
$\mathcal{L}_{X}\alpha^{\prime}_{u}=r^{\prime}_{u}\,\alpha^{\prime}_{u}$.

Reciprocally, if $\rho_{s},\rho_{u}:M\rightarrow\mathbb{R}$ are functions as
above satisfying (3.3), then
$(\alpha^{\prime}_{s},\alpha^{\prime}_{u}):=(e^{\rho_{s}}\alpha_{s},e^{\rho_{u}}\alpha_{u})$
is also a defining pair for $\Phi$.

It follows that $\mathcal{D}_{\Phi}$ is homeomorphic to

$\displaystyle\mathcal{R}:=\big{\\{}(\rho_{s},\rho_{u})\in\mathcal{C}^{0}_{X}\times\mathcal{C}^{0}_{X}:X\cdot(\rho_{u}-\rho_{s})+r_{u}-r_{s}>0\big{\\}},$

and $\mathcal{R}$ is obviously convexe, hence contractible. The proof for
Anosov flows and volume preserving Anosov flows is similar. The condition on
$\rho_{s}$ and $\rho_{u}$ becomes

$\displaystyle X\cdot\rho_{u}+r_{u}>0\qquad and\qquad X\cdot\rho_{s}+r_{s}<0$
(3.4)

if $\Phi$ is Anosov, and

$\displaystyle X\cdot(\rho_{u}-\rho_{s})+r_{u}-r_{s}>0\qquad and\qquad
X\cdot(\rho_{u}+\rho_{s})=0$ (3.5)

if $\Phi$ is volume preserving Anosov. Both conditions (3.4) and (3.5) are
convex in $(\rho_{s},\rho_{u})$. ∎

######  Remark 3.3.

If $\Phi$ is a (projectively, volume preserving) Anosov flow generated by a
vector field $X$, $f:M\rightarrow\mathbb{R}_{>0}$ is a positive function and
$\Phi^{\prime}$ is the (projectively, volume preserving) Anosov flow generated
by $fX$, then $\mathcal{D}_{\Phi}=\mathcal{D}_{\Phi^{\prime}}$. Hence, there
is a well-defined notion of defining pairs for (projectively, volume
preserving) oriented Anosov _line distributions_.

###  3.2 From Anosov flows to Anosov Liouville pairs

Throughout this section, we assume that $\Phi$ is a smooth Anosov flow on $M$
and we construct an AL pair supporting $\Phi$, proving the first part of
Theorem 1. We choose a $\mathcal{C}^{1}$ defining pair
$(\alpha_{s},\alpha_{u})\in\mathcal{D}_{\Phi}$ as in Lemma 3.1(2). Following
[Hoz22a, Section 4], we define

$\displaystyle\alpha_{-}:=\alpha_{u}+\alpha_{s},\qquad\alpha_{+}:=\alpha_{u}-\alpha_{s}.$
(3.6)

Note that $\alpha_{\pm}$ is of class $\mathcal{C}_{X}^{1}$, and the
orientation compatibility conditions of Definition 2.1 are satisfied. Let
$\mathrm{dvol}$ be the $\mathcal{C}^{1}$ volume form on $M$ defined by
$\alpha_{s}\wedge\alpha_{u}=\iota_{X}\mathrm{dvol}$. Using the elementary
identities

$\displaystyle\alpha_{s}\wedge d\alpha_{u}$
$\displaystyle=-r_{u}\,\mathrm{dvol},$ $\displaystyle\alpha_{u}\wedge
d\alpha_{s}$ $\displaystyle=r_{s}\,\mathrm{dvol},$

we easily obtain

$\displaystyle\alpha_{+}\wedge d\alpha_{+}$
$\displaystyle=(r_{u}-r_{s})\,\mathrm{dvol},$ $\displaystyle\alpha_{-}\wedge
d\alpha_{-}$ $\displaystyle=-(r_{u}-r_{s})\,\mathrm{dvol},$ $\displaystyle
d(\alpha_{-}\wedge\alpha_{+})$ $\displaystyle=2(r_{u}+r_{s})\,\mathrm{dvol}.$

Since $r_{s}<0<r_{u}$, $\alpha_{-}$ and $\alpha_{+}$ are contact forms and the
criterion of Lemma 2.3 is satisfied.222Note that
$\lambda:=e^{-s}\alpha_{-}+e^{s}\alpha_{+}$ is a Liouville form if and only if
$r_{u}>0$. Therefore, $(\alpha_{-},\alpha_{+})$ is a $\mathcal{C}^{1}$ AL pair
supporting $\Phi$.

######  Definition 3.4.

A standard AL pair supporting $\Phi$ is a $\mathcal{C}^{1}$ AL pair obtained
by the previous construction. We denote by
$\mathcal{AL}^{\mathrm{std}}_{\Phi}$ the space of these AL pairs, endowed with
the $\mathcal{C}^{1}$ topology.

There is an obvious (linear) homeomorphism between
$\mathcal{AL}^{\mathrm{std}}_{\Phi}$ and $\mathcal{D}_{\Phi}$. Since the
latter is contractible by Lemma 3.3, $\mathcal{AL}^{\mathrm{std}}_{\Phi}$ is
contractible as well.

A standard AL pair $(\alpha_{-},\alpha_{+})$ is not necessarily smooth by
definition; it is smooth exactly when the weak-stable and weak-unstable
foliations of $\Phi$ are smooth, which is a quite restrictive situation.333By
[Ghy93, Theorem 4.7.], the smoothness of the weak-(un)stable implies that
$\Phi$ is topologically equivalent to an _algebraic Anosov flow_ , i.e., the
suspension of an Anosov diffeomorphism of the 2-torus, or the geodesic flow on
a closed hyperbolic surface, up to finite cover. Nevertheless, any pair of
smooth $1$-forms $(\alpha_{-}^{\prime},\alpha_{+}^{\prime})$ sufficiently
$\mathcal{C}^{1}$-close to $(\alpha_{-},\alpha_{+})$ and satisfying
$\alpha_{\pm}^{\prime}(X)=0$ is a smooth AL pair supporting $\Phi$. This shows
the forward implication in Theorem 1.

Writing $R_{\pm}$ for the Reeb vector fields of $\alpha_{\pm}$,
straightforward computations show

$\displaystyle\alpha_{s}(R_{-})$ $\displaystyle=\frac{r_{u}}{r_{u}-r_{s}}>0,$
$\displaystyle\alpha_{u}(R_{-})$ $\displaystyle=\frac{-r_{s}}{r_{u}-r_{s}}>0,$
$\displaystyle\alpha_{s}(R_{+})$ $\displaystyle=\frac{-r_{u}}{r_{u}-r_{s}}<0,$
$\displaystyle\alpha_{u}(R_{+})$ $\displaystyle=\frac{-r_{s}}{r_{u}-r_{s}}>0.$

Therefore,

  * •

$R_{-}$ is positively transverse to $E^{ws}$ and $E^{wu}$,

  * •

$R_{+}$ is positively transverse to $E^{ws}$ and negatively transverse to
$E^{wu}$,

and this remains true for a smoothing of $(\alpha_{-},\alpha_{+})$ as
above.444These transversality properties for the Reeb vector fields are a key
feature of Anosov flows and are not satisfied for projectively Anosov flows
which are not Anosov, see [Hoz22a, Theorem 6.3.]. Since $\mathcal{F}^{ws}$ is
a _taut_ foliation, we obtain that $R_{\pm}$ has no contractible closed Reeb
orbit, thus $\xi_{\pm}=\ker\alpha_{\pm}$ is _hypertight_. This was already
observed by Hozoori [Hoz22a, Theorem 1.11.].

###  3.3 From Anosov Liouville pairs to Anosov flows

We now turn to the second part of the proof of Theorem 1. Let us assume that
$\Phi$ is a smooth non-singular flow on $M$ generated by a vector field $X$
and suppose that it is supported by an AL pair $(\alpha_{-},\alpha_{+})$. By
Proposition 2.1, $(\alpha_{-},\alpha_{+})$ defines a bi-contact structure
$(\xi_{-},\xi_{+})$ supporting $X$, so $\Phi$ is projectively Anosov and there
exists a dominated splitting
$N_{X}\cong\overline{E}^{s}\oplus\overline{E}^{u}$ as in Definition 3.1. We
shall construct a defining pair $(\alpha_{s},\alpha_{u})$ as in Lemma 3.1(2),
implying that $\Phi$ is Anosov.

The proof of [ET98, Proposition 2.2.6.] (see also [Hoz22a, Remark 3.10.])
shows that $\xi_{\pm}$ is everywhere transverse to $E^{ws}$ and $E^{wu}$. By
our orientation conventions, there exist two continuous functions
$\sigma_{s},\sigma_{u}:M\rightarrow\mathbb{R}$ such that

$\displaystyle\ker\big{\\{}e^{-\sigma_{u}}\alpha_{-}+e^{\sigma_{u}}\alpha_{+}\big{\\}}$
$\displaystyle=E^{ws},$
$\displaystyle\ker\big{\\{}e^{-\sigma_{s}}\alpha_{-}-e^{\sigma_{s}}\alpha_{+}\big{\\}}$
$\displaystyle=E^{wu}.$

Note that $\sigma_{u}$ and $\sigma_{s}$ are also continuously differentiable
along $X$. Indeed, if $\overline{e}_{s}$ is any vector field of class
$\mathcal{C}^{0}_{X}$ spanning $\overline{E}^{s}\subset N_{X}$, then

$e^{\sigma_{u}}\alpha_{+}(\overline{e}_{s})+e^{-\sigma_{u}}\alpha_{-}(\overline{e}_{s})=0,$

hence

$\sigma_{u}=\frac{1}{2}\ln\left(-\frac{\alpha_{-}(\overline{e}_{s})}{\alpha_{+}(\overline{e}_{s})}\right),$

and this quantity is continuously differentiable along $X$; the same argument
applies to $\sigma_{s}$.

We define

$\displaystyle\alpha_{u}$
$\displaystyle:=\frac{1}{2\big{(}\cosh(\sigma_{u}-\sigma_{s})\big{)}^{3/2}}\big{(}e^{-\sigma_{u}}\alpha_{-}+e^{\sigma_{u}}\alpha_{+}\big{)},$
$\displaystyle\alpha_{s}$
$\displaystyle:=\frac{1}{2\big{(}\cosh(\sigma_{u}-\sigma_{s})\big{)}^{3/2}}\big{(}e^{-\sigma_{s}}\alpha_{-}-e^{\sigma_{s}}\alpha_{+}\big{)},$

so that

$\displaystyle\ker\alpha_{u}$ $\displaystyle=E^{ws},$
$\displaystyle\ker\alpha_{s}$ $\displaystyle=E^{wu},$

and

$\displaystyle\alpha_{-}$
$\displaystyle=\frac{1}{\sqrt{\cosh(\sigma_{u}-\sigma_{s})}}\big{(}e^{\sigma_{s}}\alpha_{u}+e^{\sigma_{u}}\alpha_{s}\big{)},$
(3.7) $\displaystyle\alpha_{+}$
$\displaystyle=\frac{1}{\sqrt{\cosh(\sigma_{u}-\sigma_{s})}}\big{(}e^{-\sigma_{s}}\alpha_{u}-e^{-\sigma_{u}}\alpha_{s}\big{)}.$
(3.8)

Note that $\alpha_{u}$ and $\alpha_{s}$ are continuously differentiable along
$X$, and since $E^{ws}$ and $E^{wu}$ are invariant under $\Phi$, there exist
continuous functions $r_{s},r_{u}:M\rightarrow\mathbb{R}$ such that

$\displaystyle\mathcal{L}_{X}\alpha_{s}$ $\displaystyle=r_{s}\,\alpha_{s},$
$\displaystyle\mathcal{L}_{X}\alpha_{u}$ $\displaystyle=r_{u}\,\alpha_{u}.$

Moreover,

$\displaystyle\alpha_{-}\wedge\alpha_{+}=2\,\alpha_{s}\wedge\alpha_{u},$

so $\overline{\alpha}_{s}\wedge\overline{\alpha}_{u}>0$. We are left to show
that $r_{s}<0<r_{u}$, which will follow from Lemma 2.3. Let $\mathrm{dvol}$ be
the unique volume form on $M$ such that
$\alpha_{s}\wedge\alpha_{u}=\iota_{X}\,\mathrm{dvol}=:\tau$.

######  Lemma 3.4.

With the same notations as in Lemma 2.3, we have

$\displaystyle f_{+}$
$\displaystyle=\frac{e^{-(\sigma_{s}+\sigma_{u})}}{\cosh(\sigma_{u}-\sigma_{s})}\left(X\cdot(\sigma_{u}-\sigma_{s})+r_{u}-r_{s}\right),$
(3.9) $\displaystyle f_{-}$
$\displaystyle=\frac{e^{(\sigma_{s}+\sigma_{u})}}{\cosh(\sigma_{u}-\sigma_{s})}\left(-X\cdot(\sigma_{u}-\sigma_{s})+r_{u}-r_{s}\right),$
(3.10) $\displaystyle f_{0}$ $\displaystyle=2(r_{u}+r_{s}).$ (3.11)

###### Proof.

The quantities $\alpha_{s}$, $\alpha_{u}$, $\sigma_{s}$ and $\sigma_{u}$ are
not $\mathcal{C}^{1}$ so we cannot compute $d\alpha_{\pm}$ directly from (3.7)
and (3.8). However, these quantities are differentiable along $X$, and

$\displaystyle\alpha_{+}\wedge\mathcal{L}_{X}\alpha_{+}$
$\displaystyle=-f_{+}\,\tau,$
$\displaystyle\alpha_{-}\wedge\mathcal{L}_{X}\alpha_{-}$
$\displaystyle=f_{-}\,\tau,$
$\displaystyle\mathcal{L}_{X}\alpha_{-}\wedge\alpha_{+}+\alpha_{-}\wedge\mathcal{L}_{X}\alpha_{+}$
$\displaystyle=f_{0}\,\tau,$

and $\mathcal{L}_{X}\alpha_{\pm}$ _can_ be computed from (3.7) and (3.8). The
calculations are left to the reader. ∎

Since $f_{\pm}>0$, (3.9) and (3.10) imply

$0\leq|X\cdot\sigma|<r_{u}-r_{s},$

where $\sigma:=\sigma_{u}-\sigma_{s}$, and the inequality
$f_{0}^{2}<4f_{-}f_{+}$ gives

$(r_{u}+r_{s})^{2}\leq\cosh^{2}(\sigma)(r_{u}+r_{s})^{2}<(r_{u}-r_{s})^{2}-\big{(}X\cdot\sigma\big{)}^{2}\leq(r_{u}-r_{s})^{2},$

yielding $r_{s}<0<r_{u}$ as desired. This concludes the proof of Theorem 1.

######  Remark 3.4.

Similar computations (and Lemma A.2) show that if $\Phi$ is a non-degenerate
flow on $M$, the following are equivalent:

  1. (1)

$\Phi$ is supported by a transverse Liouville pair $(\alpha_{-},\alpha_{+})$,

  2. (2)

$\Phi$ is projectively Anosov and admits a defining pair
$(\alpha_{s},\alpha_{u})$ with $r_{u}>0$.

Note that in case, the Reeb vector fields for the standard construction of
Section 3.2 are still transverse to the weak-unstable bundle of $\Phi$, but
are not necessarily transverse to the weak-stable bundle of $\Phi$. We wish to
call $\Phi$ a semi-Anosov flow. Our techniques would also show that the space
of semi-Anosov flows on $M$ is homotopy equivalent to the space of transverse
Liouville pairs on $M$.

###  3.4 Volume preserving Anosov flows

Volume preserving Anosov flows, i.e., Anosov flows preserving a volume form,
constitute a remarkable class of Anosov flows. They are topologically
transitive, in the sense that they admit a dense orbit. A deep theorem of
Asaoka [Asa08] implies that on closed $3$-manifold, every transitive Anosov
flow is topologically equivalent to a volume preserving one. In this section,
we show some striking connections between volume preserving Anosov flows and
Anosov Liouville pairs.

######  Proposition 3.1.

Let $\Phi$ be a smooth non-singular flow on $M$. Then $\Phi$ is a volume
preserving Anosov flow if and only if it is supported by a closed AL pair.

###### Proof.

Let us first assume that $\Phi$ preserves a (smooth) volume form
$\mathrm{dvol}$, and let $\tau:=\iota_{X}\mathrm{dvol}$. Note that $\tau$ is
_closed_. Let $(\xi_{-},\xi_{+})$ be any bi-contact structure supporting
$\Phi$, and $\alpha_{\pm}$ two contact forms such that
$\ker\alpha_{\pm}=\xi_{\pm}$. There exists a smooth positive function
$\kappa:M\rightarrow\mathbb{R}_{>0}$ such that

$\alpha_{-}\wedge\alpha_{+}=\kappa\,\tau.$

Then, $\left(\alpha_{-},\frac{1}{\kappa}\alpha_{+}\right)$ is a closed AL pair
supporting $\Phi$.

Let us now assume that $\Phi$ is supported by a closed AL pair
$(\alpha_{-},\alpha_{+})$. By Theorem 1, $\Phi$ is Anosov. Let $\theta$ be any
smooth $1$-form on $M$ satisfying $\theta(X)\equiv 1$, where $X$ is the vector
field generating $\Phi$, and define
$\mathrm{dvol}:=\alpha_{-}\wedge\alpha_{+}\wedge\theta$. It is easy to check
that it is a volume form, and

$\displaystyle\mathcal{L}_{X}\mathrm{dvol}$
$\displaystyle=\mathcal{L}_{X}(\alpha_{-}\wedge\alpha_{+})\wedge\theta+\alpha_{-}\wedge\alpha_{+}\wedge\mathcal{L}_{X}\theta$
$\displaystyle=\alpha_{-}\wedge\alpha_{+}\wedge d\theta(X,\cdot\,)$
$\displaystyle=0,$

hence $\Phi$ preserves a smooth volume form. ∎

######  Remark 3.5.

A special class of closed AL pairs is given by Geiges pairs, defined in
[MNW13, Section 8.5] as pairs of contact forms $(\alpha_{-},\alpha_{+})$
satisfying

$\displaystyle\alpha_{+}\wedge d\alpha_{+}$ $\displaystyle=-\alpha_{-}\wedge
d\alpha_{-}>0,$ $\displaystyle\alpha_{+}\wedge d\alpha_{-}$
$\displaystyle=\alpha_{-}\wedge d\alpha_{+}=0.$

Geiges pairs are called $(-1)$-Cartan structures in [Hoz22], and they are
shown to be in correspondence with volume preserving Anosov flows. Here, we
note that $(\alpha_{-},\alpha_{+})$ is a $\mathcal{C}^{1}$ Geiges pair if and
only if $(\alpha_{-}-\alpha_{+},\alpha_{-}+\alpha_{+})$ is a defining pair for
the underlying volume preserving Anosov flow. As a result, the space of Geiges
pairs supporting a given flow is contractible. Not every (smooth) volume
preserving Anosov flow is supported by a smooth (or even $\mathcal{C}^{2}$)
Geiges pair, as it would imply that the weak-stable and weak-unstable bundles
are $\mathcal{C}^{2}$, so the flow would be smoothly equivalent to an
algebraic Anosov flow; see [Ghy92, Théorème A.].

The previous proof shows more: for a volume preserving Anosov flow, _any_
supporting bi-contact structure can be realized as the kernel of an AL pair.
Surprisingly, this is a characteristic feature of volume preserving Anosov
flows.

######  Theorem 3.1.

Let $\Phi$ be a smooth Anosov flow on $M$. Then $\Phi$ preserves a volume form
if and only if for every (smooth) supporting bi-contact structure
$(\xi_{-},\xi_{+})$, there exists an AL pair $(\alpha_{-},\alpha_{+})$ such
that $\xi_{\pm}=\ker\alpha_{\pm}$.

###### Proof.

The forward direction follows from the first part of the proof of Proposition
3.1. Let us assume that every (smooth) bi-contact structure
$(\xi_{-},\xi_{+})$ supporting $\Phi$ is defined by a (smooth) AL pair. Let
$A>0$ be a positive real number and $\\{\alpha^{n}_{u}\\}_{n\in\mathbb{N}}$
and $\\{\alpha^{n}_{s}\\}_{n\in\mathbb{N}}$ be sequences of smooth $1$-forms
converging to $\alpha_{u}$ and $\alpha_{s}$, respectively, in the
$\mathcal{C}^{1}$ topology. For every $n\in\mathbb{N}$, we define

$\displaystyle\alpha^{n}_{+}$
$\displaystyle:=\alpha^{n}_{u}-e^{-A}\alpha^{n}_{s},$
$\displaystyle\alpha^{n}_{-}$
$\displaystyle:=\alpha^{n}_{u}+e^{A}\alpha^{n}_{s}.$

Then, for $n$ sufficiently large (depending on $A$),
$(\alpha^{n}_{-},\alpha^{n}_{+})$ defines a (smooth) bi-contact structure
$(\xi^{n}_{-},\xi^{n}_{+})$ supporting $\Phi$. By assumption, there exists a
smooth positive function $f_{n}:M\rightarrow\mathbb{R}_{>0}$ such that
$(\alpha^{n}_{-},f_{n}\alpha^{n}_{+})$ is an AL pair defining
$(\xi^{n}_{-},\xi^{n}_{+})$. Lemma 2.3 will imply the following

#### Claim.

For every $\epsilon>0$, there exists a smooth function
$h_{\epsilon}:M\rightarrow\mathbb{R}$ such that

$\displaystyle|X\cdot h_{\epsilon}+r_{u}+r_{s}|\leq\epsilon.$ (3.12)

Assuming the Claim for now, it follows that if $\theta$ is a smooth $1$-form
such that $\theta(X)\equiv 1$, then for every closed orbit $\gamma$ of $X$,

$\int_{\gamma}(r_{u}+r_{s})\,\theta=0.$

If the flow is transitive, a classical theorem of Livšic implies that there
exists a continuous function $h:M\rightarrow\mathbb{R}$ which is
differentiable along $X$ and satisfies

$X\cdot h+r_{u}+r_{s}=0.$

Writing
$\mathrm{dvol}^{\prime}:=e^{h}\,\mathrm{dvol}=e^{h}\,\alpha_{s}\wedge\alpha_{u}\wedge\theta$,
$\mathcal{L}_{X}\mathrm{dvol}^{\prime}=0$ so $\Phi$ preserves a positive
continuous measure, and by [LMM86, Corollary 2.1.], this measure is a smooth
volume form.

It turns out that the condition (3.12) _implies_ that the flow is transitive.
We have not been able to find a proof of this fact in the literature. We refer
to Appendix B for a proof using the theory of Sinaï–Ruelle–Bowen measures.

We now prove the Claim. Let $\epsilon>0$ and choose $A>0$ such that

$\frac{\sup_{M}(r_{u}-r_{s})}{\cosh(A)}\leq\epsilon/3.$

Let $e_{s}$ and $e_{u}$ be $\mathcal{C}^{1}$ vector fields satisfying

$\displaystyle\alpha_{s}(e_{s})=1,$ $\displaystyle\alpha_{s}(e_{u})=0,$
$\displaystyle\alpha_{u}(e_{s})=0,$ $\displaystyle\alpha_{u}(e_{u})=1,$

so that
$\alpha_{s}\wedge\alpha_{u}(e_{s},e_{u})=\mathrm{dvol}(X,e_{s},e_{u})=1$.
Since $(\alpha^{n}_{-},f_{n}\alpha^{n}_{+})$ is an AL pair for $n$ large
enough, Lemma 2.3 implies the inequality

$\displaystyle\left|X\cdot\ln
f_{n}+\frac{d(\alpha^{n}_{-}\wedge\alpha^{n}_{+})(X,e_{s},e_{u})}{\alpha^{n}_{-}\wedge\alpha^{n}_{+}(e_{s},e_{u})}\right|<\frac{2\sqrt{-\left(\alpha^{n}_{-}\wedge
d\alpha^{n}_{-}(X,e_{s},e_{u})\right)\cdot\left(\alpha^{n}_{+}\wedge
d\alpha^{n}_{+}(X,e_{s},e_{u})\right)}}{|\alpha^{n}_{-}\wedge\alpha^{n}_{+}(e_{s},e_{u})|}.$

Since

$\displaystyle\lim_{n\rightarrow\infty}\,\frac{d(\alpha^{n}_{-}\wedge\alpha^{n}_{+})(X,e_{s},e_{u})}{\alpha^{n}_{-}\wedge\alpha^{n}_{+}(e_{s},e_{u})}$
$\displaystyle=r_{u}+r_{s},$
$\displaystyle\lim_{n\rightarrow\infty}\,\frac{2\sqrt{-\left(\alpha^{n}_{-}\wedge
d\alpha^{n}_{-}(X,e_{s},e_{u})\right)\cdot\left(\alpha^{n}_{+}\wedge
d\alpha^{n}_{+}(X,e_{s},e_{u})\right)}}{|\alpha^{n}_{-}\wedge\alpha^{n}_{+}(e_{s},e_{u})|}$
$\displaystyle=\frac{r_{u}-r_{s}}{\cosh(A)},$

we obtain that for $n$ large enough such that the above two quantities are
$\epsilon/3$-close to their limits,

$\displaystyle|X\cdot\ln
f_{n}+r_{u}+r_{s}|\leq\frac{r_{u}-r_{s}}{\cosh(A)}+\epsilon/3+\epsilon/3\leq\epsilon,$

hence $h_{\epsilon}:=\ln f_{n}$ satisfies the required inequality and the
Claim is proved. ∎

######  Remark 3.6.

The proof can be adapted to show that if every bi-contact structure supporting
$\Phi$ is realized as the kernel of a Liouville pair, then the determinant of
the Poincaré return map for every closed orbit of $\Phi$ is bigger or equal
than $1$. We expect that this should also imply that $\Phi$ is volume
preserving.

##  4 Spaces of Anosov Liouville pairs and bi-contact structures

This section is dedicated to the proof of Theorem 2 from the Introduction. We
first describe the main strategy in a more general setting. Let $E$ and $B$ be
topological spaces and $f:E\rightarrow B$ be a continuous map. We can assume
that $E$ and $B$ have the homotopy type of CW complexes. This is the case for
the spaces we consider (e.g., $\mathcal{AL}$, $\mathcal{BC}$, $\mathcal{AF}$,
$\mathbb{P}\mathcal{AF}$, etc.) as they are open subsets of Fréchet
spaces.555Indeed, Fréchet spaces are absolute neighborhood retracts (ANRs) by
a theorem of Dugundji; an open subset of an ANR is an ANR; every ANR has the
homotopy type of a CW complex by a theorem of Milnor and Whitehead. However,
it is known that an open subset of an infinite dimensional Fréchet space is
_not_ a CW complex. To show that $f$ is a homotopy equivalence, it is enough
to show that it is a Serre fibration with (weakly) contractible fibers.
However, it seems rather hard to show that the maps we care about (e.g.,
$\mathcal{I}$, $\mathbb{P}\mathcal{I}$) satisfy a homotopy lifting property,
as this would require a careful understanding of how the stable and unstable
bundles depend on the (projectively) Anosov flow. Instead, we choose a more
indirect approach: we first show that these maps have contractible fibers, and
we then show that they are _topological submersions_.

Figure 3: A topological submersion.

######  Definition 4.1.

$f:E\rightarrow B$ is a topological submersion if it is surjective, and for
every $x\in E$, there exists a neighborhood $U$ of $x$ in $E$ such that if we
write $y:=f(x)$ and $V:=U\cap f^{-1}(y)$, there exists a homeomorphism
$U\overset{\sim}{\longrightarrow}f(U)\times V$ making the following diagram
commute:

${U}$${f(U)\times
V}$${f(U)}$$\scriptstyle{\sim}$$\scriptstyle{f}$$\scriptstyle{\mathrm{pr}_{1}}$

Here, $\mathrm{pr}_{1}$ denotes the projection onto the first factor.

######  Lemma 4.1.

If $f:E\rightarrow B$ is a topological submersion with (weakly) contractible
fibers, then $f$ an acyclic Serre fibration.

###### Proof.

Since projections are open and openness is a local property, $f$ is open. By
[Mei02, Lemma 6], $f$ is a _homotopic submersion_ (see [Mei02, Definition 1]),
also known as a _Serre micro-fibration_ [Gro86]. The result then follows from
[Mei02, Corollary 13] (see also [Wei05, Lemma 2.2.]). ∎

###  4.1 Contractibility of fibers

In this section, we show:

######  Theorem 4.1.

Let $\Phi$ be a smooth Anosov flow on $M$. The spaces of AL pairs and weak AL
pairs supporting $\Phi$ are both contractible.

We also show a similar result for projectively Anosov flows:

######  Theorem 4.2.

Let $\Phi$ be a smooth projectively Anosov flow on $M$. The space of bi-
contact structures supporting $\Phi$ is contractible.

We obtain a version for volume preserving Anosov flows as well:

######  Theorem 4.3.

Let $\Phi$ be a smooth volume preserving Anosov flow on $M$. The space of
closed AL pairs supporting $\Phi$ is contractible.

If $\Phi$ is a smooth Anosov flow on $M$,

  * •

$\mathcal{AL}_{\Phi}$ denotes the space of smooth AL pairs on $M$ supporting
$\Phi$, endowed with the $\mathcal{C}^{\infty}$ topology,

  * •

$\mathcal{AL}^{1}_{\Phi}$ denotes the space of $\mathcal{C}^{1}$ AL pairs on
$M$ supporting $\Phi$, endowed with the $\mathcal{C}^{1}$ topology.

Smooth AL pairs supporting $\Phi$ form a dense subset of
$\mathcal{AL}^{1}_{\Phi}$. Recall that $\mathcal{D}_{\Phi}$ denotes the space
of $\mathcal{C}^{1}$ defining pairs on $M$ for $\Phi$, and
$\mathcal{AL}^{\mathrm{std}}_{\Phi}\subset\mathcal{AL}^{1}_{\Phi}$ denotes the
space of $\mathcal{C}^{1}$ standard AL pairs supporting $\Phi$, both endowed
with the $\mathcal{C}^{1}$ topology. Theorem 4.1 will be a consequence of the
contractibility of $\mathcal{AL}^{\mathrm{std}}_{\Phi}$ and the following two
lemmas.

######  Lemma 4.2.

The natural map $\mathcal{AL}_{\Phi}\rightarrow\mathcal{AL}^{1}_{\Phi}$ is a
homotopy equivalence.

###### Proof.

This follows from some standard facts in algebraic topology. We will use that
homotopy equivalences are _local_ in the following sense (see [Die71, Theorem
1.]):

#### Fact.

A continuous map $f:X\rightarrow Y$ between topological spaces is a homotopy
equivalence if there exists a _numerable_ open cover $\mathcal{U}$ of $Y$
satisfying

  1. (1)

$\mathcal{U}$ is stable under finite intersections,

  2. (2)

For every $U\in\mathcal{U}$, $f:f^{-1}(U)\rightarrow U$ is a homotopy
equivalence.

In our context, covers are automatically numerable since all the spaces under
consideration are metrizable. We can simply cover $\mathcal{AL}^{1}_{\Phi}$ by
sufficiently small $\mathcal{C}^{1}$ balls and refine this cover by taking all
possible finite intersections. The restrictions of these balls to
$\mathcal{AL}^{1}_{\Phi}$ are convex and so are finite intersections thereof,
so all of the open subsets in our cover are contractible. Since smooth AL
pairs supporting $\Phi$ are dense in $\mathcal{AL}^{1}_{\Phi}$, the preimage
of a $\mathcal{C}^{1}$ ball under $f$ is non-empty. It is also convex, and so
are the preimages of finite intersections of balls. Thus, the conditions (1)
and (2) of the Fact are trivially satisfied. ∎

######  Lemma 4.3.

$\mathcal{AL}^{\mathrm{std}}_{\Phi}$ is a strong deformation retract of
$\mathcal{AL}^{1}_{\Phi}$.

###### Proof.

Let $(\alpha_{-},\alpha_{+})\in\mathcal{AL}^{1}_{\Phi}$. As in Section 3.3,
there exist functions $\sigma_{s},\sigma_{u}:M\rightarrow\mathbb{R}$
satisfying

$\displaystyle\ker\big{\\{}e^{-\sigma_{u}}\alpha_{-}+e^{\sigma_{u}}\alpha_{+}\big{\\}}$
$\displaystyle=E^{ws},$
$\displaystyle\ker\big{\\{}e^{-\sigma_{s}}\alpha_{-}-e^{\sigma_{s}}\alpha_{+}\big{\\}}$
$\displaystyle=E^{wu}.$

If $e_{s}$ and $e_{u}$ are $\mathcal{C}^{1}$ vector fields such that
$\overline{e}_{s}$ spans $\overline{E}^{s}$ and $\overline{e}_{u}$ spans
$\overline{E}^{u}$, we can write

$\displaystyle\sigma_{s}=\frac{1}{2}\ln\left(\frac{\alpha_{+}(e_{u})}{\alpha_{-}(e_{u})}\right),~\qquad\sigma_{u}=\frac{1}{2}\ln\left(-\frac{\alpha_{-}(e_{s})}{\alpha_{+}(e_{s})}\right),$

so $\sigma_{s}$ and $\sigma_{u}$ are $\mathcal{C}^{1}$. Moreover, the map
$\mathcal{S}:(\alpha_{-},\alpha_{+})\mapsto(\sigma_{s},\sigma_{u})$ is
continuous in the $\mathcal{C}^{1}$ topology. As before, we define

$\displaystyle\alpha_{u}$
$\displaystyle:=\frac{1}{2\big{(}\cosh(\sigma_{u}-\sigma_{s})\big{)}^{3/2}}\big{(}e^{-\sigma_{u}}\alpha_{-}+e^{\sigma_{u}}\alpha_{+}\big{)},$
$\displaystyle\alpha_{s}$
$\displaystyle:=\frac{1}{2\big{(}\cosh(\sigma_{u}-\sigma_{s})\big{)}^{3/2}}\big{(}e^{-\sigma_{s}}\alpha_{-}-e^{\sigma_{s}}\alpha_{+}\big{)}.$

The computations of Section 3.3 show that
$(\alpha_{s},\alpha_{u})\in\mathcal{D}_{\Phi}$. Therefore, we obtain a
continuous map
$\mathcal{D}:(\alpha_{-},\alpha_{+})\mapsto(\alpha_{s},\alpha_{u})$. We now
define a strong deformation retraction
$r:\mathcal{AL}^{1}_{\Phi}\times[0,1]\rightarrow\mathcal{AL}^{1}_{\Phi}$. Let
$(\alpha_{-},\alpha_{+})\in\mathcal{AL}^{1}_{\Phi}$, and
$(\sigma_{s},\sigma_{u})=\mathcal{S}(\alpha_{-},\alpha_{+})$ and
$(\alpha_{s},\alpha_{u})=\mathcal{D}(\alpha_{-},\alpha_{+})$ as before. For
$t\in[0,1]$, we define

$\displaystyle\alpha^{t}_{-}$
$\displaystyle:=\frac{1}{\sqrt{\cosh((1-t)\sigma)}}\big{(}e^{(1-t)\sigma_{s}}\alpha_{u}+e^{(1-t)\sigma_{u}}\alpha_{s}\big{)},$
$\displaystyle\alpha^{t}_{+}$
$\displaystyle:=\frac{1}{\sqrt{\cosh((1-t)\sigma)}}\big{(}e^{-(1-t)\sigma_{s}}\alpha_{u}-e^{-(1-t)\sigma_{u}}\alpha_{s}\big{)},$

where $\sigma=\sigma_{u}-\sigma_{s}$. We then have

  * •

$\big{(}\alpha^{0}_{-},\alpha^{0}_{+}\big{)}=(\alpha_{-},\alpha_{+})$,

  * •

$\big{(}\alpha^{1}_{-},\alpha^{1}_{+}\big{)}=\big{(}\alpha_{u}+\alpha_{s},\alpha_{u}-\alpha_{s}\big{)}\in\mathcal{AL}^{\mathrm{std}}_{\Phi}$,

  * •

If $(\alpha_{-},\alpha_{+})\in\mathcal{AL}^{\mathrm{std}}_{\Phi}$, then
$\big{(}\alpha^{t}_{-},\alpha^{t}_{+}\big{)}=(\alpha_{-},\alpha_{+})$ for
every $t\in[0,1]$.

We claim that $\big{(}\alpha^{t}_{-},\alpha^{t}_{+}\big{)}$ is an AL pair for
every $t\in[0,1]$. Indeed, by Lemma 2.3 and Lemma 3.4, it is enough to show
the inequality

$\displaystyle\cosh^{2}\big{(}(1-t)\sigma\big{)}(r_{u}+r_{s})^{2}+(1-t)^{2}\big{(}X\cdot\sigma\big{)}^{2}<(r_{u}-r_{s})^{2}.$
(4.1)

It holds for $t=0$ since $(\alpha_{-},\alpha_{+})$ is an AL pair, and the
left-hand side is obviously a non-increasing function of $t$, so it holds for
every $t\in[0,1]$.

We finally define
$r\big{(}(\alpha_{-},\alpha_{+}),t\big{)}:=\big{(}\alpha^{t}_{-},\alpha^{t}_{+}\big{)}$
so that
$r:\mathcal{AL}^{1}_{\Phi}\times[0,1]\rightarrow\mathcal{AL}^{1}_{\Phi}$ is
continuous, and by the three bullets above, it is a strong deformation
retraction of $\mathcal{AL}^{1}_{\Phi}$ onto
$\mathcal{AL}^{\mathrm{std}}_{\Phi}$. ∎

###### Proof of Theorem 4.1.

For the case of AL pairs, combine Lemma 3.3, Lemma 4.2 and Lemma 4.3. For weak
AL pairs, the argument can be modified as follows. Lemma 4.2 can be easily
adapted to the case of weak AL pairs. Lemma 4.3 can be adapted to show that
the space of $\mathcal{C}^{1}$ weak AL pairs supporting a smooth Anosov flow
$\Phi$ deformation retracts onto the space of pairs of the form
$\alpha_{\pm}=\alpha_{u}\mp\alpha_{s}$, where $(\alpha_{s},\alpha_{u})$
satisfies the conditions of a defining pair for $\Phi$ _without the condition
$r_{s}<0$_ (but still satisfies $r_{s}<r_{u}$ and $0<r_{u}$, see Remark 3.4).
The latter space is convex, hence contractible. ∎

###### Proof of Theorem 4.2.

We only sketch how to modify the proof of Theorem 4.1 and we leave the details
to the interested reader.

First of all, we shall introduce the space of $\mathcal{C}^{0}_{X}$ bi-contact
structures. Those are continuous pairs of codimension $1$ distributions
$(\xi_{-},\xi_{+})$ which are continuously differentiable along $X$ and which
are defined by some $1$-forms $\alpha_{-},\alpha_{+}\in\Omega^{0}_{X}$
satisfying

$\displaystyle\overline{\alpha}_{-}\wedge\overline{\alpha}_{+}>0,$
$\displaystyle\overline{\alpha}_{-}\wedge\mathcal{L}_{X}\overline{\alpha}_{-}<0,$
$\displaystyle\overline{\alpha}_{+}\wedge\mathcal{L}_{X}\overline{\alpha}_{+}>0,$
(4.2)

as forms on $N_{X}$.

For the purpose of the proof, we choose an arbitrary $\mathcal{C}^{0}_{X}$
vector field $e_{u}$ such that $\overline{e}_{u}$ spans
$\overline{E}^{u}\subset N_{X}$ and defines the prescribed orientation. This
is equivalent to choosing a $1$-form $\alpha_{u}\in\Omega^{1}_{X,0}$ such that
$\mathrm{ker}\alpha_{u}=E^{ws}$ as oriented $2$-plane fields, with the
normalization $\alpha_{u}(e_{u})\equiv 1$.

We denote by $\mathcal{BC}_{\Phi}$ (resp. $\mathcal{BC}^{0}_{\Phi}$) the space
of smooth (resp. $\mathcal{C}^{0}_{X}$) bi-contact structures supporting
$\Phi$. We write
$\mathcal{BC}^{\mathrm{std}}_{\Phi}\subset\mathcal{BC}^{0}_{\Phi}$ for the
space of _standard_ bi-contact structures supporting $\Phi$, of the form
$\big{(}\ker\big{(}\alpha_{u}+\alpha_{s}\big{)},\ker\left(\alpha_{u}-\alpha_{s}\right)\big{)}$,
where $\alpha_{u}$ is fixed by the condition $\alpha_{u}(e_{u})\equiv 1$.

Lemma 4.2 can be easily adapted to show that the natural map
$\mathcal{BC}_{\Phi}\rightarrow\mathcal{BC}^{0}_{\Phi}$ is a homotopy
equivalence, using Lemma A.2.

Lemma 4.3 can be modified as follows. For
$(\xi_{-},\xi_{+})\in\mathcal{BC}^{0}_{\Phi}$, we denote by
$(\alpha_{-},\alpha_{+})$ the unique pair of $1$-forms in $\Omega^{0}_{X}$
satisfying $\ker\alpha_{\pm}=\xi_{\pm}$ and $\alpha_{\pm}(e_{u})=1$. We define
a $\mathcal{C}^{0}_{X}$ function $\sigma:M\rightarrow\mathbb{R}$ by

$\ker\big{\\{}e^{-\sigma}\alpha_{-}+e^{\sigma}\alpha_{+}\big{\\}}=E^{ws},$

so that

$\alpha_{u}=\frac{1}{2\cosh(\sigma)}\big{(}e^{-\sigma}\alpha_{-}+e^{\sigma}\alpha_{+}\big{)},$

and we define

$\alpha_{s}:=\frac{1}{2\cosh(\sigma)}\big{(}\alpha_{-}-\alpha_{+}\big{)}$

so that $\alpha_{s}\in\Omega^{1}_{X,0}$, and

$\mathrm{ker}\alpha_{s}=E^{wu}.$

This readily implies that

$\displaystyle\alpha_{-}$ $\displaystyle=\alpha_{u}+e^{\sigma}\alpha_{s},$
$\displaystyle\alpha_{+}$ $\displaystyle=\alpha_{u}-e^{-\sigma}\alpha_{s}.$

Writing $\mathcal{L}_{X}\alpha_{u}=r_{u}\alpha_{u}$ and
$\mathcal{L}_{X}\alpha_{s}=r_{s}\alpha_{s}$, where
$r_{s},r_{u}:M\rightarrow\mathbb{R}$ are continuous, the last two inequalities
in (4.2) are equivalent to

$|X\cdot\sigma|<r_{u}-r_{s}.$

Moreover,

$\alpha_{-}\wedge\alpha_{+}=2\cosh(\sigma)\alpha_{s}\wedge\alpha_{u}.$

This shows that $(\alpha_{s},\alpha_{u})$ is a defining pair for $\Phi$ that
satisfies $\alpha_{u}(e_{u})\equiv 1$. For $t\in[0,1]$, we define

$\displaystyle\alpha^{t}_{-}$
$\displaystyle:=\alpha_{u}+e^{(1-t)\sigma}\alpha_{s},$
$\displaystyle\xi^{t}_{-}:=\ker\alpha^{t}_{-}$ $\displaystyle\alpha^{t}_{+}$
$\displaystyle:=\alpha_{u}-e^{-(1-t)\sigma}\alpha_{s},$
$\displaystyle\xi^{t}_{+}:=\ker\alpha^{t}_{+}.$

These formulæ define a strong deformation retraction of
$\mathcal{BC}^{0}_{\Phi}$ onto $\mathcal{BC}^{\mathrm{std}}_{\Phi}$. Moreover,
$\mathcal{BC}^{\mathrm{std}}_{\Phi}$ is homeomorphic to the space of defining
pairs $(\alpha_{s},\alpha_{u})$ for $\Phi$ satisfying $\alpha_{u}(e_{u})\equiv
1$, and one easily checks that this space is contractible. ∎

###### Proof of Theorem 4.3.

The result essentially follows from Theorem 4.2. Let $\mathrm{dvol}$ be a
smooth volume form preserved by $\Phi$ and $\tau:=\iota_{X}\mathrm{dvol}$. If
$(\alpha_{-},\alpha_{+})$ is a closed AL pair supporting $\Phi$, there exists
a smooth positive function $\kappa:=M\rightarrow\mathbb{R}_{>0}$ such that

$\alpha_{-}\wedge\alpha_{+}=\kappa\,\tau.$

Moreover, $X\cdot\kappa=0$ and since $\Phi$ is topologically transitive,
$\kappa$ is constant. One easily checks that the space of closed AL pairs
supporting $\Phi$ is homotopy equivalent to the space of balanced pairs of
contact forms $(\alpha_{-},\alpha_{+})$ supporting $\Phi$ and satisfying
$\alpha_{-}\wedge\alpha_{+}=\tau$. We denote this space by
$\mathcal{BC}^{\tau}_{\Phi}$. There is a natural continuous map
$\mathcal{K}:\mathcal{BC}^{\tau}_{\Phi}\rightarrow\mathcal{BC}_{\Phi}$,
obtained by taking kernels, which is surjective by Theorem 3.1. One easily
checks that $\mathcal{K}$ is injective and that it is a homeomorphism. Theorem
4.2 finishes the proof. ∎

###  4.2 Homotopy equivalences

Let us recall some notations.

  * •

$\mathcal{AL}$ denotes the space of smooth AL pairs on $M$,

  * •

$\mathcal{AF}$ denotes the space of smooth Anosov flows on $M$, up to positive
time reparametrization,

  * •

$\mathbb{P}\mathcal{AF}$ denotes the space of smooth projectively Anosov flows
on $M$, up to positive time reparametrization.

Recall that there is a continuous map

$\begin{array}[]{rccc}\mathcal{I}:&\mathcal{AL}&\longrightarrow&\mathcal{AF}\\\
&(\alpha_{-},\alpha_{+})&\longmapsto&\ker\alpha_{-}\cap\ker\alpha_{+}\end{array}$

where we identify an oriented $1$-dimensional distribution with any flow
spanned by it. Similarly, there is a continuous map

$\begin{array}[]{rccc}\mathbb{P}\mathcal{I}:&\mathcal{BC}&\longrightarrow&\mathbb{P}\mathcal{AF}\\\
&(\xi_{-},\xi_{+})&\longmapsto&\xi_{-}\cap\xi_{+}\end{array}$

In this section, we show the main theorems of this article:

######  Theorem 4.4.

The map $\mathcal{I}$ is an acyclic Serre fibration.

Our argument can easily be adapted to the case of projectively Anosov flows
(this result might already be known to some experts):

######  Theorem 4.5.

The map $\mathbb{P}\mathcal{I}$ is an acyclic Serre fibration.

######  Remark 4.1.

With more work, it is possible to show that $\mathcal{I}$ is shrinkable: it is
homotopy equivalent over $\mathcal{AF}$ to
$\mathrm{id}:\mathcal{AF}\rightarrow\mathcal{AF}$. Concretely, this means that
there exists a section $s$ of $\mathcal{I}$ such that $s\circ\mathcal{I}$ is
_fiberwise homotopic_ to $\mathrm{id}$. This implies that the space sections
of $\mathcal{I}$ is non-empty and contractible. To prove this statement, one
would need to upgrade the results of Section 4.1 to hold _in family over
$\mathcal{AF}$_. A key ingredient is [LMM86, Lemma 2.1.], which shows that for
smooth Anosov flows, the Anosov splitting depends continuously on the flow. We
are not aware of a similar result for projectively Anosov flows.

We will need the following

######  Lemma 4.4.

$\mathcal{I}$ is a topological submersion.

###### Proof.

By Theorem 1, $\mathcal{I}$ is surjective. We fix some auxiliary Riemannian
metric $g$ on $M$ and identify $\mathcal{AF}$ with the space of unit Anosov
vector fields on $M$. Let
$\boldsymbol{\alpha}^{0}=(\alpha^{0}_{-},\alpha^{0}_{+})\in\mathcal{AL}$ and
let $X$ be the unit vector field generating
$\mathcal{I}(\boldsymbol{\alpha}^{0})$. We choose an arbitrary smooth $1$-form
$\theta$ such that $\theta(X)\equiv 1$. For a unit vector field $X^{\prime}$
sufficiently close to $X$ (so that $\theta(X^{\prime})>0$) and
$\boldsymbol{\alpha}=(\alpha_{-},\alpha_{+})\in\mathcal{AL}_{\Phi}$, we define

$\displaystyle\alpha^{\prime}_{\pm}$
$\displaystyle:=\alpha_{\pm}-\frac{\alpha_{\pm}(X^{\prime})}{\theta(X^{\prime})}\,\theta,$

so that $\alpha^{\prime}_{\pm}(X^{\prime})=0$. Since
$\mathcal{AL}\subset\Omega^{1}\times\Omega^{1}$ is open, we can find an open
neighborhood $\mathcal{N}_{\boldsymbol{\alpha}^{0}}$ of
$\boldsymbol{\alpha}^{0}$ in $\mathcal{AL}_{\Phi}$ and an open neighborhood
$\mathcal{N}_{\Phi}$ of $\Phi$ in $\mathcal{AF}$ such that the map

$\begin{array}[]{rccc}\psi:&\mathcal{N}_{\Phi}\times\mathcal{N}_{\boldsymbol{\alpha}^{0}}&\longrightarrow&\mathcal{AL}\\\
&\big{(}X^{\prime},\boldsymbol{\alpha}\big{)}&\longmapsto&\boldsymbol{\alpha}^{\prime}\end{array}$

is well-defined. It is continuous and satisfies
$\mathcal{I}\circ\psi=\mathrm{pr}_{1}$. Moreover, the restriction of $\psi$ to
$\\{X\\}\times\mathcal{N}_{\boldsymbol{\alpha}^{0}}$ is the inclusion
$\mathcal{N}_{\boldsymbol{\alpha}^{0}}\subset\mathcal{AL}$. One easily checks
that $\psi$ is injective, has open image and has an inverse given by
$\psi^{-1}(\boldsymbol{\alpha}^{\prime}):=\big{(}X^{\prime},\boldsymbol{\alpha}\big{)}$
where

$\displaystyle\alpha_{\pm}$
$\displaystyle:=\alpha^{\prime}_{\pm}-\alpha^{\prime}_{\pm}(X)\,\theta,$

and $X^{\prime}$ is the unit vector field spanning
$\mathcal{I}\big{(}\boldsymbol{\alpha}^{\prime}\big{)}$. Therefore,
$\psi^{-1}$ is a local trivialization of $\mathcal{I}$ around
$\boldsymbol{\alpha}^{0}$. ∎

###### Proof of Theorem 4.4.

By Theorem 4.1 and Lemma 4.4, $\mathcal{I}$ is a topological submersion with
contractible fibers, hence an acyclic Serre fibration by Lemma 4.1. ∎

###### Proof of Theorem 3.

Lemma 4.4 and its proof hold verbatim for $\mathcal{I}^{w}$ so the previous
proof applies to $\mathcal{I}^{w}$ as well; it is a homotopy equivalence by
the Whitehead theorem. ∎

###### Proof of Theorem 4.5.

By Theorem 4.2, we already know that the fibers of $\mathbb{P}\mathcal{I}$ are
contractible so it is enough to adapt Lemma 4.4. It can be done by choosing an
auxiliary smooth vector field $Z$, depending on an initial choice of
$\big{(}\xi^{0}_{-},\xi^{0}_{+}\big{)}\in\mathcal{BC}$, which is positively
transverse to $\xi^{0}_{\pm}$ and satisfies $\theta(Z)\equiv 0$. We can
uniquely choose contact forms $\alpha^{0}_{\pm}$ for $\xi^{0}_{\pm}$ by
imposing $\alpha_{\pm}(Z)\equiv 1$. The proof of Lemma 4.4 can be reproduced
with minor changes to provide a suitable trivialization near
$\big{(}\xi^{0}_{-},\xi^{0}_{+}\big{)}$. ∎

###  4.3 The kernel map

Recall that we have a continuous map

$\begin{array}[]{rccc}\underline{\ker}:&\mathcal{AL}&\longrightarrow&\mathcal{BC}\\\
&(\alpha_{-},\alpha_{+})&\longmapsto&(\ker\alpha_{-},\ker\alpha_{+})\end{array}$

######  Theorem 4.6.

The map $\underline{\ker}$ is an acyclic Serre fibration onto its image.

###### Proof.

As before, by Lemma 4.1, it is enough to show the following properties.

  1. (1)

The fibers of $\underline{\ker}$ over its image are contractible.

  2. (2)

$\underline{\ker}$ is a topological submersion onto its image.

We can simplify the situation by restricting to the space $\mathcal{AL}^{b}$
of _balanced_ AL pairs, since there is a homeomorphism

$\begin{array}[]{rccc}\vartheta:&\mathcal{C}^{\infty}(M,\mathbb{R})\times\mathcal{AL}^{b}&\overset{\sim}{\longrightarrow}&\mathcal{AL}\\\
&\big{(}\sigma,(\alpha_{-},\alpha_{+})\big{)}&\longmapsto&\big{(}e^{-\sigma}\alpha_{-},e^{\sigma}\alpha_{+}\big{)},\end{array}$

and $\underline{\ker}$ is compatible with this homeomorphism in the obvious
way.

Let us consider a bi-contact structure $(\xi_{-},\xi_{+})$ defined by a
balanced AL pair $(\alpha_{-},\alpha_{+})$, and let
$\mathrm{dvol}:=\alpha_{+}\wedge d\alpha_{+}$.

To show (1), note that any other balanced AL pair defining $(\xi_{-},\xi_{+})$
is of the form $(e^{\sigma}\alpha_{-},e^{\sigma}\alpha_{+})$ for a smooth
function $\sigma:M\rightarrow\mathbb{R}$ satisfying

$|2X\cdot\sigma+f_{0}|<2,$

where we use the notation of Lemma 2.3. By assumption, $|f_{0}|<2$, so the
space of $\sigma$ such that $(e^{\sigma}\alpha_{-},e^{\sigma}\alpha_{+})$ is a
balanced AL pair defining $(\xi_{-},\xi_{+})$ is convex, hence contractible.

To show (2), we consider an open neighborhood $\mathcal{V}$ of
$(\alpha_{-},\alpha_{+})$ in $\mathcal{AL}^{b}$. We can find a smaller
neighborhood $\mathcal{V}^{\prime}\subset\mathcal{V}$ such that for every
$(\alpha^{\prime}_{-},\alpha^{\prime}_{+})\in\mathcal{V}^{\prime}$, the pair

$\left(\widetilde{\alpha}^{\prime}_{-},\widetilde{\alpha}^{\prime}_{+}\right):=\left(\frac{1}{\sqrt{f^{\prime}}}\alpha^{\prime}_{-},\frac{1}{\sqrt{f^{\prime}}}~\alpha^{\prime}_{+}\right)$

is in $\mathcal{V}$, where

$\alpha^{\prime}_{\pm}\wedge d\alpha^{\prime}_{\pm}=\pm
f^{\prime}\,\mathrm{dvol}.$

Note that
$\left(\widetilde{\alpha}^{\prime}_{-},\widetilde{\alpha}^{\prime}_{+}\right)$
is a balanced AL pair satisfying $\widetilde{\alpha}^{\prime}_{+}\wedge
d\widetilde{\alpha}^{\prime}_{+}=\mathrm{dvol}$. The image of
$\mathcal{V}^{\prime}$ under $\underline{\ker}$ contains some open
neighborhood $\mathcal{U}^{\prime}$ of $(\xi_{-},\xi_{+})$ in $\mathcal{BC}$.
Let $\widetilde{\mathcal{V}}^{\prime}\subset\mathcal{V}^{\prime}$ be the
subspace of elements of the form
$\left(\widetilde{\alpha}^{\prime}_{-},\widetilde{\alpha}^{\prime}_{+}\right)$
for
$(\alpha^{\prime}_{-},\alpha^{\prime}_{+})\in\underline{\ker}^{-1}\left(\mathcal{U}^{\prime}\right)$.
One easily checks that
$\underline{\ker}:\widetilde{\mathcal{V}}^{\prime}\rightarrow\mathcal{U}^{\prime}$
is injective and open. It is surjective by definition, hence it is a
homeomorphism. By the previous paragraph, there is an open neighborhood of
$(\alpha_{-},\alpha_{+})$ in
$\underline{\ker}^{-1}\big{\\{}(\xi_{-},\xi_{+})\big{\\}}\cap\mathcal{AL}^{b}$
homeomorphic to

$\Sigma_{\epsilon}:=\left\\{\sigma:M\rightarrow\mathbb{R}:|X~\cdot\sigma|<\epsilon\right\\}$

for some small $\epsilon>0$. Therefore, after possibly shrinking $\epsilon$,
the map

$\begin{array}[]{ccc}\widetilde{\mathcal{V}}^{\prime}\times\Sigma_{\epsilon}&\longrightarrow&\mathcal{AL}^{b}\\\
\big{(}\big{(}\widetilde{\alpha}^{\prime}_{-},\widetilde{\alpha}^{\prime}_{+}\big{)},\sigma\big{)}&\longmapsto&\left(e^{\sigma}\widetilde{\alpha}^{\prime}_{-},e^{\sigma}\widetilde{\alpha}^{\prime}_{+}\right)\end{array}$

induces a local trivialization of
$\mathcal{AL}^{b}\rightarrow\underline{\ker}\left(\mathcal{AL}\right)$ around
$(\alpha_{-},\alpha_{+})$.

This proves that $\underline{\ker}$ restricted to $\mathcal{AL}^{b}$ is a
topological submersion with contractible fibers, and the same holds for
$\underline{\ker}$ on $\mathcal{AL}$ via the homeomorphism $\vartheta$. ∎

######  Remark 4.2.

The proof also shows that $\underline{\ker}$ is open, so its image has the
homotopy type of a CW complex.

There is an inclusion
$\underline{\ker}\big{(}\mathcal{AL}\big{)}\subset\mathbb{P}\mathcal{I}^{-1}\big{(}\mathcal{AF}\big{)}$
which is strict according to Theorem 3.1.666Indeed, any volume preserving
Anosov flow can be perturbed near a closed orbit in such a way that the new
Poincaré return map for this orbit has determinant different than $1$, so the
flow is not volume preserving anymore. In more concrete terms, there exist bi-
contact structures supporting Anosov flows which cannot be represented as the
kernel of a AL pairs. Nevertheless, we have:

######  Theorem 4.7.

The inclusion
$\underline{\ker}\big{(}\mathcal{AL}\big{)}\subset\mathbb{P}\mathcal{I}^{-1}\big{(}\mathcal{AF}\big{)}$
is a homotopy equivalence.

###### Proof.

It immediately follows from Theorem 4.4, Theorem 4.5, Theorem 4.6 and the
commutative diagram

${\mathcal{AL}}$${\underline{\ker}\big{(}\mathcal{AL}\big{)}}$${\mathbb{P}\mathcal{I}^{-1}\big{(}\mathcal{AF}\big{)}}$${\mathcal{BC}}$${\mathcal{AF}}$${\mathbb{P}\mathcal{AF}}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{\mathcal{I}}$$\subset$

$\scriptstyle\sim$

$\scriptstyle{\mathbb{P}\mathcal{I}}$${\lrcorner}$

$\scriptstyle\sim$

$\scriptstyle{\mathbb{P}\mathcal{I}}$

Note that the corestriction of $\mathbb{P}\mathcal{I}$ over $\mathcal{AF}$ is
also an acyclic Serre fibration, and all the spaces in this diagram have the
homotopy type of CW complexes. ∎

##  5 Linear Liouville pairs

As already explained in the Introduction, there exist other possible
definitions for Liouville pairs. The following one is used by some authors
(e.g., [Mit95, Hoz22a]).

######  Definition 5.1.

A pair of contact forms $(\alpha_{-},\alpha_{+})$ on $M$ is a linear Liouville
pair if the $1$-form

$(1-t)\alpha_{-}+(1+t)\alpha_{+}$

on $[-1,1]_{t}\times M$ is a positively oriented Liouville form.

The pair $(\alpha_{-},\alpha_{+})$ is a linear Anosov Liouville pair ($\ell$AL
pair for short) if both $(\alpha_{-},\alpha_{+})$ and
$(-\alpha_{-},\alpha_{+})$ are linear Liouville pairs.

Note that for this definition, $[-1,1]\times M$ is a Liouville _domain_
instead of a Liouville _manifold_. The results in Section 2.2 can be adapted
to $\ell$AL pairs. First of all, $\ell$AL pairs can be characterized in the
following way (see Lemma 2.3):

######  Lemma 5.1.

Let $(\alpha_{-},\alpha_{+})$ be a pair of contact forms on $M$. We write

$\displaystyle\alpha_{+}\wedge d\alpha_{+}$
$\displaystyle=f_{+}\,\mathrm{dvol},$ $\displaystyle\alpha_{-}\wedge
d\alpha_{-}$ $\displaystyle=-f_{-}\,\mathrm{dvol},$
$\displaystyle\alpha_{-}\wedge d\alpha_{+}$ $\displaystyle=g_{+}\
\mathrm{dvol},$ $\displaystyle\alpha_{+}\wedge d\alpha_{-}$
$\displaystyle=g_{-}\ \mathrm{dvol},$

where $\mathrm{dvol}$ is any volume form on $M$ and
$f_{\pm},g_{\pm}:M\rightarrow\mathbb{R}$ are smooth functions. Then
$(\alpha_{-},\alpha_{+})$ is a $\ell$AL pair if and only if

$\displaystyle|g_{-}|<f_{-}\qquad and\qquad|g_{+}|<f_{+}.$ (5.1)

###### Proof.

The pair $(\alpha_{-},\alpha_{+})$ is a linear Liouville pair if and only if
for every $t\in[-1,1]$,

$(\alpha_{+}-\alpha_{-})\wedge(d\alpha_{+}+d\alpha_{-}+t(d\alpha_{+}-d\alpha_{-}))>0,$

which is equivalent to

$f_{+}+f_{-}+g_{-}-g_{+}+t(f_{+}-f_{-}-g_{-}-g_{+})>0.$

This inequality is satisfied for every $t\in[-1,1]$ if and only if it is
satisfied for $t=-1$ and $t=1$, which is equivalent to $g_{+}<f_{+}$ and
$-g_{-}<f_{-}$. ∎

Similarly to Proposition 2.1, we also have:

######  Proposition 5.1.

Let $(\alpha_{-},\alpha_{+})$ be a $\ell$AL pair. Then it defines a bi-contact
structure $(\xi_{-},\xi_{+})=(\ker\alpha_{-},\ker\alpha_{+})$. Moreover, if
$X\in\xi_{-}\cap\xi_{+}$ is a nowhere vanishing vector field, then
$(X,R_{-},R_{+})$ is a basis at every point of $M$.

###### Proof.

To see that $\xi_{-}$ and $\xi_{+}$ are everywhere transverse, we argue by
contradiction and assume that they coincide at a point $x\in M$. With the same
notations as in the proof of 2.1, we readily get

$\displaystyle f_{+}$ $\displaystyle=d\alpha_{+}(X,Y),$ $\displaystyle f_{-}$
$\displaystyle=-\alpha_{-}(R_{+})d\alpha_{-}(X,Y),$ $\displaystyle g_{+}$
$\displaystyle=\alpha_{-}(R_{+})d\alpha_{+}(X,Y),$ $\displaystyle g_{-}$
$\displaystyle=d\alpha_{-}(X,Y),$

hence

$|g_{-}g_{+}|=|\alpha_{-}(R_{+})d\alpha_{-}(X,Y)d\alpha_{+}(X,Y)|=f_{-}f_{+},$

contradicting (5.1).

Now, assuming that $\mathrm{dvol}(X,R_{-},R_{+})=0$ at a point $x\in M$, the
computations in the proof of Proposition 2.1 show

$\displaystyle f_{-}$ $\displaystyle=-\alpha_{-}(R_{+})g_{-},$ $\displaystyle
f_{+}$ $\displaystyle=\frac{1}{\alpha_{-}(R_{+})}g_{+},$

hence

$|g_{-}g_{+}|=f_{-}f_{+},$

contradicting (5.1) once again. ∎

######  Remark 5.1.

A main difference between Anosov Liouville pairs as in Definition 1.1 and
linear Anosov Liouville pairs as in Definition 5.1 is that there does not seem
to be a natural action of $\mathcal{C}^{\infty}(M,\mathbb{R})$ on $\ell$AL
pairs. Moreover, we do not know if there is a natural modification making a
$\ell$AL pair balanced.

$\ell$AL pairs can also be characterized by their Reeb vector fields (see
Proposition 2.2):

######  Proposition 5.2.

Let $(\alpha_{-},\alpha_{+})$ be a pair of contact forms on $M$, negative and
positive, respectively. Then it is a $\ell$AL pair if and only if

$\displaystyle|\alpha_{-}(R_{+})|<1\qquad and\qquad|\alpha_{+}(R_{-})|<1.$
(5.2)

###### Proof.

If $(\alpha_{-},\alpha_{+})$ is a $\ell$AL pair, then Proposition 5.1 and the
computations in the proof of Proposition 2.1 imply

$\displaystyle g_{+}$ $\displaystyle=\alpha_{-}(R_{+})f_{+},$ $\displaystyle
g_{-}$ $\displaystyle=-\alpha_{+}(R_{-})f_{-},$

and (5.2) follows from (5.1).

Now, assuming that $(\alpha_{-},\alpha_{+})$ satisfies (5.2), it is enough to
prove that the conclusions of Proposition 5.1 are satisfied. This follows
exactly from the proof of Proposition 2.2. ∎

Combining Proposition 2.2 and Proposition 5.2, we obtain that any _balanced_
$\ell$AL pair is an AL pair. The converse is _not_ true by the following
lemma. It also implies that the two definitions of Liouville pairs (Definition
1.1 and Definition 5.1) are _different_ :

######  Lemma 5.2.

Every smooth volume preserving Anosov flow on $M$ admits a supporting balanced
AL pair which is not a $\ell$AL pair, and whose underlying bi-contact
structure is not defined by a $\ell$AL pair.

###### Proof.

Let $(\alpha_{s},\alpha_{u})$ be a defining pair for a volume preserving
Anosov flow $\Phi=\\{\phi^{t}\\}$. For $A\geq 1$, we define

$\displaystyle\alpha_{-}$ $\displaystyle:=e^{-A}\alpha_{u}+e^{A}\alpha_{s},$
$\displaystyle\alpha_{+}$ $\displaystyle:=e^{A}\alpha_{u}-e^{-A}\alpha_{s}.$

If $\mathrm{dvol}$ is such that
$\iota_{X}\mathrm{dvol}=\alpha_{s}\wedge\alpha_{u}$, where $X$ is the vector
field generating the flow, then one easily computes

$\displaystyle f_{\pm}$ $\displaystyle=r_{u}-r_{s}=2r_{u},~$ $\displaystyle
g_{\pm}$ $\displaystyle=-2\sinh(2A)r_{u},$

so $(\alpha_{-},\alpha_{+})$ is a $\mathcal{C}^{1}$ closed balanced AL pair,
but it is not a $\ell$AL pair since $|g_{\pm}|>f_{\pm}$. This remains true for
a suitable smoothing of $(\alpha_{-},\alpha_{+})$.

Let us assume for simplicity that the pair $(\alpha_{-},\alpha_{+})$ as above
is smooth. We show that there are no functions
$h_{\pm}:M\rightarrow\mathbb{R}_{>0}$ such that
$(h_{-}\alpha_{-},h_{+}\alpha_{+})$ is a $\ell$AL pair. Indeed, let us assume
by contradiction that such functions exist. By Lemma 5.1 they would satisfy
the following inequalities:

$\displaystyle|\cosh(2A)h_{+}X\cdot h_{-}-\sinh(2A)r_{u}h_{-}h_{+}|$
$\displaystyle<r_{u}h_{-}^{2},$ $\displaystyle|\cosh(2A)h_{-}X\cdot
h_{+}+\sinh(2A)r_{u}h_{-}h_{+}|$ $\displaystyle<r_{u}h_{+}^{2}.$

Writing $\rho_{\pm}:=1/h_{\pm}$, these are equivalent to

$\displaystyle|X\cdot\rho_{-}+\tanh(2A)r_{u}\rho_{-}|$
$\displaystyle<\frac{r_{u}}{\cosh(2A)}\rho_{+},$
$\displaystyle|X\cdot\rho_{+}-\tanh(2A)r_{u}\rho_{+}|$
$\displaystyle<\frac{r_{u}}{\cosh(2A)}\rho_{-}.$

Fixing a point $x\in M$, we define
$y_{\pm}:\mathbb{R}\rightarrow\mathbb{R}_{>0}$ by

$y_{\pm}(t):=\rho_{\pm}\circ\phi^{t}(x).$

There exists $C>0$ such that $0<y_{\pm}<C$. Moreover, these functions satisfy

$\displaystyle|\dot{y}_{-}+ay_{-}|$ $\displaystyle<\epsilon y_{+},$
$\displaystyle|\dot{y}_{+}-ay_{+}|$ $\displaystyle<\epsilon y_{-},$

where $a(t):=\tanh(2A)\,r_{u}\circ\phi^{t}(x)>0$ and
$\epsilon(t):=\frac{r_{u}\circ\phi^{t}(x)}{\cosh(2A)}>0$. Since $A\geq 1$, we
have $a>2\epsilon$. It follows that for every $T>0$,

$\displaystyle\int_{0}^{T}ay_{+}\,dt$ $\displaystyle\leq\int_{0}^{T}\epsilon
y_{-}\,dt+y_{+}(T)+y_{0}(T)$
$\displaystyle\leq\frac{1}{2}\int_{0}^{T}ay_{-}\,dt+2C$
$\displaystyle\leq\frac{1}{2}\int_{0}^{T}\epsilon y_{+}\,dt+3C$
$\displaystyle\leq\frac{1}{4}\int_{0}^{T}ay_{+}\,dt+3C,$

hence

$\int_{0}^{T}ay_{+}\,dt\leq 4C.$

However, $ay_{+}$ is bounded from below by some positive constant, which leads
to a contradiction.

If $(\alpha_{-},\alpha_{+})$ is only $\mathcal{C}^{1}$, this strategy still
applies to a suitable smoothing of $(\alpha_{-},\alpha_{+})$ which is
sufficiently $\mathcal{C}^{1}$-close to $(\alpha_{-},\alpha_{+})$. ∎

######  Remark 5.2.

We also expect that there exist (unbalanced) $\ell$AL pairs which are not AL
pairs, but the construction seems more delicate.

The “standard construction” of Section 3.2 yields a pair of contact forms
which is both an AL pair and a $\ell$AL pair (after smoothing). The next
proposition shows in particular that they induce equivalent Liouville
structures on $\mathbb{R}\times M$.

######  Proposition 5.3.

Let $(\alpha_{-},\alpha_{+})$ be a pair of contact forms which is both a
Liouville pair and linear Liouville pair. We consider two Liouville structures
on $\mathbb{R}_{s}\times M$:

  1. (1)

The completion of the Liouville domain $[-1,1]_{t}\times M$ with the “linear”
Liouville form $(1-t)\alpha_{-}+(1+t)\alpha_{+}$,

  2. (2)

The Liouville structure induced by the “exponential” Liouville form
$e^{-s}\alpha_{-}+e^{s}\alpha_{+}$.

These two Liouville structures are Liouville homotopic.

###### Proof.

Let $\epsilon>0$. We choose a smooth function
$\phi=\phi_{\epsilon}:\mathbb{R}\rightarrow\mathbb{R}_{\geq 0}$ satisfying

  * •

$\phi(s)=0$ for $s\leq-1-\epsilon$,

  * •

$\phi(s)>0$ for $s>-1-\epsilon$,

  * •

$\phi(s)=1+s$ for $s\geq-1+\epsilon$,

  * •

$\phi$ is non-decreasing and convex.

Figure 4: The function $\phi$.

For $\epsilon$ sufficiently small, the $1$-form
$\lambda_{0}=\phi(-s)\alpha_{-}+\phi(s)\alpha_{+}$ on $\mathbb{R}\times M$ is
Liouville, and it is Liouville homotopic to the completion of
$(1-t)\alpha_{-}+(1+t)\alpha_{+}$ defined on $[-1,1]\times M$. For
$\tau\in[0,1]$, we set

$\psi_{\tau}(s):=\tau e^{s}+(1-\tau)\phi(s),$

and

$\lambda_{\tau}:=\psi_{\tau}(-s)\alpha_{-}+\psi_{\tau}(s)\alpha_{+}.$

The family $\\{\lambda_{\tau}\\}_{\tau\in[0,1]}$ interpolates between
$\lambda_{0}$ and $\lambda_{1}=e^{-s}\alpha_{-}+e^{s}\alpha_{+}$. It is enough
to show that for every $\tau~\in(0,1)$, $d\lambda_{\tau}$ is non-degenerate.
By symmetry, it is enough to show that $d\lambda_{\tau}$ is non-degenerate for
$s\geq 0$. With the notations of Lemma 5.1, this is equivalent to

$\displaystyle
f_{+}+a_{\tau}(s)g_{-}-b_{\tau}(s)g_{+}+a_{\tau}(s)b_{\tau}(s)f_{-}>0,$ (5.3)

where

$\displaystyle a_{\tau}(s):=\frac{\psi_{\tau}(-s)}{\psi_{\tau}(s)},\qquad
b_{\tau}(s)=\frac{\psi^{\prime}_{\tau}(-s)}{\psi^{\prime}_{\tau}(s)}.$

It is easy to check that for $\tau\in(0,1)$ and $s\geq 0$, $0\leq
a_{\tau}(s)\leq 1$ and $0\leq b_{\tau}(s)\leq 1$. Since
$(\alpha_{-},\alpha_{+})$ is both an exponential and a linear Liouville pair,
we have that for every $a\in[0,1]$,

$\displaystyle f_{+}+ag_{-}-ag_{+}+a^{2}f_{-}$ $\displaystyle>0,$
$\displaystyle f_{+}+ag_{-}-g_{+}+af_{-}$ $\displaystyle>0,$

so for every $a\in[0,1]$ and $b\in[a,1]$, we have

$\displaystyle f_{+}+ag_{-}-bg_{+}+abf_{-}>0.$ (5.4)

By compactness, there exists $\delta>0$, only depending on
$(\alpha_{-},\alpha_{+})$, such that for every $a\in[0,1]$ and
$b~\in[a-\delta,1]$, the inequality (5.4) is satisfied. We claim that for
every $\tau\in(0,1)$ and $s\geq 0$, we have

$\displaystyle b_{\tau}(s)-a_{\tau}(s)\geq-\epsilon.$ (5.5)

Indeed, fixing $\tau\in(0,1)$, we consider two cases.

  * •

If $s\in[0,1-~\epsilon)\cup[1+\epsilon,\infty)$,

$\displaystyle\psi_{\tau}(s)$ $\displaystyle\geq\psi^{\prime}_{\tau}(s),$
$\displaystyle\psi^{\prime}_{\tau}(-s)$ $\displaystyle\geq\psi_{\tau}(-s),$

and (5.5) follows trivially since the left-hand side is non-negative.

  * •

If $s\in[1-\epsilon,1+\epsilon)$,

$\psi_{\tau}(s),\psi^{\prime}_{\tau}(s)\geq 1,$

and

$\displaystyle\psi^{\prime}_{\tau}(-s)\psi_{\tau}(s)-\psi^{\prime}_{\tau}(s)\psi_{\tau}(-s)$
$\displaystyle=\begin{multlined}(1-\tau)\Big{\\{}\tau\Big{(}\,\overset{(1)}{\overbrace{e^{s}\big{(}\phi^{\prime}(-s)-\phi(-s)\big{)}+se^{-s}}}\,\Big{)}\\\
+(1-\tau)\Big{(}\,\underset{(2)}{\underbrace{(1+s)\phi^{\prime}(-s)-\phi(-s)}}\,\Big{)}\Big{\\}}\end{multlined}(1-\tau)\Big{\\{}\tau\Big{(}\,\overset{(1)}{\overbrace{e^{s}\big{(}\phi^{\prime}(-s)-\phi(-s)\big{)}+se^{-s}}}\,\Big{)}\\\
+(1-\tau)\Big{(}\,\underset{(2)}{\underbrace{(1+s)\phi^{\prime}(-s)-\phi(-s)}}\,\Big{)}\Big{\\}}$

Since $\phi^{\prime}(-s)\geq 0$ and $0\leq\phi(-s)\leq\epsilon$,

$\displaystyle\mathrm{(1)}$
$\displaystyle\geq-e^{1+\epsilon}\epsilon+(1-\epsilon)e^{-(1+\epsilon)},$
$\displaystyle\mathrm{(2)}$ $\displaystyle\geq-\epsilon.$

For $\epsilon$ small enough, say $\epsilon\leq 1/100$, $\mathrm{(1)}\geq 0$,
and (5.5) follows.

This shows that for $\epsilon$ small enough, only depending on
$(\alpha_{-},\alpha_{+})$, the inequality (5.3) is satisfied for every
$\tau\in(0,1)$ and $s\geq 0$. The case $s\leq 0$ can be treated similarly. ∎

We do not know if Theorem 4.1 and Theorem 4.4 remain true for linear Anosov
Liouville pairs. The proof of Theorem 4.4 would straightforwardly adapt to the
linear case, provided that the space of $\ell$AL pairs supporting a given flow
is (weakly) contractible.

##  Appendix A Smoothing lemmas

This appendix concerns useful smoothing lemmas which are required to extend
the results of this paper to Anosov flows generated by $\mathcal{C}^{1}$
vector fields, as their weak-stable and weak-unstable bundles are not
necessarily $\mathcal{C}^{1}$. The approach can also be used to bypass
Hozoori’s delicate approximation techniques in [Hoz22a, Section 4]. We state
the results in greater generality than needed. $M$ now denotes a closed
$n$-dimensional manifold ($n\geq 1$) and $X$ denotes a non-singular vector
field on $M$ of class $\mathcal{C}^{k}$, $1\leq k\leq\infty$ (without any
Anosovity condition). We fix an arbitrary auxiliary metric on $M$.

The first smoothing lemma is essentially [Hoz22a, Lemma 4.3.] and the proof is
the same up to some minor changes.

######  Lemma A.1.

Let $f:M\rightarrow\mathbb{R}$ be a continuous function which is continuously
differentiable along $X$. Then for every $\epsilon>0$, there exists a
$\mathcal{C}^{k}$ function $f^{\epsilon}:M\rightarrow\mathbb{R}$ satisfying

$\displaystyle|f^{\epsilon}-f|_{\mathcal{C}^{0}}\leq\epsilon\qquad
and\qquad|X\cdot f^{\epsilon}-X\cdot f|_{\mathcal{C}^{0}}\leq\epsilon.$

In other words, with the notations of Definition 3.3, $\mathcal{C}^{k}$ is
dense in $\mathcal{C}^{0}_{X}$. The same holds with $\mathcal{C}^{\ell}_{X}$
in place of $\mathcal{C}^{0}_{X}$, for $0\leq\ell\leq k-1$. We will need a
similar result for $1$-forms on $M$.

######  Lemma A.2.

The space of $\mathcal{C}^{k}$ $1$-forms on $M$ vanishing along $X$ is dense
in $\Omega_{X,\ell}^{1}$ for $0\leq\ell\leq k-1$.

###### Proof.

This is a straightforward adaptation of the proof of [Hoz22a, Lemma 4.3.].

By compactness of $M$, we can find a positive real number $\tau>0$ and a
finite collection $\\{(U_{i},V_{i},\phi_{i})\\}_{1\leq i\leq N}$ where

  * •

$V_{i}\subset U_{i}\subset M$ are open subsets of $M$,

  * •

$\\{V_{i}\\}_{1\leq i\leq N}$ is a covering of $M$,

  * •

$\phi_{i}:U_{i}\rightarrow(-2\tau,2\tau)_{t}\times D$ is a $\mathcal{C}^{k}$
diffeomorphism such that $\phi_{i}(V_{i})=(-\tau,\tau)\times D$ and
$d\phi_{i}(X)=\partial_{t}$. Here, $D$ denotes the open unit disk in
$\mathbb{R}^{n-1}$.

Such a collection can be obtained by taking a finite collection of
sufficiently small flow-boxes for $X$ covering $M$.

Let $\\{\psi_{i}\\}_{1\leq i\leq N}$ be a partition of unity subordinate to
the open covering $\\{V_{i}\\}_{1\leq i\leq N}$. For every $i$, the support of
$\psi_{i}$ is contained in $V_{i}$ so we can find $0<r<1$ such that the
support of $\psi_{i}\circ\phi_{i}^{-1}$ is contained in $(-\tau,\tau)\times
D_{r}$, where $D_{r}$ denotes the open disk of radius $r$.

Let $h:\mathbb{R}\rightarrow\mathbb{R}_{\geq 0}$ be a smooth bump function
satisfying

  * •

For $|t|\leq\tau$, $h(t)=1$,

  * •

For $|t|\geq 2\tau$, $h(t)=0$,

  * •

$h$ is non-decreasing on $(-\infty,0)$ and non-increasing on $(0,\infty)$.

Figure 5: The nested open sets in the argument. The support of
$\alpha^{\prime}_{i}$ is contained in the hashed region.

Let $\alpha\in\Omega^{1}_{X,\ell}$. By definition,
$\alpha=\sum_{i=1}^{N}\psi_{i}\alpha$. For every $i$, we write
$\alpha_{i}:=\psi_{i}\alpha$ and
$\alpha^{\prime}_{i}:=(\phi_{i})_{*}\alpha_{i}$. We have reduced the problem
to a single flow-box $(-2\tau,2\tau)\times D$.

Let $\epsilon>0$. In what follows, the symbol “$\lesssim$” means “less or
equal than up to a constant factor that does not depend on $\epsilon$”. For a
fixed $i$, let $\beta^{0}_{i}$ be a smooth $1$-form on $D$ with support
contained in $D_{r}$ satisfying

$\left|\beta^{0}_{i}-{\alpha_{i}^{\prime}}_{|\\{0\\}\times
D}\right|_{\mathcal{C}^{\ell}}\leq\epsilon.$

By definition, $\mathcal{L}_{\partial t}\alpha_{i}^{\prime}$ is
$\mathcal{C}^{\ell}$ and vanishes along $\partial_{t}$. Therefore, we can find
a smooth $1$-form $\eta_{i}$ on $(-2\tau,2\tau)\times D$ with support
contained in $(-\tau,\tau)\times D_{r}$ satisfying

$\displaystyle\eta_{i}(\partial_{t})$ $\displaystyle=0,$
$\displaystyle\left|\eta_{i}-\mathcal{L}_{\partial
t}\alpha_{i}^{\prime}\right|_{\mathcal{C}^{\ell}}$
$\displaystyle\leq\epsilon.$

We can extend $\beta^{0}_{i}$ to a smooth $1$-form $\beta_{i}$ on
$(-2\tau,2\tau)\times D$ by setting

$\displaystyle\beta_{i}(\partial_{t})$ $\displaystyle:=0,$
$\displaystyle\mathcal{L}_{\partial t}\beta_{i}$ $\displaystyle:=\eta_{i}.$

Finally, we define $\beta_{i}^{\prime}:=h\beta_{i}$. This is a smooth $1$-form
with support in $(-2\tau,2\tau)\times D_{r}$. Note that we have

$\displaystyle\beta_{i}-\alpha_{i}^{\prime}=\beta_{i}^{0}-{\alpha_{i}^{\prime}}_{|\\{0\\}\times
D}+\int_{0}^{t}\left(\eta_{i}-\mathcal{L}_{\partial
t}\alpha_{i}^{\prime}\right)dt,$

so

$\displaystyle\left|\beta_{i}-\alpha_{i}^{\prime}\right|_{\mathcal{C}^{\ell}}$
$\displaystyle\leq\left|\beta_{i}^{0}-{\alpha_{i}^{\prime}}_{|\\{0\\}\times
D}\right|_{\mathcal{C}^{\ell}}+2\tau\left|\eta_{i}-\mathcal{L}_{\partial
t}\alpha_{i}^{\prime}\right|_{\mathcal{C}^{\ell}}\lesssim\epsilon,$
$\displaystyle\left|\mathcal{L}_{\partial t}\beta_{i}-\mathcal{L}_{\partial
t}\alpha_{i}^{\prime}\right|_{\mathcal{C}^{\ell}}$
$\displaystyle=\left|\eta_{i}-\mathcal{L}_{\partial
t}\alpha_{i}^{\prime}\right|_{\mathcal{C}^{\ell}}\leq\epsilon.$

Since the support of $\alpha_{i}^{\prime}$ is contained in $(-\tau,\tau)\times
D_{r}$, we readily get

$\displaystyle\left|{\beta_{i}}_{|((-2\tau,-\tau)\cup(\tau,2\tau))\times
D}\right|_{\mathcal{C}^{\ell}}\lesssim\epsilon.$

Moreover,

  * •

On $((-2\tau,-\tau)\cup(\tau,2\tau))\times D$,

$\displaystyle\beta_{i}^{\prime}-\alpha_{i}^{\prime}$
$\displaystyle=h\beta_{i},$ $\displaystyle\mathcal{L}_{\partial
t}\beta_{i}^{\prime}-\mathcal{L}_{\partial t}\alpha_{i}^{\prime}$
$\displaystyle=(\partial_{t}h)\beta_{i},$

  * •

On $(-\tau,\tau)\times D$,

$\displaystyle\beta_{i}^{\prime}-\alpha_{i}^{\prime}$
$\displaystyle=\beta_{i}-\alpha_{i}^{\prime},$
$\displaystyle\mathcal{L}_{\partial t}\beta_{i}^{\prime}-\mathcal{L}_{\partial
t}\alpha_{i}^{\prime}$ $\displaystyle=\eta_{i}-\mathcal{L}_{\partial
t}\alpha_{i}^{\prime},$

and we obtain

$\displaystyle\left|\beta_{i}^{\prime}-\alpha_{i}^{\prime}\right|_{\mathcal{C}^{\ell}}$
$\displaystyle\lesssim\epsilon,$ $\displaystyle\left|\mathcal{L}_{\partial
t}\beta_{i}^{\prime}-\mathcal{L}_{\partial
t}\alpha_{i}^{\prime}\right|_{\mathcal{C}^{\ell}}$
$\displaystyle\lesssim\epsilon.$

Finally, we define

$\beta:=\sum_{i}\phi_{i}^{*}\beta_{i}^{\prime},$

so that $\beta$ is a $\mathcal{C}^{k}$ $1$-form on $M$ satisfying
$\beta(X)=0$, and

$\displaystyle\left|\beta-\alpha\right|_{\mathcal{C}^{\ell}}$
$\displaystyle\leq\sum_{i=1}^{N}\left|\phi_{i}^{*}\beta_{i}^{\prime}-\phi_{i}^{*}\alpha_{i}^{\prime}\right|_{\mathcal{C}^{\ell}}\lesssim\sum_{i=1}^{N}\left|\beta_{i}^{\prime}-\alpha_{i}^{\prime}\right|_{\mathcal{C}^{\ell}}\lesssim\epsilon,$
$\displaystyle\left|\mathcal{L}_{X}\beta-\mathcal{L}_{X}\alpha\right|_{\mathcal{C}^{\ell}}$
$\displaystyle\leq\sum_{i=1}^{N}\left|\phi_{i}^{*}(\mathcal{L}_{\partial
t}\beta_{i}^{\prime})-\phi_{i}^{*}(\mathcal{L}_{\partial
t}\alpha_{i}^{\prime})\right|_{\mathcal{C}^{\ell}}\lesssim\sum_{i=1}^{N}\left|\mathcal{L}_{\partial
t}\beta_{i}^{\prime}-\mathcal{L}_{\partial
t}\alpha_{i}^{\prime}\right|_{\mathcal{C}^{\ell}}\lesssim\epsilon.$

This finishes the proof. ∎

##  Appendix B Almost volume preserving Anosov flows

In this appendix, we prove of a technical result used in the proof of Theorem
3.1. Let us recall the setup. $\Phi$ is a smooth Anosov flow on a closed
oriented $3$-manifold $M$, generated by a vector field $X$. For an adapted
metric $g$, $r_{u}>0$ and $r_{s}<0$ denote the expansion rates in the unstable
and stable directions respectively. The divergence of $X$ for this metric is
simply $\mathrm{div}_{g}X=r_{u}+r_{s}$. We say that $\Phi$ is almost volume
preserving if it satisfies one of the following equivalent conditions (compare
with (3.12)):

  * (C1)

For every $\epsilon>0$, there exists a smooth function
$f_{\epsilon}:M\rightarrow\mathbb{R}$ satisfying

$\displaystyle\left|\mathrm{div}_{g}X+X\cdot f_{\epsilon}\right|\leq\epsilon.$

  * (C2)

For every $\epsilon>0$, there exists a smooth volume form
$\mathrm{dvol}_{\epsilon}$ on $M$ satisfying

$\displaystyle|\mathrm{div}_{\epsilon}X|=\left|\frac{\mathcal{L}_{X}\mathrm{dvol}_{\epsilon}}{\mathrm{dvol}_{\epsilon}}\right|\leq\epsilon,$

where $\mathrm{div}_{\epsilon}X$ denotes the divergence of $X$ with respect to
$\mathrm{dvol}_{\epsilon}$.

######  Proposition B.1.

If $\Phi$ is a smooth almost volume preserving Anosov flow on $M$, then $\Phi$
is volume preserving.

###### Proof.

As noted in the proof of Theorem 3.1, it is enough to show that $\Phi$ is
topologically transitive, i.e., $\Phi$ has a dense orbit. We closely follow
the strategy from [Mic22] that relies on some key properties of
Sinaï–Ruelle–Bowen (SRB) measures for Anosov diffeomorphisms. Here, the time-
one map $\phi=\phi^{1}$ of $\Phi$ is not Anosov since it is neither
contracting nor expanding in the direction of the flow. Nevertheless, the main
arguments of [Mic22] can be adapted to our context with only minor
modifications.

A Borel probability measure $\mu$ on $M$ preserved by $\phi$ is a _SRB
measure_ if it has absolutely continuous conditional measures on unstable
manifolds, see [LY85, Definition 1.4.2.]. Here, the (strong) unstable
manifolds of $\Phi$ and $\phi$ coincide. By [CY05, Theorem C], $\phi$ admits a
SRB measure. Let $g$ be a Riemannian metric on $M$ adapted to $\Phi$ (so that
$r_{s}<0<r_{u}$) and $\mu$ be any Borel probability measure preserved by
$\phi$. For $\mu$-a.e. $x\in M$ and for unit vectors $v_{u}\in E^{u}(x)$ and
$v_{s}\in E^{u}(x)$, the _Lyapunov exponents_

$\displaystyle\Lambda_{u}(x)$
$\displaystyle:=\lim_{n\rightarrow\pm\infty}\frac{1}{n}\ln\|d\phi^{n}(v_{u})\|=\lim_{n\rightarrow\pm\infty}\frac{1}{n}\int_{0}^{n}r_{u}\circ\phi^{s}(x)\,ds,$
$\displaystyle\Lambda_{s}(x)$
$\displaystyle:=\lim_{n\rightarrow\pm\infty}\frac{1}{n}\ln\|d\phi^{n}(v_{s})\|=\lim_{n\rightarrow\pm\infty}\frac{1}{n}\int_{0}^{n}r_{s}\circ\phi^{s}(x)\,ds,$

exist, are finite, independent of the metric and satisfy
$\Lambda_{s}(x)<0<\Lambda_{u}(x)$. The third Lyapunov exponent corresponding
to the direction of the flow is $0$.

The hypothesis on $\Phi$ implies that $\Lambda_{u}+\Lambda_{s}=0$ $\mu$-a.e.
Indeed, let $\epsilon>0$ and choose a smooth (hence bounded) function
$f_{\epsilon}:M\rightarrow\mathbb{R}$ such that $|r_{u}+r_{s}+X\cdot
f_{\epsilon}|<\epsilon$. Then $\mu$-a.e., we have

$\displaystyle|\Lambda_{u}+\Lambda_{s}|$
$\displaystyle\leq\limsup_{n\rightarrow\infty}\frac{1}{n}\int_{0}^{n}|r_{u}+r_{s}+X\cdot
f_{\epsilon}|\circ\phi^{s}\,ds+\limsup_{n\rightarrow\infty}\frac{1}{n}\left|\int_{0}^{n}(X\cdot
f_{\epsilon})\circ\phi^{s}\,ds\right|$
$\displaystyle\leq\epsilon+\limsup_{n\rightarrow\infty}\frac{1}{n}\left|f_{\epsilon}\circ\phi^{n}-f_{\epsilon}\right|$
$\displaystyle=\epsilon.$

By [LY85, Theorem A], $\mu$ is a SRB measure for $\phi$ if and only if it
satisfies the _Pesin entropy formula_

$\displaystyle h_{\mu}(\phi)=\int_{M}\Lambda_{u}\,d\mu.$

Assuming that $\mu$ is a SRB measure for $\phi$, it follows that

$\displaystyle h_{\mu}\big{(}\phi^{-1}\big{)}=\int_{M}-\Lambda_{s}\,d\mu,$

so since $-\Lambda_{s}$ corresponds to the unstable Lyapunov exponent of
$\phi^{-1}$, $\mu$ is also a SRB measure for $\phi^{-1}$. By [Led84, Théorème
(5.5)], $\mu$ is absolutely continuous with respect to the Lebesgue measure on
$M$ (induced by some Riemannian metric). If $\Omega$ denotes the non-wandering
set of $\Phi$, then $\mu(\Omega)=1$ so the Lebesgue measure of $\Omega$ is
positive. Smale’s spectral decomposition for $\Omega$ states that $\Omega$ can
be partitioned into finitely many _basic sets_ $\Omega_{1},\dots,\Omega_{N}$.
Those are compact invariant subsets on which $\Phi$ is transitive. It follows
that there exists a basic set $\Omega_{i}$ with positive Lebesgue measure. By
[BR75, Corollary 5.7. (b)], $\Omega_{i}=M$ and $\Phi$ is transitive. This
concludes the proof. ∎

######  Remark B.1.

The proof can be adapted to show that any almost volume preserving Anosov flow
of class $\mathcal{C}^{2}$ on any closed manifold is volume preserving. In
higher dimension, $\Lambda_{u}$ should be replaced by the sum of the positive
Lyapunov exponents (counted with multiplicity) and $\Lambda_{s}$ by the sum of
the negative ones.

######  Remark B.2.

The main result of [Mic22] asserts that an Anosov _diffeomorphism_ of class
$\mathcal{C}^{2}$ on a closed manifold $M$ satisfying that at every periodic
point, the Poincaré return map has determinant one is volume preserving. The
same result remains true for an Anosov flow whose Poincaré return map at every
closed orbit has determinant one. As in the proof in [Mic22], this condition
and Anosov’s _shadowing property_ imply that $\Lambda_{u}+\Lambda_{s}=0$
$\mu$-a.e., which is enough to conclude.

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