2. Dynamics on moduli spaces

Let $M=\mathbf{Reeb}(P,\phi)$ be an IP 5-fold, $\alpha \in \mathbf{Reeb}(P,\phi)$ , and $\xi=\ker \alpha$ with Reeb vector field $R_\alpha$ . Let $\theta_B \;:\; B\backslash L\rightarrow S^1$ be a concrete open book on B adapted to $\alpha_B=\alpha\vert_B$ , and $\theta_M \;:\; M\backslash B\rightarrow S^1$ a concrete open book adapted to $\alpha$ . We denote by $P_\theta=\theta_M^{-1}(\theta)$ , $F_\theta=\theta_B^{-1}(\theta)$ , the $\theta$ -pages.

Holomorphic foliations. For the sake of brevity, we will streamline the arguments which have already appeared throughout the literature and give the appropiate references. We proceed to the construction for the proof of Theorem A. We may construct the fibers of a Lefschetz fibration $\pi_\theta \;:\; P_\theta \rightarrow \mathbb{D}^2_\theta$ on each page $P_\theta$ , with regular fiber F, in such a way that these fibrations form a well-defined $S^1$ -family in M; and so that such fibers foliate the contact manifold M, and are symplectic submanifolds in each page. We can do so as follows.

Finite energy foliation over B. First, we may construct an almost complex structure $J_B$ on the 4-dimensional symplectisation $\mathbb{R}\times B$ , together with an $\mathbb{R}$ -invariant and Fredholm regular $J_B$ -holomorphic finite energy foliation $\mathcal{M}_B$ of $\mathbb{R}\times B$ , by punctured holomorphic curves which are asymptotically cylindrical Liouville completions $\widehat F_\theta$ of $F_\theta$ , asymptotic to L, so that $\widehat F_\theta$ projects to $F_\theta$ under the projection $\pi_B \;:\; \mathbb{R}\times B \rightarrow B$ . Moreover, we have $\mathcal{M}_B^q \;:\!=\; \mathcal{M}_B/\mathbb{R}\cong S^1$ . This construction of a “holomorphic” open book has appeared in several forms in the literature (e.g. [ Reference ChrisW10a , Reference CasimA11 ]), and therefore we will omit details. Nevertheless, there is a slight technicality: unlike e.g. in [ Reference ChrisW10a ], the symplectic form in $\mathbb{R}\times M$ is a priori given as $\omega_B=d(e^t \alpha_B)$ . In order to construct the holomorphic foliation, we need to deform $(\alpha_B,d\alpha_B)$ to a stable Hamiltonian structure (SHS) $\mathcal{H}_B=(\lambda_B, d\alpha_B)$ which is tangent to pages away from the binding, and the resulting $J_B$ will be compatible with the symplectisation of $\mathcal{H}_B$ , i.e. will map the $\mathbb{R}$ -direction to the Reeb vector field of $\mathcal{H}_B$ . The resulting holomorphic curves, while not strictly holomorphic for an almost complex structure compatible with $\alpha_B$ , will still be symplectic submanifolds for $\omega_B$ , and this is what we ultimately care about for Theorem A.

Codimension-2 foliation over M. We may then do the same construction, but two-dimensions higher, as follows. Since $\xi$ is supported by the open book $\theta_M$ , away from B we have an isomorphism $TP_\theta\cong \xi\vert_{P_\theta}$ , and on B we have $\xi\vert_B=\xi_B \oplus \xi_B^\perp$ where $\xi_B=\ker \alpha_B$ , and $\xi_B^\perp$ the $d\alpha$ -symplectic complement inside $\xi$ of $\xi_B$ . For each page, one then extends $J_B$ to an almost complex structure $J_\theta$ on $P_\theta$ (viewed as a degenerate Liouville filling of B), compatible with $\omega_\theta=d\alpha\vert_{P_\theta}$ , and generic as a 1-parameter family. One can then deform $(\alpha,d\alpha)$ to a stable Hamiltonian structure $\mathcal{H}=(\lambda, d\alpha)$ with kernel $\xi_\mathcal{H}=\ker \lambda$ deforming the kernel of $\alpha$ , by simply deforming the 1-form, so that $\lambda$ interpolates between $\alpha$ near the binding and $d\theta_M$ away from it. We then have $\xi_\mathcal{H}\cong \xi$ ; $\xi_\mathcal{H}$ is tangent to $P_\theta$ away from a neighbourhood of B, so we still have the property $\xi\vert_{P_\theta}\cong \xi_\mathcal{H}\vert_{P_\theta} \cong TP_\theta$ ; and $\xi_\mathcal{H}=\xi$ near B. Moreover, the Reeb vector field of $\mathcal{H}$ still coincides with $R_\alpha$ . One can then induce an $\mathcal{H}$ -compatible almost complex structure J on $\mathbb{R}\times M$ (i.e. it preserves $\xi_\mathcal{H}$ where it is $d\alpha$ -compatible, and maps $\partial_t$ to $R_\mathcal{\alpha}$ ), with the extra properties that $J\vert_{\mathbb{R}\times B}=J_B$ , the splitting $\xi\vert_B=\xi_B \oplus \xi_B^\perp$ is J-complex, and $J\vert_{\xi_\mathcal{H}\vert_{P_\theta}}$ corresponds to $J_\theta$ under the isomorphism $\xi_\mathcal{H}\vert_{P_\theta} \cong TP_\theta$ . Note that we also have a natural $J^\xi$ which is compatible with $\alpha$ , deforming J, for which the isomorphism $\xi\cong \xi_\mathcal{H}$ becomes complex, and $J^\xi(\partial_t)=R_\alpha$ . One then uses the $\mathcal{H}$ -compatible J to make the open book $\theta_M$ holomorphic, i.e. we obtain a codimension-2 holomorphic foliation $\mathcal{L}$ of $\mathbb{R}\times M$ whose leaves are $\mathcal{L}_B=\mathbb{R}\times B$ and Liouville completions $\widehat P_\theta$ of the pages $P_\theta$ . We have that $\widehat P_\theta$ projects to $P_\theta$ under the projection $\pi_M \;:\; \mathbb{R}\times M\rightarrow M$ , and is asymptotically cylindrical to the $\mathbb{R}$ -invariant hypersurface $\mathcal{L}_B$ in the sense of [ Reference Agustin and RichardMS ]. Moreover, the complex hypersurface $(\widehat P_\theta,J\vert_{\widehat P_\theta})$ is biholomorphic to $(P_\theta,J_\theta)$ . See Figure 6 for a sketch.

Fig. 6. The codimension-2 holomorphic foliation $\mathcal{F}$ .

Moduli space over M. The moduli space $\mathcal{M}_B$ of holomorphic curves in $\mathcal{L}_B$ , which is a copy of $\mathbb{R}\times S^1$ , naturally extends to a moduli space $\mathcal{M}$ in $\mathbb{R}\times M$ . Note that every curve in $\mathcal{M}$ , while J-invariant but not $J^\xi$ -invariant, is $d\alpha$ -symplectic. Moreover, an application of both 4-dimensional and higher-dimensional Siefring intersection theory as in [ Reference Agustin and RichardMS , Reference AgustinMo ] implies that: either the image of a curve in $\mathcal{M}$ lies completely in $\mathcal{L}_B$ and is a leaf of $\mathcal{M}_B$ ; or it lies completely in a leaf of $\mathcal{H}$ (and in particular is disjoint from $\mathcal{L}_B$ except “at infinity”, intersecting it along L). For convenience of the reader, we outline the higher-dimensional version of the argument (as fully explained in [ Reference Agustin and RichardMS ]), which first restricts the behaviour of holomorphic curves to lie inside the leaves of the codimension-2 foliation, so that the 4-dimensional techniques then apply. We remark that the higher-dimensional and 4-dimensional argument is exactly the same.

Controlling curves: Siefring intersection theory. As explained in [ Reference Agustin and RichardMS ], there is a well-defined and homotopy-invariant Siefring intersection pairing $u*H$ for a curve $u\in \mathcal{M}$ and a hypersurface $H\in \mathcal{F}$ . We claim that this pairing vanishes: using homotopy invariance with respect to H completely analogously as e.g. [ Reference Janko and ChrisLW , lemma 4·9], we have

\[u*H=u*(\mathbb{R}\times B).\]

From homotopy invariance with respect to u, it suffices to compute the above in the case where $u\in \mathcal{M}_B$ lies completely in $\mathbb{R}\times B$ . Then we can appeal to the following intersection formula [ Reference RichardSie ]:

\[u*(\mathbb{R}\times B)=\frac{1}{2}(\mu_N^\tau(u)-\#\Gamma(u)_{odd}),\]

where $\mu_N^\tau(u)$ is the total normal Conley-Zehnder index of u with respect to a trivialisation $\tau$ of the symplectic normal bundle to $\mathbb{R}\times B$ , and $\#\Gamma(u)_{odd}$ is the number of asymptotics of u which have odd normal Conley–Zehnder index. In our case, each asymptotic of u has normal index equal to 1, and the claim that $u*H=0$ follows. Moreover, if u did not lie in a leaf of $\mathcal{F}$ , this pairing would be strictly positive, by the fact that interior intersections contribute positively, and intersections at infinity all vanish (the asymptotic winding numbers of u all vanish, and these are the extremal ones). This would be a contradiction, and so we get the desired restrictions of elements in $\mathcal{M}$ .

SFT-Gromov Compactification. We may then consider the SFT-Gromov compactification (where SFT stands for Symplectic Field Theory) $\overline{\mathcal{M}}$ of $\mathcal{M}$ by adding strata of nodal curves, as well as its pointed version $\overline{\mathcal{M}}_*$ by adding a single marked point in the domain of each curve, together with the resulting evaluation map $ev \;:\; \overline{\mathcal{M}}_* \rightarrow \mathbb{R}\times M$ which evaluates a curve at its marked point, and the forgetful map $\pi_* \;:\; \overline{\mathcal{M}}_*\rightarrow \overline{\mathcal{M}}$ which forgets the marked point. Moreover, one can apply Siefring intersection theory to the compactified moduli space too, and still conclude that the components of each element lie in the holomorphic hypersurfaces. Applying the results in [ Reference ChrisW10b ] (namely, theorem 7 and its parametric version theorem 8), one concludes that each $P_\theta$ admits the claimed generic Lefschetz fibration $\pi_\theta\;:\;P_\theta \rightarrow \mathbb{D}^2_\theta$ , each inducing the original open book at B. Note that each such fibration is minimal (i.e. there are no contractible vanishing cycles), due to exactness of the symplectic form, and moreover every curve is immersed. In particular, non-nodal curves are embedded, and the nodal curves have embedded components, intersecting each other transversely; the same is true even after projecting via $\pi_M$ .

This means that $\overline{\mathcal{M}}_*$ naturally admits a stratification $\overline{\mathcal{M}}_*=\mathcal{M}_*^0\bigsqcup \mathcal{M}_*^1$ , where $\mathcal{M}_*^i$ consists of pointed curves with precisely i nodes, and having closure $\overline{\mathcal{M}}_*^i=\bigcup_{j\leq i} \mathcal{M}_*^i$ . In particular $\mathcal{M}_*^0$ is the top open stratum consisting of non-nodal curves, and $\mathcal{M}_*^1$ is closed. We similarly have a stratification for the unmarked moduli space $\overline{\mathcal{M}}=\mathcal{M}^0\bigsqcup \mathcal{M}^1$ , where $\mathcal{M}^i=\pi_*(\mathcal{M}^i_*)$ . The expected dimension of $\mathcal{M}_*^i$ is $6-2i$ (and that of $\mathcal{M}^i$ is $4-2i$ ), and from a similar analysis as carried out in [ Reference AgustinMo , section 4·7] for curves lying in holomorphic hypersurfaces, one can show that each nodal strata $\mathcal{M}^i$ is Fredholm regular.

Structural diagrams. The upshot is that we have a diagram of the form

where ev is a diffeomorphism since elements in $\overline{\mathcal{M}}$ together with trivial cylinders over orbits in L foliate $\mathbb{R}\times M$ , and so $\overline{\mathcal{M}}_*^q \;:\!=\; \overline{\mathcal{M}}_*/\mathbb{R}\cong M\backslash L$ , and we denote $\pi=\pi_* \circ ev^{-1}$ . We also used that $\overline{\mathcal{M}}\cong \mathbb{R}\times S^3$ , and so $\overline{\mathcal{M}}^q \;:\!=\; \overline{\mathcal{M}}/\mathbb{R}\cong S^3$ . Namely, $\overline{\mathcal{M}}^q$ is naturally equipped with the trivial open book $\overline{\mathcal{M}}^q=\mathbf{OB}(\mathbb{D}^2,\unicode{x1d7d9})$ , with $\theta$ -page $P_\theta^\mathcal{M} \cong \mathbb{D}^2_\theta$ corresponding to the 2-disk $\mathbb{D}^2_\theta$ at the base of the Lefschetz fibration $\pi_\theta$ , via $\mathbb{D}_\theta^2=(\pi_\theta \circ ev^q \circ (\pi_*^q)^{-1})(P_\theta^\mathcal{M})$ , where $ev^q$ and $\pi^q_*$ are the quotient maps induced by ev and $\pi_*$ , respectively. Its binding is identified with $\mathcal{M}_B^q \;:\!=\; \mathcal{M}_B/\mathbb{R}\cong S^1$ . By quotienting out the $\mathbb{R}$ -action, and denoting $\pi^q=\pi_*^q \circ (ev^q)^{-1}$ , we obtain a similar diagram

See Figure 7, where we denote $\mathcal{M}_*^{q,i}=\mathcal{M}_*^{i}/\mathbb{R}$ , $\mathcal{M}^{q,i}=\mathcal{M}^{i}/\mathbb{R}=\pi_*(\mathcal{M}_*^{q,i})$ , for $i=0,1$ . We denote by $\theta_\mathcal{M} \;:\; \overline{\mathcal{M}}^q\backslash \mathcal{M}_B^q\rightarrow S^1$ the open book fibration on the quotient moduli space, which we call the shadowing open book. This finishes the construction which proves Theorem A.

Fig. 7. The forgetful map, and the stratification in the quotient moduli spaces. In the above picture, the boundary of each page is identified to each other.

Symplectic and contact forms on moduli spaces. We now proceed to the construction underlying Theorem B. Let

\[\mathcal{P}=\{\varphi \in C^\infty(\mathbb{R},(0,1)) \;:\; \varphi^\prime>0\}\]

be the space of orientation-preserving diffeomorphisms between $\mathbb{R}$ and (0, 1). For each $\varphi \in \mathcal{P},$ we can induce a symplectic form $\omega_{\mathcal{M}}^\varphi$ on $\overline{\mathcal{M}}$ and a ( $\varphi$ -independent) contact form $\alpha_\mathcal{M}$ on $\overline{\mathcal{M}}^q$ , such that

\[(\overline{\mathcal{M}}\cong \mathbb{R}\times S^3,\omega^\varphi_{\mathcal{M}}=d(e^{\varphi(t)} \alpha_{\mathcal{M}}))\]

is the symplectisation of the contact manifold $(\overline{\mathcal{M}}^q\cong S^3, \alpha_{\mathcal{M}}).$ Indeed, let

\[\omega^\varphi \;:\!=\; d(e^{\varphi(t)}\alpha) \in \Omega^2(\mathbb{R}\times M)\]

for $\varphi \in \mathcal{P}$ . Denote $\omega^\varphi_* \;:\!=\; ev^*\omega^\varphi \in \Omega^2(\overline{\mathcal{M}}_*)$ , and $F_u=\pi_*^{-1}(u)$ the fiber over $u \in \overline{\mathcal{M}}$ (its domain). By construction, $\omega^\varphi_*$ is a symplectic form in $\overline{\mathcal{M}}_*$ , which is symplectic on the fibers of $\pi_*$ , i.e. $\omega_u^\varphi=\omega^\varphi_*\vert_{F_u}$ is an area form on the domain of u. Moreover, there is a tautological notion of a symplectic connection for $\pi_*$ , so that every vector $v \in T_u \overline{\mathcal{M}}$ has a horizontal lift $\tilde{v} \in T_{(u,z)}\overline{\mathcal{M}}_*$ , for every $(u,z) \in F_u$ (where z lies in the domain of u). Indeed, a tangent vector field $v \in T_u \overline{\mathcal{M}}$ is simply a vector field along $F_u$ , taking values in the (generalised; see below) normal bundle $N_u$ and lying in the kernel of the normal linearised Cauchy–Riemann operator $\mathbf{D}_u^N$ at u. The horizontal lift $\widetilde{v}$ of v at (u, z) is simply $\widetilde{v}=v(z) \in N_u\vert_z$ , i.e. the vector field itself. This also makes sense along nodal curves. Indeed, recall that the tangent space to a fixed (smooth) stratum $\mathcal{M}^i$ of the moduli space along a nodal curve $u=(u_1,\dots,u_{i+1})$ consists of tuples $\eta=(\eta_1,\dots,\eta_{i+1})$ of normal sections $\eta_j$ along each component $u_j$ , lying in the kernel of $\bigoplus_{j=1;EV}^{i+1}\mathbf{D}^N_{u_j}$ , the fiber-product (under the linear evaluation map EV at the nodes) of the normal linearised CR-operators at each $u_j$ ; so in particular each element in the tuple $\eta$ agrees at each nodal pair. Tangent vectors along a nodal curve which are not tangent to the fixed stratum containing it can also be thought of as vector fields along the curve (as follows from standard gluing analysis). So the above notion of a horizontal lift carries through immediately.

For $v,w \in T_u \overline{\mathcal{M}}$ , let $\omega_*^\varphi(v,w) \;:\!=\; \omega^\varphi_*(\widetilde{v},\widetilde{w})$ viewed as a function on $F_u$ , and define $\omega^\varphi_{\mathcal{M}} \in \Omega^2(\overline{\mathcal{M}})$ via

\[(\omega^\varphi_{\mathcal{M}})_{u}(v,w)=\int_{z\in F_u} \omega_*^\varphi(v(z),w(z))dz,\]

where by simplicity we denote $dz=\omega_u^\varphi$ . We remark that this same construction for the special case of closed and immersed curves (not including nodal degenerations) has been carried out in [ Reference Joseph, Liat and ÁlvaroCKP ], although with a slightly different language. In general, generically and in dimension 6, the locus of non-immersed curves consists of isolated curves with a single simple critical point (see [ Reference ChrisW19 , Appendix A] in for the closed case). This, and the fact that there is still a well-defined normal bundle at critical points –the generalised normal bundle [ Reference ChrisW10c ]–, implies that the above integral is well-defined. In our special setup, we have already seen that every curve is immersed, so this is not even an issue. While we moreover have punctures in our setup, the finiteness of the integral is ensured by our choice of a diffeomorphism $\varphi$ . The fact that $\omega^\varphi_{\mathcal{M}}$ is symplectic follows by adapting the arguments in [ Reference Joseph, Liat and ÁlvaroCKP ] (except the nondegeneracy property: here J is not necessarily integrable, and so the kernel of $\mathbf{D}^N_u$ is not necessarily a complex space. But we understand precisely what the kernel is, due to the foliation property and Fredholm regularity, and hence immediately see that non-degeneracy follows; see below for more details). The contact form on $\overline{\mathcal{M}}^q$ is obtained from the symplectic form in the obvious way, i.e. as

\[\alpha_{\mathcal{M}}=(e^{\varphi}\varphi^\prime)^{-1}i_{\partial_t}\omega^\varphi_{\mathcal{M}} \in \Omega^1(\overline{\mathcal{M}}^q).\]

By construction, this contact form is independent on the choice of $\varphi$ . Indeed, we can give a more explicit description as follows. First note that $d\alpha\vert_{F_u}$ is a symplectic form on the interior of $F_u$ (which degenerates at the boundary since $\partial F_u=L$ is a collection of $\alpha$ -orbits). The volume of $F_u$ with respect to $d\alpha\vert_{F_u}$ is, by Stokes’ theorem, given by

\[\mbox{vol}(F_u)=\int_{F_u}d\alpha\vert_{F_u}=\int_{L}\alpha \;=\!:\; A_\alpha>0,\]

the total $\alpha$ -action of L, which is independent of u. We then have

\[(\alpha_\mathcal{M})_u(v)=\int_{z\in F_u}\alpha_z(v(z))dz,\]

\[(d\alpha_\mathcal{M})_u(v,w)=\int_{z\in F_u}d\alpha_z(v(z),w(z))dz,\]

for $v,w\in T\overline{\mathcal{M}}^q$ , where for simplicity we use the notation dz to indicate that we integrate over z with respect to the area form $d\alpha\vert_{F_u}$ , and where we use the same letter u to denote the projection of $u \in \mathcal{M}$ to $\mathcal{M}^q=\mathcal{M}\backslash \mathbb{R}$ (which is also embedded). The contact structure $\xi_{\mathcal{M}}=\ker \alpha_{\mathcal{M}}$ can be understood, over u, as the average of the contact structures $\xi_z$ for $z\in F_u$ ; see Lemma 2·1 below. We call $\xi_\mathcal{M}$ , the shadowing contact structure. Moreover, it is supported by the trivial open book on $S^3$ , and hence it is isotopic to the standard contact structure on $S^3$ . This can be understood as follows.

The complex normal bundle. We can give the following explicit description of the complex normal bundle $N_u$ for each $u\in \overline{\mathcal{M}}$ . Let $\mbox{Hor}_\theta$ denote the $\omega_\theta$ -symplectic complement to u inside $P_\theta$ , which is a symplectic connection for $\pi_\theta$ , i.e. $TP_\theta=\mbox{Hor}_\theta \oplus \mbox{Vert}_\theta$ , $\mbox{Vert}_\theta=\ker d\pi_\theta$ , $\mbox{Hor}_\theta \cong T\mathbb{D}^2_\theta$ under $d\pi_\theta$ ; this splitting for $TP_\theta$ is also $J_\theta$ -complex. Using the complex isomorphism $(TP_\theta,J_\theta)\cong (\xi_\mathcal{H}\vert_{P_\theta}, J\vert_{\xi_\mathcal{H}\vert_{P_\theta}})$ , the $J_\theta$ -complex splitting $TP_\theta=\mbox{Hor}_\theta \oplus \mbox{Vert}_\theta$ then induces a J-complex splitting $\xi_\mathcal{H}\vert_{P_\theta}= \mbox{Hor}_\theta^\mathcal{H} \oplus \mbox{Vert}^\mathcal{H}_\theta$ , and similarly a $J^\xi$ -complex splitting $\xi =\mbox{Hor}_\theta^\mathcal{\xi} \oplus \mbox{Vert}^\mathcal{\xi}_\theta$ . Under $d\pi \;:\; T(M\backslash L)\rightarrow T\overline{\mathcal{M}}^q$ , $\mbox{Hor}_\theta$ projects to $TP_\theta^\mathcal{M}$ . See Figure 8. If we denote by $\widehat N_\theta$ the $\omega$ -symplectic complement of $\widehat P_\theta$ , then, if $u\subset \widehat P_\theta$ , we have $N_u=\mbox{Hor}_\theta\vert_u \oplus \widehat N_\theta\subset u^*T(\mathbb{R}\times M)$ . Moreover, the tangent space $T_u\overline{\mathcal{M}}=\ker \mathbf{D}_u\subset W^{1,2}(N_u)$ is a 4-dimensional subspace consisting of smooth sections of $N_u$ , and we have the explicit description $\ker \mathbf{D}_u=\langle e_1,e_2 \rangle \oplus \langle \partial_t,n\rangle$ , where $e_1,e_2=J_\theta(e_1)$ point-wise span $\mbox{Hor}_\theta\vert_u$ (or equivalently span $P_\theta^\mathcal{M}\vert_u$ when viewed as tangent to $\mathcal{M}^q$ ), and n corresponds to the Hopf direction. In other words, vectors in $\mbox{Hor}_\theta$ correspond to nearby fibers of the Lefschetz fibration in the same page, $\partial_t$ corresponds to $\mathbb{R}$ -translation, and n, to curves in nearby pages. We have an analogous description in the case when $u\subset \mathcal{M}_B$ .

Fig. 8. The complex splitting $TP_\theta=\mbox{Hor}_\theta \oplus \mbox{Vert}_\theta$ , and the induced splitting $\xi_\mathcal{H}=\mbox{Hor}^\mathcal{H}_\theta \oplus \mbox{Vert}^\mathcal{H}_\theta$ . The symplectic connection $\mbox{Hor}_\theta$ projects to $T_uP_\theta^\mathcal{M}$ . Near the binding B, we have $\mbox{Hor}^\xi_\theta=\mbox{Hor}_\theta^\mathcal{H}$ and $\mbox{Vert}^\xi_\theta=\mbox{Vert}_\theta^\mathcal{H}$ . Away from B, $\mbox{Hor}_\theta=\mbox{Hor}_\theta^\mathcal{H}$ and $\mbox{Vert}_\theta=\mbox{Vert}_\theta^\mathcal{H}$ . This structure naturally induces a stable Hamiltonian structure deformation $\xi_{\mathcal{H}_\mathcal{M}}$ of $\xi_\mathcal{M}$ with Reeb vector field $R_\mathcal{M}$ . The contact structure $\xi_\mathcal{M}$ is the average plane in the cone $C_\xi=d\pi(\xi)$ , and similarly, $\xi_{\mathcal{H}_\mathcal{M}}$ is the average plane in $C_\mathcal{H}=d\pi(\xi_\mathcal{H})$ ; see Lemma 2·1 below.

The shadowing open book supports the shadowing contact structure. Since the open book $M=\mathbf{OB}(P,\phi)$ supports the contact structure $\xi$ , away from the binding B we have an isomorphism $\xi\vert_{P_\theta} \cong TP_\theta$ , for each $\theta$ . Therefore, the splitting $TP_\theta=\mbox{Vert}_\theta \oplus \mbox{Hor}_\theta$ induces a splitting $\xi\vert_{P_\theta} = \mbox{Vert}_\theta^\xi \oplus \mbox{Hor}_\theta^\xi$ . Then $\xi_{\mathcal{M}}$ is given along the interior of the $\theta$ -page $P_\theta^\mathcal{M}$ as the average

\[\xi_{\mathcal{M}}\vert_{P_\theta^\mathcal{M}}=\int_{z\in F_u}d\pi^q(\mbox{Hor}^\xi_\theta(z))dz.\]

See Remark 2·2 below for details on how to order to interpret this expression.

Since $\mbox{Hor}_\theta \cong T\mathbb{D}^2_\theta$ , $\xi_{\mathcal{M}}\vert_{P_\theta^\mathcal{M}}$ is isomorphic to $TP_\theta^\mathcal{M}=d\pi^q(T\mathbb{D}^2_\theta)$ . This, together with the fact that its binding $\mathcal{M}_B^q$ is a Reeb orbit (see the discussion of the holomorphic shadow below), means that the trivial open book supports the contact structure $\xi_\mathcal{M}$ . Note that, for $u \in \mathcal{M}^q_B$ , the contact structure $\xi_{\mathcal{M}}\vert_u$ is identified with $\xi_B^\perp\vert_u$ . Here, $\xi_B^\perp$ is the symplectic normal bundle of $\xi_B$ inside $\xi\vert_B$ with respect to $d\alpha\vert_{\xi}$ , which is symplectically trivial along u. One can then intepret the distribution $\mbox{Hor}^\xi$ across the binding as $\mbox{Hor}^\xi\vert_B=\xi_B^\perp$ .

The shadowing almost complex structure. We have that $J^\xi$ induces a shadowing almost complex structure $J^\xi_\mathcal{M}$ on $\ker \mathbf{D}_u$ , as follows. Consider the Hilbert space $W^{1,2}(N_u)$ of $W^{1,2}$ -sections of the normal bundle of u, endowed with the inner product

\[g_u(v,w)=\int_{z\in F_u} g_z(v(z),w(z))dz,\]

where $g_z=d\alpha_z(\cdot,J^\xi \cdot)+\alpha_z\otimes \alpha_z+dt \otimes dt$ . By varying u, we can view g as a Riemannian metric on the Banach bundle $W^{1,2}(N_u)\hookrightarrow \mathcal{W} \rightarrow \mathcal{M}$ . We denote by $g^q$ the metric induced by g on the quotient Banach bundle $W^{1,2}(N_u\backslash \mathbb{R})\hookrightarrow \mathcal{W}^q=\mathcal{W}\backslash \mathbb{R} \rightarrow \mathcal{M}^q=\mathcal{M}\backslash \mathbb{R}$ . The almost complex structure $J^\xi$ gives an endomorphsim

\[J^\xi_u \;:\; W^{1,2}(N_u)\Longrightarrow W^{1,2}(N_u),\]

\[J^\xi_u(v(z))=J^\xi_z(v(z)),\]

for $z\in F_u$ , compatible with $g_u$ . Then $\omega_u(v,w)=g_u(v,J^\xi_u(w))$ is a symplectic form on $W^{1,2}(N_u)$ , which can be viewed as a fiber-wise symplectic form in the Banach bundle $\mathcal{W}$ . Let

\[P_u \;:\; W^{1,2}(N_u)\Longrightarrow \ker \mathbf{D}_u\]

denote the $L^2$ -orthogonal projection with respect to $g_u$ . Then $J^\xi_\mathcal{M}$ is defined via

\[(J^\xi_\mathcal{M})_u(v(u))(z)=P_u(J_u^\xi(v(z))),\]

for $z\in F_u$ , and $v\in \ker \mathbf{D}_u$ . It is compatible with $\omega_\mathcal{M}^\varphi$ . The splitting $\ker \mathbf{D}_u=\xi_\mathcal{M}\oplus \langle \partial_t,R_\mathcal{M}\rangle$ is $J_\mathcal{M}^\xi$ -complex, where $R_\mathcal{M}$ denotes the Reeb vector field of $\alpha_\mathcal{M}$ , which follows from the fact that $P_u(R_\alpha\vert_u)$ is a multiple of $R_\mathcal{M}$ (see Equation (2·2) below).

The shadowing stable Hamiltonian structure. We have a shadowing stable Hamiltonian structure $\mathcal{H}_\mathcal{M}=(\lambda_\mathcal{M},d\alpha_\mathcal{M})$ on $\overline{\mathcal{M}}^q$ whose kernel $\xi_{\mathcal{M}_H}$ deforms $\xi_\mathcal{M}$ , exactly as for $\mathcal{H}$ and $\xi$ , i.e. $\xi_{\mathcal{H}_\mathcal{M}}$ is tangent to $TP_\theta^\mathcal{M}$ away from $\mathcal{M}_B^q$ , and the Reeb vector field of $\mathcal{H}_\mathcal{M}$ is $R_\mathcal{M}$ , the Reeb vector field of $\alpha_\mathcal{M}$ . The splitting $\ker \mathbf{D}_u=\mathcal{H}_\mathcal{M}\oplus \langle \partial_t,R_\mathcal{M}\rangle$ is $J_\mathcal{M}$ -complex, where $J_\mathcal{M}$ is induced from J analogously as above, i.e. via $(J_\mathcal{M})_u(v(z))=P^\mathcal{H}_u(J_z(v(z)))$ , where $P^\mathcal{H}_u \;:\; W^{1,2}(N_u)\rightarrow \ker \mathbf{D}_u$ is the orthogonal projection with respect to

\[g^\mathcal{H}_u(v,w)=\int_{z\in F_u} g^\mathcal{H}_z(v(z),w(z))dz,\]

with $g^\mathcal{H}_z=d\alpha_z(\cdot,J \cdot)+\lambda_z\otimes \lambda_z+dt \otimes dt$ .

The (pseudo-)holomorphic shadow. We call the Reeb flow of $\alpha_\mathcal{M}$ on $\overline{\mathcal{M}}^q$ , the holomorphic shadow of the Reeb flow of $\alpha$ on M, generated by $R_\mathcal{M}$ . In fact, $R_{\mathcal{M}}\vert_u$ is a positive reparametrisation of the $L^2$ -orthogonal projection of $R_\alpha\vert_u$ (a vector field in M normal to u, i.e. in $W^{1,2}(N_u)$ ) to the kernel of $\mathbf{D}_u^N$ , i.e. the tangent space to the moduli space. This can be proved as follows.

By construction, the Reeb vector field $R_\mathcal{M}\in T\overline{\mathcal{M}}^q$ of $\alpha_\mathcal{M}$ is defined by the equations

\begin{equation*} \mathbf{D}_uR_\mathcal{M}=0, \end{equation*}

(2·1) \begin{equation}1=(\alpha_\mathcal{M})_u(R_\mathcal{M}(u))=\int_{z\in F_u}\alpha_z(R_\mathcal{M}(z))dz,\end{equation}

\[0=(d\alpha_\mathcal{M})_u(R_\mathcal{M}(u),\cdot)=\int_{z\in F_u}d\alpha_z(R_\mathcal{M}(z),\cdot)dz.\]

We claim that

(2·2) \begin{equation}R_\mathcal{M}(u)=\frac{P_u(R_\alpha\vert_u)}{(\alpha_\mathcal{M})_u(P_u(R_\alpha\vert_u))}.\end{equation}

This can be checked as follows. Note that the metric $g^q_\mathcal{M}=g^q\vert_{T\mathcal{M}^q}$ , where we view $T\overline{\mathcal{M}}^q\subset \mathcal{W}^q$ as a rank-3 subbundle, can in fact be expressed as

\[g^q_\mathcal{M}=d\alpha_\mathcal{M}(\cdot,J^\xi_\mathcal{M}\cdot)+\Vert R_\mathcal{M}\Vert^2\alpha_\mathcal{M}\otimes \alpha_\mathcal{M},\]

where

\[\Vert R_\mathcal{M}(u) \Vert^2=(g^q_\mathcal{M})_u(R_\mathcal{M}(u),R_\mathcal{M}(u))=\int_{z\in F_u} \left[\alpha_z(R_\mathcal{M}(z))\right]^2dz.\]

We then consider a basis $e_1,e_2=J^\xi_\mathcal{M}(e_1) \in \xi_{\mathcal{M}}, e_3 = 1/{\Vert R_\mathcal{M}\Vert} R_\mathcal{M}$ of $T_u\mathcal{M}^q$ , which is orthonormal with respect to $g^q_\mathcal{M}$ . Then

\[P_u(R_\alpha\vert_u)=\sum_{i=1}^3g_u(R_\alpha\vert_u,e_i(u))e_i(u).\]

Moreover,

\[g_u(R_\alpha\vert_u,e_i(u))=\int_{z\in F_u} g_z(R_\alpha(z),e_i(z))dz=\int_{z\in F_u} \alpha_z(e_i(z))dz=(\alpha_\mathcal{M})_u(e_i(u)),\]

and therefore

\[P_u(R_\alpha\vert_u)=\frac{R_\mathcal{M}(u)}{\Vert R_\mathcal{M}(u)\Vert^2},\]

with $(\alpha_\mathcal{M})_u(P_u(R_\alpha\vert_u))={1}/{\Vert R_\mathcal{M}(u)\Vert^2}$ . This proves the claim.

The shadow as a flow on M preserving the foliation. To gain some more insight on the holomorphic shadow, we need better understanding of the tangent space to $\overline{\mathcal{M}}^q$ . If u lies in a hypersurface $H \in \mathcal{L}$ , there is a natural splitting of the normal operator $\mathbf{D}_u^N=\mathbf{D}_u^{T_H} \oplus \mathbf{D}_u^{N_H}$ into tangent and normal components with respect to H (cf. [ Reference AgustinMo , section 4·7]). Here, $N_H$ is the normal bundle to H, which is a choice of holomorphic complement to H which near infinity is $\mathbb{R}$ -invariant and coincides with $\xi_B^\perp \subset \xi\vert_B$ . If H is not cylindrical and projects to the $\theta$ -page $P_\theta$ , then $\ker \mathbf{D}_u^{N_H}$ is 2-dimensional, spanned by the $\mathbb{R}$ -direction $\partial_t$ and a vector field $n_M\vert_u$ which takes values in $N_H\vert_u$ ; while $\ker \mathbf{D}_u^{T_H}$ is identified under $d\pi_\theta\vert_u$ with $T_{\pi_\theta(u)}\mathbb{D}^2_\theta$ , parametrising the fibers of the Lefschetz fibration $\pi_\theta$ near to u. If $H=\mathcal{L}_B$ is cylindrical over B, then $\ker \mathbf{D}_u^{N_H}=0$ while $\ker \mathbf{D}_u^{T_H}$ is spanned by $\partial_t$ and a vector field $n_B\vert_u$ which takes values in $N_u\cap TB$ . We may choose the vector fields $n_B$ and $n_M$ so that they glue smoothly together to a vector field n, in such a way that, when we identify $\mathcal{M}^q\cong S^3$ , the flow of n is precisely the Hopf flow on $S^3$ . The projection of $R_\alpha\vert_u$ to $\ker \mathbf{D}_u^N$ can then be written as

\[R_{\mathcal{M}}\vert_u=\left\{\begin{array}{c@{\quad}c} F(u)n\vert_u, & \mbox{ if } u \in \mathcal{M}_B^q \\[5pt] F(u)n\vert_u+R_{\mathcal{M}}^T\vert_u & \mbox{ if } u \in \overline{\mathcal{M}}^q\backslash \mathcal{M}_B^q\end{array}\right.,\]

where $F \;:\; \overline{\mathcal{M}}^q\rightarrow \mathbb{R}^+$ is a smooth positive function, and $R_{\mathcal{M}}^T\vert_u \in \ker \mathbf{D}^{T_H}_u\cong T_{\pi_\theta(u)} \mathbb{D}^2$ , where $H \in \mathcal{H}$ projects to $P_\theta$ and contains u. This fully completes the proof that the trivial open book in $\overline{\mathcal{M}}^q=\mathbf{OB}(\mathbb{D}^2,\unicode{x1d7d9})$ is adapted to the Reeb dynamics of $\alpha_\mathcal{M}$ , its binding $\mathcal{M}^q_B$ being a Reeb orbit. This completes the proof of Theorem B.

The shadowing cone. We define the shadowing cone by $C_\alpha \;:\!=\; d\pi(\!\ker d\alpha)\subset T\overline{\mathcal{M}}^q$ , where $\pi \;:\; M\backslash L \rightarrow \overline{\mathcal{M}}^q$ is the quotient map to the leaf space. We also define the cones $C_\xi \;:\!=\; d\pi(\xi)$ , and $C_\mathcal{H} \;:\!=\; d\pi(\xi_\mathcal{H})$ .

Remark 2·2. Formula (2·3) can be interpreted as follows. The intersection of $C_\alpha$ with the $g^q_\mathcal{M}$ -unit sphere $S_\mathcal{M}\subset T\overline{\mathcal{M}}^q$ , which parametrises the directions in $C_\alpha$ , is, away from $\mathcal{M}^q_B$ , a closed subset of the upper hemisphere $S_\mathcal{M} \cap \{d\theta_\mathcal{M}> 0\}$ of $S_\mathcal{M}$ . This subset can therefore be globally parametrised by two angles $(\phi_1,\phi_2)$ (longitude and latitude); the direction $\ker d\alpha_\mathcal{M}$ then corresponds to

\[(\phi_1(R_\mathcal{M}),\phi_2(R_\mathcal{M}))=\frac{R_\mathcal{M}}{\Vert R_\mathcal{M} \Vert}=\int_{z\in F_u}(\phi_1(z),\phi_2(z))dz.\]

Along $\mathcal{M}_B^q$ , equation (2·3) also holds, where the average direction is the only direction, i.e.

\[\ker d\alpha_\mathcal{M}\vert_{\mathcal{M}_B^q}=T^+\mathcal{M}^q_B=C_\alpha\vert_{\mathcal{M}_B^q},\]

where $T^+\mathcal{M}^q_B$ are the vectors in $T\mathcal{M}^q_B$ which point in the positive direction according to the orientation of $\mathcal{M}^q_B=S^1$ . The expressions in (2·4) can be understood in a similar fashion. Namely, the metric $g^q_\mathcal{M}$ induces a diffeomorphism between the Grassmannian of co-oriented 2-planes in $T\overline{\mathcal{M}}^q$ and $S_\mathcal{M}$ , by mapping such a 2-plane to the unique unit vector which is positively orthogonal to the plane; the above considerations then apply.

Proof of Lemma 2·1. For the first part, we need to show that $(\alpha_\mathcal{M})_u(d_z\pi(R_\alpha(z)))$ is positive for $z\in F_u$ , $u=\pi(z)$ . If $e_1,e_2=J^\xi_\mathcal{M}(e_1) \in \xi_{\mathcal{M}}, e_3=R_\mathcal{M}$ is the $g^q_\mathcal{M}$ -orthogonal basis for $T_u\mathcal{M}^q$ considered above, we have

\[d_z\pi(R_\alpha(z))=\sum_{i=1}^3 g_z(R_\alpha(z),e_i(z))e_i(u)=\sum_{i=1}^3 \alpha_z(e_i(z))e_i(u),\]

and so

\[(\alpha_\mathcal{M})_u(d_z\pi(R_\alpha(z)))=\alpha_z(R_\mathcal{M}(z)),\]

which is strictly positive. This proves the first claim. The second claim follows immediately from the fact that $R_\alpha$ is positively transverse to $P_\theta$ for every $\theta$ , away from B. The third and fifth follows by construction. The fourth follows from the fact that B is invariant under $R_\alpha$ . This proves the lemma.

A non-trivial example: the Katok example in dimension 5. Now we look at the 5-dimensional instances of the well-known examples by Katok [ Reference KatokK73 ], of Finsler metrics on $S^3$ with only finitely many simple closed geodesics. It turns out that the holomorphic shadow construction can be understood explicitly, and corresponds to the Reeb dynamics on the boundary of suitable irrational ellipsoids.

We follow the discussion in appendix A in [ Reference Agustin and OttoMvK2 ]. Namely, we consider the Brieskorn manifold

\[\Sigma^{5} \;:\!=\; \left\{ (z_0,z_1,z_2,z_3) \in \mathbb{C}^{4} \;\Bigg|\;\sum_j z_j^2 = 0 \right\}\cap S^7,\]

equipped with the contact form $\alpha =\frac{i}{2}\sum_j z_j d\bar z_j -\bar z_j dz_j$ . This space is contactomorphic to $S^*S^3$ with its canonical contact structure. We consider the unitary change of coordinates:

\[w_0=z_0,w_1=z_1,w_{2}=\frac{\sqrt 2}{2} (z_{2}+i z_{3}),w_{3}=\frac{i \sqrt 2}{2} (z_{2}-i z_{3}).\]

For $\epsilon>0$ small and irrational, define the function

\[H_\epsilon(w)=\Vert w \Vert^2+\epsilon(|w_{2}|^2-|w_3|^2),\]

and perturb the contact form as $\alpha_\epsilon = H_\epsilon^{-1} \cdot \alpha.$ The Reeb flow is seen to be

\[(w_0,w_1,w_2,w_3)\longmapsto(e^{2\pi it}w_0,e^{2\pi it}w_1,e^{2\pi it(1+\epsilon)}w_2,e^{2\pi it(1-\epsilon)}w_3).\]

This flow has only 4 periodic orbits by irrationality of $\epsilon$ . We have a supporting open book for the contact form $\alpha_\epsilon$ given by

\[\pi_0 \;:\; \Sigma^{5} \longrightarrow \mathbb{C},\]

\[(w_0,w_1,w_2,w_3) \longmapsto w_0.\]

The zero set of $\pi_0$ defines the binding $\mathbb{R}P^3$ , and the pages are the sets of the form $P_\theta=\{\arg \pi_0=\theta\}$ , $\theta \in S^1$ , which are all copies of $\mathbb{D}^*S^{2}$ . As in [ Reference Agustin and OttoMvK2 ], only two of the four orbits correspond to interior fixed points of the resulting return map, which actually has no boundary fixed points; the other two orbits lie in the binding, but have periods which differ from the return time. Similarly, define the maps

\[\pi_i \;:\; \Sigma^5 \longrightarrow \mathbb{C},\]

\[(w_0,w_1,w_2,w_3) \longmapsto w_i,\]

for $i=1,2,3$ . The Reeb flow then preserves the set of fibers of $\pi_i$ . For $i=1$ , $\pi_1$ is the standard Lefschetz fibration on each page $P_\theta=\mathbf{LF}(\mathbb{D}^*S^1,\tau_P^2)$ , where $\tau_P$ is the Dehn twist. We will refer to the flow induced on the fiber-space of $\pi_i$ as the ith holomorphic shadow.

Recall that the ellipsoid E(a, b) is defined as

\[E(a,b)=\left\{(u,v)\in \mathbb{C}^2 \;:\; \frac{\pi|u|^2}{a}+\frac{\pi|u|^2}{b}\leq 1\right\},\]

which is a star-shaped domain in $\mathbb{C}^2$ and so its boundary $S(a,b) \;:\!=\; \partial E(a,b)\cong S^3$ , which we call a spheroid to avoid confusion (although this is also usually called an ellipsoid in the literature), inherits a contact form whose Reeb flow is $\phi_t(u,v)=(e^{2\pi i a t}u,e^{2 \pi i bt}v)$ . One may view this as a flow on $S^3=S(1,1)$ , which is adapted to the trivial open book given by $(u,v)\mapsto {v}/{|v|}$ , and the return map on the disk-like pages is simply the rotation by angle $2\pi{a}/{b}$ . We say that the ellipsoid/spheroid is irrational if a and b are rationally independent. In this case, the first return map at each page has the origin as the unique interior fixed point, and no boundary fixed points.

From the above discussion, we easily conclude:

One can similarly choose the open book to be a $\pi_i$ , and the rest of the $\pi_j$ to be the Lefschetz fibrations on a fixed page of $\pi_i$ . The dynamics on the irrational spheroid $S(1+\epsilon,1-\epsilon)$ is then the 3rd-holomorphic shadow with respect to $\pi_2$ . We remark that, rather interestingly, this is precisely the one which doubly covers the 3-dimensional instance of Katok’s construction on $\mathbb{R}P^3$ [ Reference Peter, Hansjörg and KaiAGZ , theorem 2·3, remark 2·17] (and the latter we see in the binding of $\pi_0$ ).

Proof of Lifting theorem. We prove surjectivity of $\mathbf{HS}$ .

Denote by $\alpha_{0}$ the contact form on $S^3$ whose Reeb dynamics is adapted to the concrete open book $\pi_0 \;:\; S^3\backslash L\rightarrow S^1$ , where $L \cong S^1$ , having disk-like $\theta$ -page $P_\theta^0$ . Let $\xi_0=\ker \alpha_0$ be the induced contact distribution.

We fix the following choices:

1. (i) a contact form $\alpha_B \in \mathbf{Reeb}(F,\phi_F)$ (e.g. via the Thurston-Wilkelnkemper construction of a Giroux form), adapted to some choice of concrete open book on B;

2. (ii) an exact symplectic form $\omega_\theta=d\lambda_\theta$ on each page $P_\theta$ in such a way that $(P_\theta,\omega_\theta)$ is a (degenerate) Liouville filling of $(B,\alpha_B)$ for every $\theta$ ;

3. (iii) an $\omega_\theta$ -compatible almost complex structure $J_\theta$ on $P_\theta$ , which is generic as a 1-parameter family.

4. (iv) an almost complex structure $J_B$ on $\mathbb{R}\times M$ which makes the concrete open book on B holomorphic.

This data can be used to construct a Giroux form $\alpha \in \mathbf{Reeb}(\mathbf{LF}(F,\phi_F),\phi)$ , satisfying $\alpha_B=\alpha\vert_B$ and $\omega_\theta=d\alpha\vert_{P_\theta}$ for each $\theta$ , which is auxiliary. This gives an auxiliary concrete contact distribution $\xi=\ker \alpha$ on M, supported by the IP open book. One can also construct an almost complex structure J, such that $J\vert_{\xi\vert_{P_\theta}}$ corresponds to $J_\theta$ under the isomorphism $\xi\vert_{P_\theta}\cong TP_\theta$ , and such that $J\vert_B=J_B$ ; we use this J to construct the holomorphic foliation $\overline{\mathcal{M}}^q$ associated to $\alpha$ .

Having made these choices, away from B, we have a symplectic splitting

\[(TP_\theta,\omega_\theta)=(\mbox{Vert}_\theta,\omega_\theta^v)\oplus(\mbox{Hor}_\theta,\omega_\theta^h) \;:\!=\; (\!\ker d\pi_\theta,\omega_\theta\vert_{\ker d\pi_\theta})\oplus (\mbox{Vert}_\theta^{\perp_{\omega_\theta}},\omega_\theta\vert_{\mbox{Vert}_\theta^{\perp_{\omega_\theta}}}),\]

together with a symplectic isomorphism $(\xi\vert_{P_\theta},d\alpha\vert_{\xi\vert_{P_\theta}})\cong (TP_\theta,\omega_\theta)$ . This induces a symplectic splitting

\[(\xi\vert_{P_\theta},d\alpha\vert_{\xi\vert_{P_\theta}})=(\mbox{Vert}^{\xi}_\theta,\omega_\theta^{v,\xi})\oplus(\mbox{Hor}^{\xi}_\theta,\omega_\theta^{h,\xi}),\]

where $\omega_{\theta}^{v,\xi}=d\alpha\vert_{\mbox{Vert}^{\xi}_\theta},$ $\omega_\theta^{h,\xi}=d\alpha\vert_{\mbox{Hor}^{\xi}_\theta}$ . The symplectic distribution $(\mbox{Hor}^{\xi},\omega^{h,\xi})$ extends to B as $(\xi_B^{\perp_M},d\alpha\vert_{\xi_B^{\perp_{M}}})$ , where the symplectic complement is with respect to $d\alpha$ .

The holomorphic foliation $\overline{\mathcal{M}}^q$ and concrete open book on M provide a diffeomorphism $\mathcal{F} \;:\; S^3 \rightarrow \overline{\mathcal{M}}^q$ , so that we have an induced concrete trivial open book $\pi_\mathcal{M}\;:\;\overline{\mathcal{M}}^q\backslash \mathcal{M}^q_B\rightarrow S^1$ with $\theta$ -page $P_\theta^\mathcal{M}$ together with a contact form $\alpha_{\mathcal{M}}\in \Omega^1(\overline{\mathcal{M}}^q)$ and a concrete contact distribution $\xi_{\mathcal{M}}=\ker \alpha_{\mathcal{M}}$ supported by the open book. Via the pullback connection $\mbox{Hor}_\theta^\mathcal{M}=(ev^q)^*(\mbox{Hor}_\theta)\subset T\overline{\mathcal{M}}^q_*\vert_{\pi_*^{-1}(P_\theta^{\mathcal{M}})}$ for the forgetful map $\pi_*$ , away from B we have an induced identification $TP_\theta^\mathcal{M}\vert_u=\mbox{Hor}_\theta\vert_u$ as a two-plane distribution on M along u. We also have a two-plane distribution $\mbox{Hor}_\theta^{\xi_\mathcal{M}}\subset TM\vert_{P_\theta}$ , which is identified with $\xi_\mathcal{M}^\theta \;:\!=\; \xi_\mathcal{M}\vert_{P_\theta^\mathcal{M}}$ , as well as, away from B, an induced linear isomorphism $\mbox{Hor}_\theta\cong \mbox{Hor}_\theta^{\xi_\mathcal{M}}$ covering the identity, coming from the analogous isomorphism $TP_\theta^\mathcal{M}\cong \xi_\mathcal{M}^\theta$ away from $\mathcal{M}_B^q$ . This induces a symplectic form $\omega_\theta^{h,\xi_{\mathcal{M}}}$ on $\mbox{Hor}_\theta^{\xi_\mathcal{M}}$ , coming from $\omega_\theta^{h}$ . Moreover, we can view $d\alpha_{\mathcal{M}}$ as a non-degenerate 2-form on $\mbox{Hor}^{\xi_\mathcal{M}}$ . For $u \in \mathcal{M}_B^q$ , $\xi_{\mathcal{M}}\vert_u$ gets identified with some 2-plane distribution $\xi_B^{\perp_{\mathcal{M}}}\subset TM\vert_B$ transverse to TB which also comes with the non-degenerate 2-form $d\alpha_\mathcal{M}$ . Note that $\mbox{Hor}_\theta^{\xi_\mathcal{M}}$ might not necessarily be a subbundle of the concrete contact distribution $\xi=\ker \alpha$ , i.e. it does not necessarily agree with $\mbox{Hor}_\theta^{\xi}$ . To remedy this, we may take an isotopy $\{\alpha^t\}_{t \in [0,1]}$ of Giroux forms, with $\alpha^0=\alpha$ , satisfying $d\alpha^t\vert_{P_\theta}=\omega_\theta$ and $\alpha^t\vert_{B}=\alpha_B$ independently of t, together with the corresponding splitting

\[(\xi^t\vert_{P_\theta},d\alpha^t\vert_{\xi^t\vert_{P_\theta}})=(\mbox{Vert}^{\xi^t}_\theta,\omega_\theta^{v,\xi^t})\oplus(\mbox{Hor}^{\xi^t}_\theta,\omega_\theta^{h,\xi^t}),\]

so that $\mbox{Hor}^{\xi^1}=\mbox{Hor}^{\xi_\mathcal{M}}$ (and in particular $\xi_B^{\perp_M}=\xi_B^{\perp_\mathcal{M}} \;=\!:\; \xi_B^\perp$ along B). This induces an accompanying homotopy of almost complex structures $\{J_t\}$ , so that $J_t\vert_B=J_B$ , and $J_t$ is independent of t under the isomorphism $\xi^t\vert_{P_\theta}\cong TP_\theta$ (i.e. it corresponds to $J_\theta$ ). The corresponding foliations $\overline{\mathcal{M}}^q_t$ are then t-independent.

Up to this modification, we have $d\alpha_{\mathcal{M}}=f \omega^{h,\xi}$ for a unique positive smooth function $f\;:\;M\backslash L \rightarrow \mathbb{R}^+$ . Since $\mathcal{M}^q_B$ is invariant under the flow of $R_\mathcal{M}$ , we have $df\vert_{\xi_B^\perp}=0$ . We then consider the contact form $\alpha^f \;:\!=\; f\alpha$ , and its Reeb vector field $R_\alpha^f$ . By construction, we have $\ker \alpha^f=\xi$ , $d\alpha^f\vert_{\mbox{Hor}^{\xi}_\theta}=d\alpha_{\mathcal{M}}\vert_{\mbox{Hor}^{\xi}_\theta}$ , and $R_\alpha^f\vert_B\in TB$ so that B is invariant under the flow of $R_\alpha^f$ .

Now, the Reeb vector field $R_{\mathcal{M}}$ of $\alpha_{\mathcal{M}}$ can be viewed as the unique vector field on M characterised by the equations:

1. (i) $\mathbf{D}_u^N R_{\mathcal{M}}\vert_u=0$ for each $u \in \overline{\mathcal{M}}$ ;

2. (ii) $d\alpha_{\mathcal{M}}(R_\mathcal{M},\cdot)=0$ ;

3. (iii) $\alpha_{\mathcal{M}}(R_\mathcal{M})=1$ .

While $R_{\mathcal{M}}$ is positively transverse to $TP_\theta,$ as well as to $\mbox{Hor}_\theta^{\xi}$ , a priori it might fail to be positively transverse to $\mbox{Vert}_\theta^{\xi}$ . Since we still have freedom in the vertical directions, we can then further modify the Giroux form $\alpha$ via an isotopy as above, without changing the horizontal distribution and relative B, and without changing the foliation $\overline{\mathcal{M}}^q$ , so that $\eta \in T\overline{\mathcal{M}}^q$ satisfies $\alpha_{\mathcal{M}}(\eta)>0$ if and only if $\alpha(\eta)>0$ pointwise, when viewed as a vector field in M. If $P_u \;:\; W^{1,2}(N_u)\rightarrow \ker \mathbf{D}_u^N$ denotes the orthogonal projection with respect to the metric induced by J and $d\alpha$ , the projection $P_u(R^f_\alpha\vert_u)$ of $R^f_\alpha\vert_u$ satisfies the first two equations above, and $g\vert_u \;:\!=\; \alpha_\mathcal{M}(P_u(R^f_\alpha\vert_u))>0$ defines a smooth positive function $g\;:\;M\backslash L \rightarrow \mathbb{R}^+$ . Then $R_\mathcal{M}=g^{-1}P_u(R^f_\alpha\vert_u)$ , so that $R_{\mathcal{M}}$ is the holomorphic shadow of $R^f_\alpha$ . Note that $R^f_\alpha$ is positively transverse to the pages: if it were tangent to a page at a point, since pages are invariant under J and symplectic for $d\alpha$ , $R_{\mathcal{M}}$ would also be tangent to the page at that point, which is a contradiction. Then $\alpha^f \in \mbox{Reeb}(P,\phi)$ satisfies $\mathbf{HS}(\alpha^f,J)=\alpha_{\mathcal{M}}$ .

3. Transverse paths and symplectic tomographies

Alternatively to the construction of the holomorphic shadow, we may keep track of which holomorphic curves are intersected by each Reeb orbit in M, without changing the original dynamics. Namely, for $p \in M\backslash L$ , we may consider the path

\[\gamma_p(t)=\pi^q(\phi_t^M(p))\in \overline{\mathcal{M}}^q,\]

where $\phi_t^M$ is the flow of $R_\alpha$ . If $p \in B\backslash L$ , this is a parametrisation of $\mathcal{M}^q_B \cong S^1$ ; if $p \in M\backslash B$ , this is a path in $S^3$ which is positively transverse to each disk-like page $P_\theta^\mathcal{M}$ as well as to the contact structure $\xi_\mathcal{M}$ , and is by construction an orbit of the shadowing cone $C_\alpha$ . Note that different choices of p might induce paths which intersect each other (corresponding to their orbits intersecting the same holomorphic curve), and even self-intersect (corresponding to an orbit intersecting the same holomorphic curve multiple times), so these paths are not orbits of an autonomous flow. See Figure 9. We will refer to the collection

\[\mathbf{TS}(\alpha,J)=\{\gamma_p \;:\; p \in M\backslash L\}\]

as the transverse shadow of the Reeb flow of $\alpha$ on M, with respect to J, which is by definition the collection of those orbits of $C_\alpha$ coming from orbits of $\alpha$ on M.

One may further choose to “package” these transverse paths in different ways, e.g. by considering those paths induced by points on a section of the Lefschetz fibration at a given page, as follows.

Fig. 9. The qualitative image of a transverse path $\gamma_p$ under the defining map $\Phi_0\;:\;S^3\rightarrow \mathbb{C}$ for the trivial open book $\pi_0=\Phi_0/\vert \Phi_0\vert$ .

Fix the 0-page $P_0$ in M, and consider a two disk $D \cong \mathbb{D}^2$ satisfying:

1. (i) $D\subset P_{0}$ ;

2. (ii) $\partial D \subset \partial P_{0}=B$ is a loop which is disjoint from the binding L of the concrete open book in B, and transverse to the interior of each of its pages and to the contact structure $\xi_B$ ;

3. (iii) $D_0$ is a symplectic section of the Lefschetz fibration $\pi_0$ , i.e. D intersects each fiber of $\pi_0$ precisely once, and hence $D=im(s)$ for $s \;:\; \mathbb{D}_0^2\rightarrow P_0$ satisfying $\pi_0 \circ s=id$ .

We refer to such a disk D as a (horizontal) symplectic tomography for the Reeb dynamics on M. Note that, if $\partial D$ is a Reeb orbit of $\alpha_B$ which is linked with L, then $f(\partial D)=\partial D$ is invariant under the return map. This is closely related to the counts of holomorphic sections with Lagrangian boundary condition of a given Lefschetz fibration, as considered e.g. by Seidel in [ Reference PaulSei08 ], with the difference that we consider the more flexible class of symplectic ones.

For each such symplectic tomography D, we have an associated return map

\[f_D \;:\; P_0^\mathcal{M}\Longrightarrow P_0^\mathcal{M}\]

on the 0-page of the moduli space, as follows. We identify $P_0^\mathcal{M}$ with $\mathbb{D}_0^2$ via $\mathbb{D}_0^2=(\pi_0 \circ ev \circ \pi_*^{-1})(P_0^\mathcal{M})$ , and define $f_D$ by

\[f_D(u)=\gamma_{s(u)}(\tau(u,D)) \in P_0^\mathcal{M},\]

where $\tau(u,D)=\min\{t>0 \;:\; \gamma_{s(u)}(t)\in P_0^\mathcal{M}\}$ is the first return time of the transverse path $\gamma_{s(u)}$ to the 0-page $P_0^\mathcal{M}$ . See Figure 10.

Fig. 10. The return map $f_D$ associated to the tomography D. Open tangencies of f(D) with the vertical foliation (which could, a priori, theoretically arise from “foldings” of the disk f(D)) might prevent in general that $f_D$ preserves area. This does not happen perturbatively, i.e. when we perturb a foliation-preserving map, however. Note that $f(L)=L$ .

The symplectic disk $(D,d\alpha\vert_D)$ is symplectomorphic to $(P_0^\mathcal{M},d\alpha_{\mathcal{M}}\vert_{P_0^\mathcal{M}})$ , and both have finite symplectic area. In general, $f_D$ might a priori decrease area. Indeed, f(D) is a symplectic disk in $P_0$ with the same symplectic area as D, but it might have an open set of vertical tangencies, i.e. intersecting a fiber along an open subset of positive area (as opposed to D, which intersects fibers at a single point). Nevertheless, this is not possible for perturbative situations where one perturbs a fiber-wise preserving map, in which case the perturbed $f_D$ still preserves area.

On the other hand, if $f_D$ is easily seen to be surjective, i.e. every holomorphic fiber of $\pi_0$ is intersected by the symplectic disk f(D). Indeed, since f is homotopic to the identity by a smooth homotopy which preserves the boundary, f(D) is homologous to D relative boundary. Moreover, $\partial f(D)$ is a push-off in the Reeb direction of $\partial D$ (they agree if $\partial D$ is an orbit, as observed above), hence they can be homotoped to each other via the Reeb flow of $R_B$ , and in particular away from L (the boundary of the holomorphic fibers). It follows that the homological intersection number of f(D) with the fibers agrees with that of D, i.e. it is 1. In general, $f_D$ is not necessarily injective. However, this is certainly true in the case where f is close to a fiber-wise preserving map, since otherwise f(D) would have vertical tangencies.

Perturbative case. As observed in the above discussion, if f is sufficiently close to a fiber-wise preserving map, then $f_D$ is an area-preserving homeomorphism of the 2-disk for every tomography D. By Brouwer’s translation theorem, we find an interior fixed point for $f_D$ ; by construction this corresponds to an (interior) fiber-wise 1-recurrent point in the fixed page $P_0$ . Varying vertically the tomography D along $P_0$ , we obtain infinitely many such points. If $k\geq 1$ , fiber-wise k-recurrent points correspond to interior fixed points of the return map

\[f_{k,D}(u)=\gamma_{s(u)}(\tau_k(u,D)),\]

where $\tau_k(u,D)$ is the k-th return time of the transverse path $\gamma_{s(u)}$ to $P_0^\mathcal{M}$ . Note that this map is in general different from $f_D^k$ ; recall that $f_D$ is not the return map of an autonomous flow. Having fixed k, using that D and $P_0$ vary in compact families, we can take a sufficiently small perturbation so that $f_{l,D}$ is still an area-preserving homeomorphism for every $l\leq k$ , D and any choice of page, and apply the same argument. This finishes the proof of Theorem D.

Discussion and outlook. Roughly speaking, one might expect that the results here presented open the possibility of obtaining information on a flow on dimension 5, from a flow in dimension 3 (more specifically, on $S^3$ ). The flow of the holomorphic shadow, by construction, maps holomorphic curves to holomorphic curves, i.e. it is a flow in the moduli space. While the ”best approximation” of the original flow with this property, it potentially alters the dynamics of the original Reeb flow in a significant way. It also forgets dynamical information in the vertical directions, as well as most of the interesting dynamical information at B (it is adapted to study spatial problems rather than planar ones); note that in dimension 3, the shadow, when seen as a flow on B, is just a reparametrisation of the original one. Moreover, while the holomorphic shadow admits periodic orbits away from the binding $\mathcal{M}^q_B$ (one or infinitely many, by combining Frank’s theorem with Brouwer’s translation theorem; and in fact if $\alpha_\mathcal{M}$ is $C^\infty$ -generic the union of their images is dense in $S^3$ [ Reference KeiIrie ]), these may bear no relationship whatsoever with spatial periodic orbits of the original Reeb flow $R_\alpha$ ; unless the latter vector field satisfies the linearised CR-equation, in which case $R_\alpha=R_\mathcal{M}$ , which one expects to happen only for certain integrable systems as the (rotating) Kepler problem. Also, it would be surprising to obtain holomorphic curves which are invariant under the original Reeb flow away from the integrable case (or perhaps perturbations thereof); if such curves existed e.g. in the case of the three-body problem where $F=\mathbb{D}^*S^1$ is an annulus and we have twisting at the boundary, one would be able to conclude via the classical Poincaré–Birkhoff theorem that there are infinitely many periodic points with arbitrary large period.

On the other hand, one might hope to “lift” dynamical properties from the shadow to the original dynamics. For instance, one can envision extracting information related to the complexity of the original flow by studying its shadow, at least generically. For instance, from the semi-conjugation (1·1), one obtains that the topological entropy of $\phi_t^{M;\mathcal{M}}$ is no smaller than that of $\phi_t^{S^3;\mathcal{M}}$ . Moreover, $C^\infty$ -generically, Reeb flows on $S^3$ have positive topological entropy [ Reference Vincent, Pierre, Umberto and AnaCDHR ]. Since heuristically $\phi_t^{M;\mathcal{M}}$ is “less chaotic” than the original Reeb flow, this suggests that generic adapted Reeb flows on any IP contact 5-fold might have positive topological entropy, which is moreover generated by spatial orbits (note that if the planar problem has positive entropy, so does the spatial one, since it is an invariant subset). Of course, this is not the case for the integrable limit situations here considered. Note that, the shadow of the 3-dimensional flow, when viewed as a flow on B, has the same topological entropy as the original flow; but, since every homeomorphism of the circle has vanishing entropy, we obtain no interesting information when we semi-conjugate to the moduli space.

Alternatively, the notion of the transverse shadow, while still forgets dynamical information in the vertical directions (and so in principle it might not be possible to recover the original Reeb flow by knowledge of the collection of all horizontal tomographies for each page), is more reliable than its holomorphic counterpart. The clear disadvantage is that it is not a flow. However, for each choice of symplectic tomography, the failure of the associated transverse shadow to be a (non-autonomous) flow is precisely the existence of fiber-wise intersections. Note that an invariant surface for the return map f is a point $u \in P_0^\mathcal{M}$ which is a fixed point for $f_D$ , for every symplectic tomography D. Note that the notion of a vertical tomography is complicated by the fact that there is no globally defined projection to a fiber.