2.1 (Projectively) Anosov flows and the associated expansion rates

Anosov flows in dimension 3 are non-singular flows, whose action on the tangent space of the ambient manifold exhibits exponential expansion and contraction in two distinct transverse directions.

Definition 2.1. Let $\phi ^t$ be the flow generated by the non-vanishing $C^1$ vector field X. We call $\phi ^t$ Anosov if there exists a continuous invariant splitting $TM\simeq E^{ss} \oplus E^{uu} \oplus \langle X\rangle $ , such that for some positive constant C and a norm $\|.\|$ , we have

$$ \begin{align*}\|\phi^t_*(u)\|\leq e^{-Ct}\|u\|\quad\text{and}\quad \|\phi^t_*(v)\|\geq e^{Ct}\|v\|\end{align*} $$

for any $u\in E^{ss}$ and $v\in E^{uu}$ . We call $E^{ss}$ and $E^{uu}$ strong stable and unstable directions, respectively.

The classical examples of such flows are the geodesic flows on the unit tangent bundle of hyperbolic surfaces and the suspension of Anosov diffeomorphisms of torus. However, various surgery operations on Anosov flows have yielded many more examples, including infinitely many examples on hyperbolic manifolds. These include surgeries of Handel and Thurston [Reference Handel and Thurston29], Fried [Reference Fried23], Goodman [Reference Goodman28], Foulon and Hassleblatt [Reference Foulon and Hasselblatt21], and more recently, the bi-contact geometric surgeries introduced by Salmoiraghi [Reference Salmoiraghi45, Reference Salmoiraghi46], which manage to reproduce, up to orbit equivalence, the previous operations in many cases.

One important generalization of Anosov flows, which bridges Anosov dynamics to contact geometry, is the following.

Definition 2.3. Let $\tilde {\phi }^t$ be the Poincaré linear flow as in Remark 2.2. We call $\phi ^t$ projectively Anosov if there exists a continuous invariant splitting $TM/\langle X\rangle \simeq E^{s} \oplus E^{u}$ , such that for some positive constant C and a norm $\|.\|$ , we have

$$ \begin{align*}\|\tilde{\phi}^t_*(v)\|/\|\tilde{\phi}^t_*(u)\|\leq e^{Ct}\|v\|/\|u\|\end{align*} $$

for any $u\in E^{s}$ and $v\in E^{u}$ . We call $E^s$ and $E^u$ stable and unstable directions, respectively.

In other words, a flow is projectively Anosov if its induced Poincaré linear flow admits a dominated splitting. That is, a continuous and invariant splitting into two line bundles on which the action of the flow is relatively expanding in one direction with respect to the other.

Abusing notation, we also refer to $\pi ^{-1}(E^s)$ and $\pi ^{-1}(E^u)$ , which are a priori $C^0$ two-dimensional sub bundles of $TM$ , by $E^s$ and $E^u$ , respectively, and call them the weak stable and unstable bundles, respectively (see Remark 2.5).

In the case where a projectively Anosov flow is Anosov, the weak stable and unstable bundles are known to be $C^1$ [Reference Hasselblatt30]. It is worth mentioning that that unlike the Anosov case (as discussed in Remark 2.2), in the case of projectively Anosov flows, the splitting of the normal bundle $TM/\langle X\rangle $ cannot necessarily be lifted to the tangent space $TM$ [Reference Noda40].

Although it is not a priori obvious if such a class of dynamics is strictly larger than the class of Anosov flows, we know that projectively Anosov flows are abundant. See [Reference Eliashberg and Thurston18, Reference Mitsumatsu39] for examples on torus bundles, [Reference Bowden, Bonatti and Potrie13] for non-Anosov examples on hyperbolic manifolds, and [Reference Asaoka, Dufraine and Noda7] for a more general construction.

To build the bridge from the above definitions to the world of differential and contact geometry, it is very useful for us to measure the infinitesimal expansion or contraction of the length of vectors in the invariant bundles. Note that without loss of generality, we can assume the norm involved in the definition of (projectively) Anosov flows is $C^\infty $ .

Definition 2.4. Using the above notation and considering $TM/\langle X\rangle \simeq E^{s} \oplus E^{u}$ , let ${\tilde {e}_s\in E^s}$ and $\tilde {e}_u\in E^u$ be the unit vectors with respect to some X-differentiable norm $\|.\|$ on $TM/\langle X\rangle $ . We call

$$ \begin{align*}r_s:=\frac{\partial}{\partial t} \ln{\|\tilde{\phi}_*^t (\tilde{e}_s)\|}\bigg|_{t=0}\quad\text{and} \quad r_u:=\frac{\partial}{\partial t} \ln{\|\tilde{\phi}_*^t (\tilde{e}_u)\|}\bigg|_{t=0}\end{align*} $$

the expansion rates of the stable and unstable directions, respectively, with respect to $\|.\|$ .

We remark that similar notions have been previously used in the study of various aspects of Anosov flows [Reference Coles and Sharp14, Reference Hasselblatt30, Reference Simić48].

Remark 2.5. Note that the norm used in the definition of expansion rates is defined on the normal bundle $TM/\langle X\rangle $ . However, it is easy to describe these quantities based on the tangent bundle $TM$ . First, given a norm on $TM/\langle X\rangle $ , consider some metric on this vector bundle, which induces the norm. Notice that given any transverse $C^1$ plane field $\eta $ , there exists a natural isomorphism $\eta \simeq TM/\langle X\rangle $ via the projection $\pi :TM\rightarrow \eta \simeq TM/\langle X\rangle $ . Therefore, a Riemannian metric on $TM/\langle X\rangle $ induces a Riemannian metric on $\eta $ , which can be naturally extended to $TM$ by letting $\|X\|=1$ and $X\perp \eta $ , where $\|.\|$ is the norm on $TM$ induced from such a metric. If $\tilde {e}_s \in E^s \subset TM/\langle X\rangle $ and $\tilde {e}_u\in E^u\subset TM/\langle X\rangle $ are the unit vector fields, their image $e_s\in \pi ^{-1}(E^s)\cap \eta $ and $e_u\in \pi ^{-1}(E^u)\cap \eta $ , under the isomorphism, are unit vector fields with respect to the norm induced on $TM$ . It is easy to compute

$$ \begin{align*}\mathcal{L}_X e_s=-r_se_s +q_s X\quad\text{and}\quad \mathcal{L}_X e_u=-r_ue_u +q_u X,\end{align*} $$

where $q_s$ and $q_u$ are real functions, depending on our choice of $\eta $ . Note that we will have $q_s=q_u=0$ when $\eta $ is preserved by X (see §3 of [Reference Hozoori33] for a more thorough discussion). Since we are assuming $\eta $ to be $C^1$ , this only happens for an Anosov flow when it is contact or a suspension flow [Reference Foulon and Hasselblatt20].

Note that this remark also justifies the abuse of notation we adopted in the beginning of this section, that is, referring to $\pi ^{-1}(E^s)$ by $E^s$ (and similarly for $E^u$ ). To be clearer, the correspondence between the metrics on $TM$ and $TM/\langle X\rangle $ discussed above allows us to talk about the unit vector $e_s\in E^s$ and, therefore, the expansion rate of the stable direction unambiguously, whether we consider $E^s\subset TM$ or $E^s \subset TM/\langle X \rangle $ . The same holds for $e_u\in E^u$ and the expansion rate of the unstable direction.

Remark 2.6. Alternatively, one can characterize the expansion rates in terms of differential forms. Consider a metric on $TM/\langle X \rangle $ and define $\tilde {\alpha _s}$ as a differential form on $TM/\langle X \rangle $ by letting $\ker {\tilde {\alpha _s}}=E^u\subset TM/\langle X \rangle $ and $\tilde {\alpha _s}(\tilde {e}_s)=1$ , where $\tilde {e}_s\in E^s\subset TM/\langle X \rangle $ is a unit vector field with respect to the chosen metric. Similarly, define $\tilde {\alpha }_u$ and easily compute

$$ \begin{align*}\mathcal{L}_X \tilde{\alpha}_s=r_s\tilde{\alpha}_s\quad\text{and}\quad \mathcal{L}_X \tilde{\alpha}_u=r_u\tilde{\alpha}_u,\end{align*} $$

where $r_s$ and $r_u$ are the expansion rates of the stable and unstable directions with respect to such a metric.

As we will see in the remainder of the paper, it is usually desirable to work with differential forms on $TM$ . Therefore, we can define $\alpha _s:=\pi ^*\tilde {\alpha }_s$ and $\alpha _u:=\pi ^*\tilde {\alpha }_u$ , where $\pi :TM\rightarrow TM/\langle X\rangle \simeq \eta $ is the natural projection, and note that

$$ \begin{align*}\mathcal{L}_X \alpha_s=r_s\alpha_s\quad\text{and}\quad \mathcal{L}_X \alpha_u=r_u\alpha_u.\end{align*} $$

In fact, the above expressions can be taken as the definition of the expansion rates $r_s$ and $r_u$ . More precisely, if we take the unit vector fields $e_s\in E^s\cap \eta \subset TM$ and $e_u\in E^u\cap \eta \subset TM$ , induced after choosing a transverse plane field $\eta $ as in Remark 2.5, and define the differential form $\alpha _s$ by letting $\ker {\alpha _s}=E^u\oplus \langle X\rangle $ and $\alpha _s(e_s)=1$ , this matches the definition above. The same holds for $\alpha _u$ . This means that, as in Remark 2.5, the choice of the transverse plane field $\eta $ does not affect the geometry of expansion in terms of differential forms on $TM$ .

Not surprisingly, the (relative) exponential expansion for (projectively) Anosov flows can be easily characterized in terms of such expansion rates (see [Reference Hozoori33] for more details and proofs).

Proposition 2.7. Let X be a projectively Anosov vector field and $r_s$ and $r_u$ , the expansion rates of stable and unstable directions, respectively, with respect to any Riemannian metric satisfying the metric condition of Definition 2.3, which is X-differentiable. Then,

$$ \begin{align*}r_u-r_s>0.\end{align*} $$

While Proposition 2.7 expresses the relative expansion in the unstable direction with respect to the stable direction, the following proposition realizes when we have absolute expansion and contraction in those directions, which by Doering’s result [Reference Doering17] yields Anosovity (see Remark 2.2).

Proposition 2.8. Let X be a projectively Anosov vector field and $r_s$ and $r_u$ . Then X is Anosov if and only if, with respect to some Riemannian metric, we have

$$ \begin{align*}r_u>0>r_s.\end{align*} $$

2.2 Relation to (bi-)contact geometry

Recall that a $C^1$ 1-form $\alpha $ is a contact form on M if $\alpha \wedge d\alpha $ is a non-vanishing volume form on M. If $\alpha \wedge d\alpha>0$ (compared to the orientation on M), we call $\alpha $ a positive contact form and otherwise, a negative one. We call the $C^1$ plane field $\xi :=\ker {\alpha }$ a (positive or negative) contact structure on M. Notice that by the Frobenuis theorem, contact structures can be thought of as maximally non-integrable $C^1$ plane fields, that is, the extreme opposite of foliations.

For example, $\xi _{std}:=\ker {dz-ydx}$ ( $\xi _{std}:=\ker {dz+ydx}$ ) is called the standard positive (negative) contact structure on $\mathbb {R}^3$ , while $\xi _n:=\ker {\{\cos {2\pi n z}dx-\sin {2\pi n} dy \}}$ on $\mathbb {T}^3\,{=}\,\mathbb {R}^3/\mathbb {Z}^3$ gives an infinite family of distinct positive (negative) contact structures, when $n\in \mathbb {Z}>0$ ( $n<0$ ).

Although we do not go toward the topological aspects of contact structures in this paper, it is worth mentioning that positive (negative) contact structures do not have any local invariant, thanks to the Darboux theorem that states that any two positive (negative) contact structures are locally contactomorphic (that is, locally look like the standard model on $\mathbb {R}^3$ ). Furthermore, Gray’s theorem shows that any homotopy of a contact structure through contact structures is induced by an isotopy of the ambient manifold.

Associated to any contact structure, there is an important class of flows, which we will use in this paper. Given any contact form $\alpha $ for a contact structure $\xi :=\ker {\alpha }$ , there exists a unique vector field $R_{\alpha }$ , satisfying

$$ \begin{align*}d\alpha(R_{\alpha},.)=0\quad\text{and}\quad \alpha(R_\alpha)=1.\end{align*} $$

Such a vector field is called a Reeb vector field and it is easy to check $\mathcal {L}_{R_\alpha }\alpha =0$ . This implies $\mathcal {L}_{R_\alpha }\alpha \wedge d\alpha =0$ . In particular, Reeb vector fields are volume preserving. Furthermore, Reeb vector fields preserve $\xi $ , and are transverse to the underlying contact structure. It is easy to observe that conversely, given a contact structure, any transverse vector field preserving the contact structure $\xi $ is a Reeb vector field for an appropriate choice of contact form.

A natural and well-studied interplay of contact geometry and Anosov dynamics happens when a Reeb vector field is Anosov, that is, the case of contact Anosov flows (see for instance [Reference Foulon and Hasselblatt21]). However, in this paper, we are interested in a more general relation between the two theories, thanks to the following proposition, first observed by Mitsumatsu [Reference Mitsumatsu39] and Eliashberg and Thurston [Reference Eliashberg and Thurston18], which characterizes projectively Anosov flows in terms of contact geometry. We remind the reader that, as mentioned in the introduction of the paper, we are assuming the underlying manifold to be oriented and the invariant bundles for the projectively Anosov flows to be transversely orientable.

In other words, considering a projectively Anosov flow, the bi-sectors of $E^s$ and $E^u$ can be seen to be a pair of positive and negative contact structures, possibly after a perturbation to make them $C^1$ . And conversely, any vector field directing the intersection of such transverse pair is projectively Anosov.

We note that the above proposition also shows that the (periodic) orbits of a projectively Anosov flows are Legendrian (knots), that is, tangent, for both of the underlying contact structures.

Using Proposition 2.9, we can easily give examples of non-Anosov projectively Anosov flows. For instance, $(\xi _m:=\ker {\{dz+\epsilon (\cos {2\pi m z}dx-\sin {2\pi m} dy)\}},\xi _n:=\ker {\{dz+\epsilon '(\cos {2\pi n z}dx-\sin {2\pi n} dy) \}})$ is a pair of positive and negative transverse contact structures, whenever $m<0<n$ are integers and $\epsilon \neq \epsilon '$ . Therefore, any flow, whose generating vector field lies in the intersection $\xi _m\cap \xi _n$ , is a projectively Anosov flow on $\mathbb {T}^3=\mathbb {R}^3/\mathbb {Z}^3$ by Proposition 2.9. Note that there are no Anosov flows on $\mathbb {T}^3$ .

We call such pair of transverse negative and positive contact structures $(\xi _-,\xi _+)$ a bi-contact structure, supporting the underlying projectively Anosov flow. It turns out that by enriching a bi-contact structure with additional contact geometric structures, one can also characterize Anosov flows purely in terms of contact geometry [Reference Hozoori33].

5.1 Taut contact hyperbolas and a Reeb dynamical characterization

As it can be seen in the discussion above, additional geometric symmetry can be observed in the case of volume preserving Anosov flows. In this section, we describe this extra structure in terms of the theory of contact hyperbolas developed by Perrone [Reference Perrone42], following the similar theory of contact circles by Geiges and Gonzalo [Reference Geiges and Gonzalo25, Reference Geiges and Gonzalo26].

A contact hyperbola on M is a pair of positive and negative contact forms $(\alpha _1,\alpha _2)$ such that $\alpha _a:=a_1\alpha _1+a_2\alpha _2$ is also a contact form for any $a:=(a_1,a_2)\in \mathbb {H}_r^1$ , where $H_r^1=\{(a_1,a_2)|a_1^2-a_2^2=r\}$ for $r\in \{-1,1\}$ . Furthermore, a contact hyperbola is called taut if $\alpha _a\wedge d\alpha _a=r\alpha _1\wedge d\alpha _1$ (or equivalently, $\alpha _a\wedge d\alpha _a=-r\alpha _2\wedge d\alpha _2$ ) for any $a\in \mathbb {H}_r^1$ . It is easy to show [Reference Perrone42] that $(\alpha _1,\alpha _2)$ is a taut contact hyperbola if and only if

$$ \begin{align*}\alpha_1\wedge d \alpha_1=-\alpha_2\wedge d\alpha_2\quad\text{and}\quad \alpha_1\wedge d \alpha_2=-\alpha_2\wedge d\alpha_1.\end{align*} $$

Notice that if $(\alpha _1,\alpha _2)$ is a taut contact hyperbola, then $\ker {\alpha _1}$ and $\ker {\alpha _2}$ form a bi-contact structure if transverse, with any flow directing the intersection of them being projectively Anosov. It is known that the converse is not true, that is, there are projectively Anosov flows which do not come from a contact hyperbola.

As it is seen above, for a volume preserving Anosov flow, the supporting bi-contact structure $(\ker { \alpha _-}=\ker {\{\alpha _u+\alpha _s\}},\ker {\alpha _+}:=\ker {\{ \alpha _u-\alpha _s\} })$ exists, where $\alpha _u|_{E^s}=\alpha _s|_{E^u}=0$ , and the induced positive volume forms and the expansion rates satisfy $\Omega ^{\alpha _-}=\Omega ^{\alpha _+}=\alpha _s\wedge \alpha _u\wedge \alpha _X$ (for any 1-form $\alpha _X$ with $\alpha _X(X)=1$ ) and $r_s^+=r_s^-=-r_u^+=-r_u^-$ , respectively (see Remarks 2.5 and 2.6). By Lemma 4.2, we have

$$ \begin{align*}\alpha_+\wedge d\alpha_+=(r^+_u-r^+_s)\Omega^{\alpha_+}=(r^-_u-r^-_s)\Omega^{\alpha_-}=-\alpha_-\wedge d \alpha_-.\end{align*} $$

Moreover, the discussion in the beginning remarks of this section shows that $R_{\alpha _+}\subset \xi _-$ and $R_{\alpha _-}\subset \xi _+$ , yielding

$$ \begin{align*}\alpha_+\wedge d\alpha_-=-\alpha_-\wedge d\alpha_+=0.\end{align*} $$

Therefore, $(\alpha _-,\alpha _+)$ is a taut contact hyperbola in this case. In fact, it satisfies the stronger condition of $\alpha _+\wedge d\alpha _-=-\alpha _-\wedge d\alpha _+=0$ . A taut contact hyperbola with this property is called a $(-1)$ -Cartan structure [Reference Perrone42]. It turns out that not only can this be improved to a geometric characterization of volume-preserving flows, it will also give us a characterization purely in terms of the underlying Reeb flows.

Proof. The above discussion shows that item (1) implies item (2). We can also conclude item (3) from item (2), by noticing that $\alpha _-\wedge d\alpha _+=-\alpha _+\wedge d\alpha _-=0$ yields $R_{\alpha _-}\subset \xi _+$ and $R_{\alpha _+}\subset \xi _-$ . Therefore, the only remaining part is to show that a projectively Anosov flow is volume preserving, with the assumptions of item (3).

We first note that any projectively Anosov flow with $R_{\alpha _-}\subset \xi _+$ and $R_{\alpha _+}\subset \xi _-$ is Anosov, thanks to Theorem 6.3 of [Reference Hozoori33]. Let $\alpha _-$ and $\alpha _+$ be the contact forms in item (3). As in Remark 4.1, we consider the expansion rates $r_s$ and $r_u$ , induced from the decomposition $\alpha _+=\alpha _u-\alpha _s$ . Notice that we can write $\alpha _-=f\alpha _u+g\alpha _s$ for positive X-differentiable functions $f,g>0$ . One can easily check by solving $d\alpha _+(R_{\alpha _+},X)=0$ and using the fact that $\alpha _+(R_{\alpha _+})=1$ (or as it is done in §6 of [Reference Hozoori33]), we can write ( $q_+$ being a real function)

(1) $$ \begin{align} R_{\alpha_+}=\frac{1}{r_u-r_s} (-r_se_u-r_ue_s)+q_+ X, \end{align} $$

and from $\alpha _-(R_{\alpha _+})=0$ , we get

(2) $$ \begin{align} g=-\frac{fr_s}{r_u}. \end{align} $$

Also, using $\alpha _+(R_{\alpha -})=0$ and $\alpha _-(R_{\alpha _-})=1$ , we can write ( $q_-$ being a real function)

(3) $$ \begin{align} R_{\alpha_-}=\frac{1}{f+g}(e_u+e_s)+q_- X. \end{align} $$

However, we have

(4) $$ \begin{align} 0=d\alpha_-(R_{\alpha_-},X)&=d\alpha_-(e_s+e_u,X)=-X\cdot\alpha_-(e_s,e_u)-\alpha_-([e_s+e_u,X])\nonumber\\ &\quad\Rightarrow X\cdot f+X\cdot g+gr_s+fr_u=0. \end{align} $$

Using equation 2 in the above, we get

$$ \begin{align*}X\cdot f+X\cdot g+(f+g)(r_s+r_u)=0,\end{align*} $$

which yields

(5) $$ \begin{align} r_s+r_u=-X\cdot (f+g).\end{align} $$

Finally, to show that the flow is volume preserving, it suffices to define $\Omega :=e^{f+g}\Omega ^{\alpha _+}$ and use Lemma 4.2 and equation (5) to compute

$$ \begin{align*}\mathrm{div}_X\Omega&=X\cdot (f+g)e^{f+g}\Omega^{\alpha_+} + e^{f+g} (\mathrm{div}_X\Omega^{\alpha_+})\Omega^{\alpha_+}\\&=e^{f+g}(X\cdot (f+g)+r_s+r_u)\Omega^{\alpha_+}=0.\\[-3pc]\end{align*} $$

Theorem 5.1 gives examples of $(-1)$ -Cartan structures and taut contact hyperbolas whenever we have volume preserving flows, including the case of algebraic Anosov flows, and more interestingly, we get new examples on hyperbolic manifolds using the examples of contact Anosov flows on those manifolds [Reference Foulon and Hasselblatt21]. The construction also yields new examples of $(-1)$ -Cartan structures on any manifold supporting a transitive Anosov flow, many of which are not contact [Reference Asaoka5, Reference Béguin, Bonatti and Yu10].

5.2 From the viewpoint of Liouville geometry

In [Reference Hozoori33], we have shown how from an Anosov flow X on a 3-manifold M, we can construct two Liouville pairs, $(\alpha _-,\alpha _+)$ and $(-\alpha _-,\alpha _+)$ , where $(\xi _-:=\ker {\alpha _-},\xi _+:=\ker {\alpha _+})$ is a supporting bi-contact structure for X. That is, $\omega _1:=d\alpha _1$ and $\omega _2:=d\alpha _2$ are exact symplectic structures on $[-1,1]_t\times M$ , where $\alpha _1:=(1-t)\alpha _-+(1+t)\alpha _+$ and $\alpha _2:=(1-t)\alpha _--(1+t)\alpha _+$ .

Recall that $(W,d\alpha )$ is an exact symplectic 4-manifold if W is an oriented 4-manifold (with boundary) and $d\alpha $ is an exact symplectic structure on W, that is, $d\alpha \wedge d\alpha>0$ . For any exact symplectic manifold $(W,d\alpha )$ , there exists a unique vector field Y, such that $\iota _Y d\alpha =\alpha $ , or equivalently $\mathcal {L}_Y d\alpha =d\alpha $ . Such a vector field is called a Liouville vector field if it points in the outward direction on $\partial W$ , and the pair $(W,Y)$ is called a Liouville structure. We note that this is the case for an exact symplectic manifold constructed from Anosov 3-flows above, since it is a symplectic filling for the contact manifold $(M,\xi _+)\cup (-M,\xi _-)$ .

The relation between the associated Liouville vector field and the underlying Anosov vector field is more subtle in the general case of $C^1$ Anosov flows. However, for $C^{1+}$ volume preserving Anosov flows, such a connection becomes very straightforward thanks to the symmetries implied by the existence of an invariant volume form and the fact that in this case, the weak stable and unstable bundles are $C^1$ [Reference Hasselblatt30, Reference Hurder and Katok35].

In what follows, we let $(\alpha _-=\alpha _u+\alpha _s,\alpha _+=\alpha _u-\alpha _s)$ be the $(-1)$ -Cartan structure of Theorem 5.1 (in particular, $\Omega ^{\alpha _+}=\alpha _s\wedge \alpha _u\wedge \alpha _X$ is a positive volume form for any 1-form $\alpha _X$ with $\alpha _X(X)=1$ ). We have the 1-form $\alpha _1=(1-t)\alpha _-+(1+t)\alpha _+=2\alpha _u-2t\alpha _s$ on $ [-1,1]_t\times M$ , and compute

$$ \begin{align*}d\alpha_1\wedge d\alpha_1&=\{ 2d\alpha_u-2dt\wedge \alpha_s-2td\alpha_s \} \wedge \{ 2d\alpha_u-2dt\wedge \alpha_s-2td\alpha_s \}\\&=-4dt\wedge \alpha_s\wedge d\alpha_u=4r_udt\wedge \Omega^{\alpha_+},\end{align*} $$

implying that $( [-1,1]_t\times M,d\alpha _1)$ is an exact symplectic manifold. Similarly, we can show $d\alpha _2\wedge d\alpha _2=-4r_sdt\wedge \Omega ^{\alpha _-}$ , yielding another exact symplectic structure on $[-1,1]_t\times M$ . Notice that with the above assumptions, we have $r_u=-r_s>0$ and $\Omega ^{\alpha _+}=\Omega ^{\alpha _-}$ . However, the fact that the weak stable and unstable bundles are $C^1$ in this case, thanks to the Hölder continuity of the derivatives of the flow, plays a crucial role in preserving the symmetry when going from a metric description of the underlying Anosov flow to a contact geometric one, and hence, significantly simplifying the construction of the above Liouville pairs, compared to Anosov flows of lower regularity studied in [Reference Hozoori33]. We also remark that unless X is an algebraic Anosov flow [Reference Ghys27], the $(-1)$ -Cartan structure is a priori only $C^1$ , and therefore, the 2-forms $d\alpha _1$ and $d\alpha _2$ above are exact symplectic structures, only in the $C^0$ sense.

Now, if we define the vector field $Y_1:=({1}/{r_u})X+2t\partial _t$ , we can compute

$$ \begin{align*}\iota_{Y_1} d\alpha_1=\frac{2}{r_u} \iota_X d\alpha_u -4t\alpha_s-\frac{2t}{r_u} \iota_Xd\alpha_s=2\alpha_u-4t\alpha_s+2t\alpha_s=\alpha_1.\end{align*} $$

Therefore, $Y_1$ is the Liouville vector field for $([-1,1]_t\times M,d\alpha _1)$ . Notice that a similar computation helps us compute the Liouville vector field of $([-1,1]_t\times M,d\alpha _2)$ . It can be seen [Reference Simić48] that the 1-forms $\alpha _u$ and $\alpha _s$ can be chosen such that $r_u$ and $r_s$ are $C^1$ . In that case, it is noteworthy and surprising that although the constructed symplectic structures above are a priori only $C^0$ , their corresponding Liouville vector fields are $C^1$ .

Now, we can consider the vector field $X_L:=({1}/{r_u})X$ , which generates a reparameterization of the original flow. Its associated expansion rates are $r^{\prime }_u={r_u}/{r_u}=1$ and $r^{\prime }_s={r_s}/ {r_u}=-1$ (see Remark 3.18 of [Reference Hozoori33]) and we have $Y_1|_{\{ 0\}\times M}=X_L$ . We call such $X_L$ the Liouville reparameterization of a volume preserving Anosov 3-flow (in the sense of [Reference Simić48], this is the synchronization of the flow with respect to both stable and unstable directions, simultaneously).

The following theorem proves that the Liouville reparameterization of a volume preserving Anosov flow has even a closer relation to the underlying Reeb dynamics of Theorem 5.1.

Proof. The above argument yields item (2). To prove item (1), let ( $\alpha _-=\alpha _u+\alpha _s,\alpha _+=\alpha _u-\alpha _s$ ) be the $(-1)$ -Cartan structure of part (2) in Theorem 5.1. We have

$$ \begin{align*}\mathcal{L}_{X_L}\alpha_+=\mathcal{L}_{X_L}(\alpha_u-\alpha_s)=\alpha_u+\alpha_s=\alpha_-.\end{align*} $$

Similarly, one can show $\mathcal {L}_{X_L}\alpha _-=\alpha _+$ . Define the 1-form $\alpha _{X_L}$ by letting $\alpha _{X_L}(X_L)=1$ and $\alpha _{X_L}(\langle R_{\alpha _-},R_{\alpha _+}\rangle )=0$ . The goal is to prove $\mathcal {L}_{X_L}\alpha _{X_L}=0$ . Note that by construction, $\alpha _{X_L}$ is differentiable along the flow and $(\mathcal {L}_{X_L}\alpha _{X_L})\wedge \alpha _{X_L}=0$ .

Also, by plugging the basis $(R_{\alpha _-},R_{\alpha _+},X_L)$ , we can observe

$$ \begin{align*}d\alpha_+=\alpha_{X_L} \wedge \alpha_- \quad\text{and}\quad d\alpha_-=\alpha_{X_L}\wedge \alpha_+,\end{align*} $$

which implies

$$ \begin{align*}(\mathcal{L}_{X_L}\alpha_{X_L})\wedge \alpha_-&=\mathcal{L}_{X_L}(\alpha_{X_L}\wedge \alpha_-)-\alpha_{X_L}\wedge \mathcal{L}_{X_L}\alpha_-\\&=\mathcal{L}_{X_L} d\alpha_+-\alpha_{X_L}\wedge\alpha_+=d(\mathcal{L}_{X_L} \alpha_+)-\alpha_{X_L}\wedge\alpha_+=d\alpha_--d\alpha_-=0.\end{align*} $$

Similarly, we have $(\mathcal {L}_{X_L}\alpha _{X_L})\wedge \alpha _+=0$ . This yields $\mathcal {L}_{X_L}\alpha _{X_L}=0$ , completing the proof.