A generalized Poincaré–Birkhoff theorem

Abstract

We prove a generalization of the classical Poincaré–Birkhoff theorem for Liouville domains, in arbitrary even dimensions. This is inspired by the existence of global hypersurfaces of section for the spatial case of the restricted three-body problem (Moreno and van Koert in Global hypersurfaces of section in the spatial restricted three-body problem, 2020).

Appendices

Appendix A: Hamiltonian twist maps: examples and non-examples

We will now discuss some examples that help clarify the nature of the Hamiltonian twist condition.

1.1 Examples

The following construction, an adaptation of a standard one, further illustrates that the Hamiltonian twist condition is not localized at B.

Proposition A.1

For each $k,\ell \in {\mathbb {N}}$, there are strict contact manifolds $(Y_k,\alpha _{k,\ell })$ carrying adapted open books $(B_k= B,\pi _k)$, $\pi _k: Y_k\backslash B \rightarrow S^1$, with fixed page $\Sigma$, such that the following holds:

• The return maps $\tau _k$ all agree in a collar neighborhood of $B=\partial \Sigma$, and are generated by Hamiltonians $H_k$;

• Furthermore, there is a symplectomorphism $\phi _k$ from $\mathring{\Sigma }$, the interior of the page $\Sigma$, to the open subset $W_2$ of the Liouville completion ${{\widehat{W}}}$ of a fixed Liouville domain $(W,\lambda )$ with $\partial W=B$, where

  $$\begin{aligned} W_2= W \cup _\partial ([1,2)\times B, d (r\alpha _B)\, ), \end{aligned}$$

  and $\alpha _B=\lambda \vert _B$ is the contact form at B, and $r \in [1,2)$.

• The return map $\phi _k\circ \tau _k \circ \phi _k^{-1}$ extends to a Hamiltonian diffeomorphism ${{\bar{\tau }}}_k$ on the closure $\bar{W}_2$, generated by Hamiltonians ${{\bar{H}}}_k$.

• The Hamiltonian twist condition holds for ${{\bar{H}}}_k$ for $k\le \ell$, but not for $k>\ell$.

Proof

Consider a Liouville domain $(W,\lambda )$ with a $2\pi$-periodic Reeb flow on its boundary (e.g., $D^*S^2$). We identify a collar neighborhood $\nu _W(B)$ of $B=\partial W$ with $(1/2,1]\times B$, where $B=\{r=1\}$, via a diffeomorphism $\varepsilon : (1/2,1]\times B \longrightarrow \nu _W(B)\subset W$. We assume $\lambda =r\alpha _B$ along $\nu _W(B)$, $\alpha _B=\lambda \vert _B$. Define the smooth Hamiltonian

$$\begin{aligned} H(x)={\left\{ \begin{array}{ll} 0, &{} \text {if }x\notin \nu _W(B), \\ f(r), &{} \text {if }x=\varepsilon (r,b) \in \nu _W(B). \end{array}\right. } \end{aligned}$$

Here, f is a smooth, decreasing function with the property

• $f(1/2)=0$;

• $f'(r)\ge -2\pi$ and $f'(r)=-2\pi$ near $r=1$.

The Hamiltonian vector field of H is given by

$$\begin{aligned} X_H(x)={\left\{ \begin{array}{ll} 0 &{} \text {if }x\notin \nu _W(B), \\ f'(r) R_\alpha &{} \text {if }x=\varepsilon (r,b) \in \nu _W(B). \end{array}\right. } \end{aligned}$$

Define the fibered Dehn twist by $\tau (x)=Fl^{X_H}_1(x)$, where $Fl^{X_H}_t$ is the Hamiltonian flow of H with respect to $d\lambda$. We have $\tau ^*\lambda =\lambda - {\text {d}}U$, where we choose the primitive U to be a negative function: with a computation, we can show that it is possible to choose $U(1)=-2\pi$, and will do so. The iterate $\tau ^k$ is generated by $H_k=kH$, and $(\tau ^k)^*\lambda =\lambda -{\text {d}}U_k$, with $U_k=\sum _{j=0}^{k-1}(U \circ \tau ^j)$.

We consider the associated open book

$$\begin{aligned} Y_k=OB(W,\tau ^k):= B\times D^2 \cup _\partial W_{\tau ^k}, \end{aligned}$$

where $W_{\tau ^k}=W\times {\mathbb {R}}/(x,t)\sim (\tau ^k(x),t+U_k(x))$ is the mapping torus. The manifold $Y_k$ carries an adapted contact form $\alpha _{k,\ell }$ which looks like $\alpha _{k,\ell }=\lambda +d\theta$ along $W_{\tau ^k}$, and $\alpha _{k,\ell }=h_1(\rho )\alpha _B+h_2(\rho )d\theta$ along $B\times D^2$. Here, $(\rho ,\theta ) \in D^2$, and $h_1$ and $h_2=h_{2,k,\ell }$ are suitable profile functions, which we will fix now. Choose $h_1$ and $h_2$, such that

• they do not depend on k for $\rho \le 1/2$;

• $h_1^\prime \le 0$ with equality only at $\rho =0$. We may take $h_1(\rho )=2-\rho ^2$ near $\rho =0$ (this is not essential but very convenient);

• near $\rho =0$, we have $-\frac{h_2^\prime }{h_1^\prime }(\rho )=\ell + \epsilon >0$ (non-singular) for some small $\epsilon \in (0,1)$.

• $h_2\equiv k$, $h_1=-\rho +2$ near $\rho =1$ (so $h_2$ depends on k on the interval (1/2, 1]).

Note that, in the definition of $Y_k$, the binding model is glued to the mapping torus using the gluing map

$$\begin{aligned} \begin{aligned} \Phi _{glue}: B\times D^2_{\rho >1/2}&\longrightarrow W_{\tau ^k}\\ (b;\rho ,\theta )&\longmapsto \left( 2-\rho ,b;\frac{-U_k(1)\theta }{2\pi }\right) = (2-\rho ,b;k \theta ). \end{aligned} \end{aligned}$$

This pulls back $d\theta + \lambda$ to $kd\theta +(2-\rho ) \alpha _B$. This explains the above choices.

The global hypersurface of section, i.e., a fixed page, is $\Sigma =W \cup _\partial B \times [0,1]$, with coordinate $\rho \in [0,1]$, and we can compute the return map $\tau _k$ explicitly. We find

$$\begin{aligned} \tau _k(x)={\left\{ \begin{array}{ll} \tau ^k(x), &{} \text {if }x \in W, \\ (Fl^R_{-2\pi h_2'(\rho )/h_1'(\rho )}(b),\rho ), &{} \text {if }x=(b,\rho ) \in B\times [0,1], \end{array}\right. } \end{aligned}$$

where $Fl^R_t$ is the Reeb flow of $\alpha _B$ at B. The Hamiltonian generating $\tau _k$ can be obtained by patching $H_k$ on W to a Hamiltonian that generates $\tau _k$ along $B\times [0,1]$; we need to match the slopes on the boundary, which can be done by rewriting $\tau _k(b,\rho )=(Fl^R_{-2\pi (h_2'(\rho )/h_1'(\rho )+k)}(b),\rho )$. Then, $H_k$ extends to $\Sigma$ via $\bar{H}_k(r)=-2\pi \int _1^\rho (h_2'(s)/h_1'(s)+k)h_1^\prime (s)ds + f(1)$ along $B\times [0,1]$. Note that the form $d\lambda$ also extends along $B\times [0,1]$ via $d\alpha _{k,\ell }\vert _\Sigma =h_1^\prime (\rho )d\rho \wedge \alpha _B$. Therefore, $H_k$ generates $\tau _k$, and $\tau _k$ is independent of k on the collar neighborhood $B\times [0,1/2]$.

To complete the proof, we first note that the 2-form $d (h_1(\rho ) \alpha _B)$ is degenerate on $\partial \Sigma$. However, the map

$$\begin{aligned} \begin{aligned} \phi _k: {\mathring{\Sigma }}&\longrightarrow W_2= W \cup _\partial ([1,2) \times B, d (r\alpha _B)\, ),\\ w&\longmapsto {\left\{ \begin{array}{ll} w &{} w \in W \\ (h_1(\rho ),b) &{} w =(b,\rho )\in B \times (0,1] \end{array}\right. } \end{aligned} \end{aligned}$$

is a symplectomorphism, and the closure of $W_2$ is an actual Liouville domain. Furthermore, due to our explicit choice $h_1(\rho )=2-\rho ^2$ near $\rho =0$, we find $\rho =\sqrt{2-r}$, so we can compute the conjugated return map $\phi _k \circ \tau _k \circ \phi _k^{-1}$ near $r=0$ as

$$\begin{aligned} \phi _k \circ \tau _k \circ \phi _k^{-1}(r,b)= (r, Fl^R_{2\pi (\ell +\epsilon )}(b) ). \end{aligned}$$

This map extends to a symplectomorphism ${{\bar{\tau }}}_k:{{\bar{W}}}_2 \rightarrow {{\bar{W}}}_2$. Here, note that $\phi _k^{-1}$ is not smooth at $r=2$, but this is resolved by the explicit form of $\tau _k$, which does not contain any $\rho$ dependence in the B-direction near $\rho =0$. This extended map is still Hamiltonian, and satisfies the twist condition for $k\le \ell$, but not for $k>\ell$.

Therefore, it satisfies the claim of the proposition. $\square$

Remark A.2

Given a return map $\tau$ that is Hamiltonian, we point out that the Hamiltonian family generating $\tau$ is not unique, and more importantly, that various dynamical properties depend on the choice of Hamiltonian. For example, on the disk $(D^2, rdr\wedge d\theta )$, the return map $\tau ={{\,\mathrm{id}\,}}$ is generated by the autonomous Hamiltonians $H_k=k\pi r^2$. For given k, the Robbin–Salamon index of the 1-periodic orbit at $\partial D^2$ is 2k, i.e., k-dependent. The associated paths of symplectic matrices have the same endpoints, but are not homotopic rel endpoints. This also illustrates the interpretation of the RS-index as a winding number. Note that $D^2$ has a Hamiltonian circle action that extends over the whole space. We do not know whether the same type of phenomenon occurs for more general symplectic manifolds (i.e., without a global Hamiltonian circle action).

1.2 Non-examples: Katok examples

In [25], Katok constructed examples of non-reversible Finsler metrics on $S^n$ with only finitely many simple closed geodesics. Here is a description of such examples using Brieskorn manifolds. We consider

$$\begin{aligned} \Sigma ^{2n-1} := \left\{ (z_0,\ldots ,z_n) \in {\mathbb {C}}^{n+1} ~\Bigg |~\sum _j z_j^2 = 0 \right\} \cap S^{2n+1}_1, \end{aligned}$$

equipped with the contact form $\alpha =\frac{i}{2}\sum _j z_j d\bar{z}_j -{{\bar{z}}}_j dz_j$. These spaces are contactomorphic to $S^*S^n$ with its canonical contact structure. The given contact form is actually the prequantization form.

We describe the setup in detail when $n=2m+1$ is odd. We group the coordinates in pairs, and make the following unitary coordinate transformation:

$$\begin{aligned} w_0=z_0, w_1= & {} z_1, w_{2j}=\frac{\sqrt{2}}{2} (z_{2j}+i z_{2j+1}), w_{2j+1}\\= & {} \frac{i \sqrt{2}}{2} (z_{2j}-i z_{2j+1}) \text { for }j=1,\dots ,m. \end{aligned}$$

Because this is a unitary transformation, the form $\alpha$, expressed in w-coordinates, still has the form

$$\begin{aligned} \alpha =\frac{i}{2} \sum _j w_j d{{\bar{w}}}_j -{{\bar{w}}}_j dw_j. \end{aligned}$$

For a tuple $\epsilon =(\epsilon _1,\dots , \epsilon _m) \in (-1,1)^m$, define the function $H_\epsilon$ on a neighborhood of $\Sigma ^{2n-1}$ via

$$\begin{aligned} H_\epsilon (w)=\Vert w \Vert ^2+\sum _{j} \epsilon _j( |w_{2j}|^2-|w_{2j+1}|^2 ). \end{aligned}$$

For $\epsilon$ sufficiently small, this function is positive, so we define a perturbed contact form by

$$\begin{aligned} \alpha _\epsilon = H_\epsilon ^{-1} \cdot \alpha . \end{aligned}$$

The Reeb vector field of $\alpha _\epsilon$ is

$$\begin{aligned} R_\epsilon = X_\epsilon +{\overline{X}}_\epsilon , \end{aligned}$$

where

$$\begin{aligned} X_\epsilon= & {} i w_0\frac{\partial }{\partial w_0} + i w_1\frac{\partial }{\partial w_1} +\sum _j\left( i(1+\epsilon _j) \frac{\partial }{\partial w_{2j}} + i(1-\epsilon _j) \frac{\partial }{\partial w_{2j+1}}\right) , \\ {\overline{X}}_\epsilon= & {} -i {\overline{w}}_0\frac{\partial }{\partial {\overline{w}}_0} - i {\overline{w}}_1\frac{\partial }{\partial {\overline{w}}_1} -\sum _j\left( i(1+\epsilon _j) \frac{\partial }{\partial {\overline{w}}_{2j}} + i(1-\epsilon _j) \frac{\partial }{\partial {\overline{w}}_{2j+1}}\right) . \end{aligned}$$

The Reeb flow is therefore given by

$$\begin{aligned}&(w_0,\dots ,w_n)\longmapsto (e^{2\pi it}w_0,e^{2\pi it}w_1,e^{2\pi it(1+\epsilon _1)}w_2,e^{2\pi it(1-\epsilon _1)}w_3,\\&\qquad \dots , e^{2\pi it(1+\epsilon _m)}w_{n-1},e^{2\pi it(1-\epsilon _m)}w_{n}). \end{aligned}$$

This flow has only $n+1$ periodic orbits if all $\epsilon _j$ are rationally independent. These are given by

$$\begin{aligned} \begin{aligned} \gamma _0(t)=&\left( \frac{1}{\sqrt{2}}e^{2\pi it},\frac{i}{\sqrt{2}}e^{2\pi it},0,\dots ,0\right) , t \in [0,1]\\ \beta _0(t)=&\left( \frac{1}{\sqrt{2}}e^{2\pi it},-\frac{i}{\sqrt{2}}e^{2\pi it},0, \dots , 0\right) , t \in [0,1]\\ \gamma _j(t)=&\left( 0,0,\dots , e^{2\pi it(1+\epsilon _j)},0,\ldots ,0,0\right) , t \in [0,1/(1+\epsilon _j)]\\ \beta _j(t)=&\left( 0,0,\dots , 0, e^{2\pi it(1-\epsilon _j)},\ldots ,0,0\right) , t \in [0,1/(1-\epsilon _j)]\\ \end{aligned} \end{aligned}$$

for $j=1,\dots ,m$.

Remark A.3

As stated, we see that there are only finitely many periodic orbits. Furthermore, since the unperturbed system, i.e., $\epsilon =0$, describes the geodesic flow on the round sphere, and the perturbation $\alpha _\epsilon$ is $C^2$-small for small $\epsilon$, it follows that the Reeb flow of the contact form $\alpha _\epsilon$ corresponds to the geodesic flow of a Finsler metric. In Sect. A.3, we describe how to obtain an explicit relation with the famous Katok examples for $S^*S^2$.

We construct a supporting open book for the contact form $\alpha _\epsilon$ using the map

$$\begin{aligned} \Theta : \Sigma ^{2n-1} \longrightarrow {\mathbb {C}}, (w_0,w_1,\ldots ,w_n) \longmapsto w_0. \end{aligned}$$

The zero set of $\Theta$ defines the binding, the pages are the sets of the form $P_\theta =\{\arg \Theta =\theta \}$, $\theta \in S^1$, which are all copies of ${\mathbb {D}}^*S^{n-1}$, and the monodromy is $\tau ^2$ where $\tau$ is the Dehn–Seidel twist. The (boundary extended) return map for the page $P_0=\Theta ^{-1}({\mathbb {R}}_{>0})\cong {\mathbb {D}}^*S^{n-1}$ is

$$\begin{aligned} \begin{aligned} \Phi : P_0&\longrightarrow P_0, \\ p=(r_0,w_1,w_2,w_3,\ldots ,w_{n-1},w_{n})&\longmapsto (r_0, w_1,e^{2\pi i\epsilon _1}w_2,e^{-2\pi i \epsilon _1}w_3, \ldots ,\\&e^{2\pi i\epsilon _{m}}w_{n-1}, e^{-2\pi i\epsilon _{m}}w_n). \end{aligned} \end{aligned}$$

Here, $w_0=r_0 \in {\mathbb {R}}_{\ge 0}$ is a real non-negative number, and note that the first return time is constant equal to 1 (which follows by looking at the first coordinate). If all $\epsilon _j$ are irrational and rationally independent, this map has only two periodic points, both actually fixed, given by

$$\begin{aligned} \begin{aligned} p_0=&\left( \frac{1}{\sqrt{2}},\frac{i}{\sqrt{2}},0,\dots ,0\right) \\ q_0=&\left( \frac{1}{\sqrt{2}},-\frac{i}{\sqrt{2}},0, \dots , 0\right) .\\ \end{aligned} \end{aligned}$$

Note that $p_0,q_0$ are both interior fixed points, and irrationality of the $\epsilon _j$ implies that there are no boundary fixed points. We will explain now why this map is Hamiltonian with boundary preserving Hamiltonian flow. The symplectic form on the interior of the page $P_0$ is the restriction of $d\alpha _{\epsilon }$. To manipulate this, let us define

$$\begin{aligned} H= \Vert w \Vert ^2,\quad \Delta _\epsilon =\sum _{j} \epsilon _j( |w_{2j}|^2-|w_{2j+1}|^2), \end{aligned}$$

so $H_\epsilon =H+\Delta _\epsilon$. Observe that the return map $\Phi$ is generated by the $2\pi$-flow of the vector field

$$\begin{aligned} X=i\sum _{j=1}^m\epsilon _j\left( w_{2j}\frac{\partial }{\partial w_{2j}}-{{\bar{w}}}_{2j}\frac{\partial }{\partial {{\bar{w}}}_{2j}} - w_{2j+1}\frac{\partial }{\partial w_{2j+1}}+\bar{w}_{2j+1}\frac{\partial }{\partial {{\bar{w}}}_{2j+1}} \right) . \end{aligned}$$

This vector field is tangent to the page and preserves H and $\Delta$, and hence also $H_\epsilon$. Plug X in into $d\alpha _\epsilon$. We find

$$\begin{aligned} \begin{aligned} \iota _X(d H_\epsilon ^{-1}\wedge \alpha +H_\epsilon ^{-1} d\alpha )&= -\alpha (X) d H_\epsilon ^{-1}+H_{\epsilon }^{-1} \iota _X d\alpha \\&=-\Delta _\epsilon d H_\epsilon ^{-1} -H_\epsilon ^{-1} d \Delta _\epsilon = -d (H_\epsilon ^{-1} \Delta _\epsilon ). \end{aligned} \end{aligned}$$

This means that the Hamiltonian generating the return map is $H_\epsilon ^{-1} \Delta _\epsilon$. Moreover, index-positivity follows, by observing that it holds for the round metric on $S^2$ and the fact that it is an open condition. It follows from Theorem B that $\Phi$ does not satisfy the twist condition for any Liouville structure on ${\mathbb {D}}^*S^2$.

Remark A.4

The setup for n even is very similar: we drop the $w_0$-coordinate.

1.3 Relation with the Katok examples

We explain how to see that the above dynamical systems indeed correspond to the Katok examples in case of $S^*S^2$ (i.e., $n=2$). More precisely, we will show that the geodesic flow of the Katok examples is conjugated to the Reeb flow of $\alpha _\epsilon$. We need some preparation, which applies to all dimensions, before we specialize to dimension 3. First of all, we fix positive weights $(a_1,\ldots , a_n)\in {\mathbb {R}}_{>0}^n$. Then, we define the 1-forms on the sphere $S^{2n-1}$ given by

$$\begin{aligned} \beta _0 =\iota ^*\left( \frac{\sum _j x_j dy_j-y_j dx_j}{\sum _k a_k (x_k^2+y_k^2)}\right) =\iota ^*\left( \frac{1}{\sum _k a_k \vert z_j\vert ^2}\frac{i}{2}\sum _j z_j d{{\bar{z}}}_j-{{\bar{z}}}_j dz_j\right) \end{aligned}$$

and

$$\begin{aligned} \beta _1= \iota ^*\left( \sum _j \frac{1}{a_j}(x_j dy_j-y_j dx_j)\right) =\iota ^*\left( \frac{i}{2}\sum _j \frac{1}{a_j}\left( z_j d{{\bar{z}}}_j-{{\bar{z}}}_j dz_j\right) \right) , \end{aligned}$$

where $\iota$ is the inclusion map $S^{2n-1}\rightarrow {\mathbb {R}}^{2n}$. We will show that the first form is a contact form and that it is strictly contactomorphic to the latter. For this, consider the map

$$\begin{aligned} \begin{aligned} \psi : S^{2n-1}&\longrightarrow S^{2n-1} \\ (z_1,\ldots ,z_n)&\longmapsto \left( \sqrt{ \frac{a_1}{\sum _k a_k(x_k^2+y_k^2)} }z_1,\ldots , \sqrt{ \frac{a_n}{\sum _k a_k(x_k^2+y_k^2)} }z_n \right) . \end{aligned} \end{aligned}$$

We find

$$\begin{aligned} \begin{aligned} \psi ^*\beta _1&= \sum _j \frac{1}{a_j}\sqrt{ \frac{a_j}{\sum _k a_k(x_k^2+y_k^2)}}\\&\quad \times \left( \sqrt{ \frac{a_j}{\sum _k a_k(x_k^2+y_k^2)}} ( x_jdy_j-y_jdx_j )\right. \\&\quad \left. +(x_jy_j -y_jx_j) d\left( \sqrt{ \frac{a_j}{\sum _k a_k(x_k^2+y_k^2)} } \right) \right) \\&=\beta _0. \end{aligned} \end{aligned}$$

This also shows that $\beta _0$ is a contact form, as $\psi$ is a diffeomorphism. We have shown:

Lemma A.5

The form $\beta _0$ is a contact form, and it is strictly contactomorphic to $\beta _1$. The Reeb field for $\beta _k$ for $k=0,1$ is given by

$$\begin{aligned} R=\sum _j a_j \left( x_j \frac{\partial }{\partial y_j}-y_j \frac{\partial }{\partial x_j} \right) . \end{aligned}$$

We now specialize to the 3-dimensional situation, for which in w-coordinates (cf. Remark A.4), we have

$$\begin{aligned} \Sigma ^3=\left\{ (w_1,w_2,w_3)\in {\mathbb {C}}^3:w_1^2-2iw_2w_3=0\right\} \cap S^3. \end{aligned}$$

Consider the explicit covering map

$$\begin{aligned} \pi : S^3 \longrightarrow \Sigma ^3, (z_0,z_1) \longmapsto \left( w_1=\sqrt{2}z_0 z_1,w_2=z_0^2,w_3=-iz_1^2\right) . \end{aligned}$$

We quickly verify that this is a covering map

• we have $w_1^2-2i w_2w_3=2z_0^2z_1^2-2z_0^2z_1^2=0$;

• we have $|w_1|^2+|w_2|^2+|w_3|^2=2|z_0|^2|z_1|^2+|z_0|^4+|z_1|^4=(|z_0|^2+|z_1|^2)^2=1$;

• the map is two to one, since all entries are quadratic.

We compute the pullback $\pi ^*\alpha _\epsilon$

$$\begin{aligned} \begin{aligned} \pi ^*\alpha _\epsilon&= \frac{1}{2|z_0|^2|z_1|^2+|z_0|^4+|z_1|^4 +\epsilon |z_0|^4-\epsilon |z_1|^4} 2(|z_0|^2+|z_1|^2) \frac{i}{2}\sum _j (z_j d{{\bar{z}}}_j -{{\bar{z}}}_j d z_j)\\&= \frac{2(|z_0|^2+|z_1|^2)}{(|z_0|^2+|z_1|^2)((1+\epsilon ) |z_0|^2 +(1-\epsilon ) |z_1|^2)} \frac{i}{2}\sum _j (z_j d{{\bar{z}}}_j -{{\bar{z}}}_j d z_j)\\&= \frac{2}{((1+\epsilon ) |z_0|^2 +(1-\epsilon ) |z_1|^2)} \frac{i}{2}\sum _j (z_j d{{\bar{z}}}_j -{{\bar{z}}}_j d z_j). \end{aligned} \end{aligned}$$

By Lemma A.5, the form $\pi ^* \alpha _\epsilon$ to strictly contactomorphic to the contact form $\beta _1$ with weights $a_1=1+\epsilon$ and $a_2=1-\epsilon$, which is just the ellipsoid model for the contact 3-sphere. To complete the argument, we use a result due to Harris and Paternain [20, Sect. 5], which relates the ellipsoids to the Katok examples.

Appendix B: Symplectic homology of surfaces

Let us consider connected Liouville domains in dimension 2. The simplest such Liouville domain is $D^2$, which has vanishing symplectic homology. For all other surfaces, note:

Lemma B.1

Let $(W,\lambda )$ be a connected Liouville domain of dimension 2. Assume that W is not diffeomorphic to $D^2$. Take a periodic Reeb orbit $\delta$ on one of the boundary components of W. Then, $[\delta ] \in {{\tilde{\pi }}}_1(W)$ is non-trivial. Furthermore, if $\delta _1$ and $\delta _2$ are periodic Reeb orbits on different boundary components, then $[\delta _1]\ne [\delta _2]$ as free homotopy classes. $\square$

Assume $W\ne D^2$, and denote the completion by ${{\widehat{W}}}$. Then, the chain complex for an admissible Hamiltonian ${{\widehat{H}}}$ that is both negative and $C^2$-small on W has the form

$$\begin{aligned} CF_\bullet ({{\widehat{H}}}) =\bigoplus _{\delta \in {{\tilde{\pi }}}_1(W)} CF^{\delta }_\bullet ({{\widehat{H}}}), \end{aligned}$$

where $CF_\bullet ^{\delta }({{\widehat{H}}})$ is generated by 1-periodic orbits in the free homotopy class $\delta$. The direct summand corresponding to contractible orbits needs as least as many generators as ${{\,\mathrm{rk}\,}}H_\bullet (W)$ by the Morse inequalities.

Lemma B.2

For each class $\delta$, the direct summand $CF_\bullet ^{\delta }({{\widehat{H}}})$ forms a subcomplex, and so, we have a splitting

$$\begin{aligned} HF_\bullet ({{\widehat{H}}})=\bigoplus _{\delta \in {{\tilde{\pi }}}_1(W)} HF_\bullet ^\delta ({{\widehat{H}}}). \end{aligned}$$

In addition, as ungraded modules, we have

$$\begin{aligned}&HF_\bullet ^\delta ({{\widehat{H}}})\\&\quad \cong {\left\{ \begin{array}{ll} {\mathbb {Z}}^2 &{} \text {if }\delta \text { is a positive boundary class}, \text {and }\text{ slope }({{\widehat{H}}}) \text {is sufficiently large},\\ H_\bullet (W) &{} \text {if }\delta \text { is the trivial class,}\\ 0 &{} \text {otherwise.} \end{array}\right. } \end{aligned}$$

Here, a positive boundary class just means a homotopy class of a positive multiple of a boundary component (oriented according the positive boundary orientation).

Proof

The first assertion follows from the fact that Floer cylinders do not change the free homotopy class. For the second claim, we use:

• The Floer differential of a $C^2$-small Hamiltonian between critical points is the Morse differential, which implies the second case.

• After a suitable Morse perturbation (a Morse function on $S^1$ with precisely two critical points) breaking the $S^1$-symmetry given by time-shifts, each positive boundary class gives two generators, corresponding to the critical points of the Morse function on $S^1$; as shown in [12], the differential is the Morse differential, which vanishes. Moreover, this symmetry-breaking process preserves the homotopy classes of periodic orbits, as observed in Remark 3.12.$\square$

Corollary B.3

Suppose that W is a connected Liouville domain of dimension 2. Assume that W is not diffeomorphic to $D^2$. Then, as an ungraded module, we have

$$\begin{aligned} SH_\bullet (W) \cong H_\bullet (W) \oplus \bigoplus _{\delta \text { positive boundary class}} {\mathbb {Z}}^2. \end{aligned}$$

Appendix C: On symplectic return maps

In this appendix, for convenience of the reader, we collect some standard facts concerning return maps arising from a given Reeb dynamics on some contact manifold (cf. the construction of the Calabi homomorphism, e.g., in [28, Sect. 10.3], or [2, Sect. 3.3] for the case of the 2-disk). In particular, we show that long Hamiltonian orbits on a global hypersurface of section correspond to long Reeb orbits on the ambient contact manifold.

Consider a map $\tau :\text{ int }(\Sigma ) \rightarrow \text{ int }(\Sigma )$ defined on the interior of a 2n-dimensional Liouville domain $\Sigma$. We assume that $\Sigma$ arises as a (connected) global hypersurface of section for some Reeb dynamics on a $2n+1$-dimensional contact manifold $(M,\alpha )$, and $\tau$ is the associated return map. Let $R_\alpha$ be the Reeb vector field of $\alpha$. Denote by $B=\partial \Sigma$, which we assume to be a contact submanifold of M with induced contact form $\alpha _B=\alpha \vert _B$, so that $R_\alpha \vert _B$ is tangent to B. Let $\lambda =\alpha \vert _\Sigma$, which is a Liouville form on $\text{ int }(\Sigma )$, since $R_\alpha$ is assumed to be positively transverse to the interior of $\Sigma$. That is, the two-form $\omega =d\lambda$ is symplectic on $\text{ int }(\Sigma )$. The 1-form $\lambda _B=\lambda \vert _B$ coincides with the contact form $\alpha _B$. Note that it is degenerate along B. By Stokes’ theorem, the symplectic volume of $\Sigma$ then coincides with the contact volume of B

$$\begin{aligned} \text{ vol }(\Sigma ,\omega )=\int _\Sigma \omega ^n=\int _\Sigma {\text {d}}(\lambda \wedge {\text {d}}\lambda ^{n-1})=\int _B \alpha _B \wedge {\text {d}}\alpha _B^{n-1}=\text{ vol }(B,\alpha _B). \end{aligned}$$

Note that $\tau$ is automatically a symplectomorphism with respect to $\omega$. Indeed, denote the time-t Reeb flow by $\varphi _t$, and let $T:\text{ int }(\Sigma )\rightarrow {\mathbb {R}}^+$

$$\begin{aligned} T(x)=\min \{t>0:\varphi _t(x)\in \text{ int }(\Sigma )\} \end{aligned}$$

denote the first return time function. Then, $\tau (x)=\varphi _{T(x)}(x)$, and so, for $x \in \text{ int }(\Sigma )$, $v \in T_x\Sigma$, we have

$$\begin{aligned} d_x\tau (v)=d_xT(v)R_{\alpha }(\tau (x))+d_x\varphi _{T(x)}(v). \end{aligned}$$

Using that $\varphi _t$ satisfies $\varphi _t^*\alpha =\alpha$, we obtain

$$\begin{aligned} \begin{aligned} (\tau ^*\lambda )_x(v)&=\alpha _{\tau (x)}(d_x\tau (v))\\&=d_xT(v)+(\varphi _{T(x)}^*\alpha )_x(v)\\&=d_xT(v)+\lambda _x(v).\\ \end{aligned} \end{aligned}$$

(C.7)

Therefore

$$\begin{aligned} \tau ^*\lambda ={\text {d}}T + \lambda , \end{aligned}$$

(C.8)

which in particular implies that $\tau ^*\omega =\omega$.

Moreover, the average of the return time function gives the contact volume of M, i.e., we have the identity

$$\begin{aligned} \int _\Sigma T \omega ^n=\text{ vol }(M,\alpha ). \end{aligned}$$

(C.9)

This may be proved as follows. We have a smooth embedding

$$\begin{aligned} \psi :{\mathbb {R}}/{\mathbb {Z}}\times \text{ int }(\Sigma ) \rightarrow M, \end{aligned}$$

given by $\psi (s,x)=\varphi _{sT(x)}(x)$, which is a diffeomorphism onto $M\backslash B$. It satisfies

$$\begin{aligned}(\psi ^*\alpha )(\partial _s)=\alpha (T R_\alpha )=T,\end{aligned}$$

and, for $v \in T\text{ int }(\Sigma )$,

$$\begin{aligned}(\psi ^*\alpha )(v)=\alpha (s{\text {d}}T(v)R_\alpha +{\text {d}}\varphi _{sT}(v))=s{\text {d}}T(v)+\alpha (v).\end{aligned}$$

Then

$$\begin{aligned} \psi ^*\alpha =T {\text {d}}s + s{\text {d}}T+\lambda ={\text {d}}(sT)+\lambda , \end{aligned}$$

and so

$$\begin{aligned} \psi ^*(\alpha \wedge d\alpha ^n)=({\text {d}}(sT)+\lambda )\wedge {\text {d}}\lambda ^n=T {\text {d}}s \wedge \omega ^n. \end{aligned}$$

Integrating, and using the fact that B is codimension 2 in M, we obtain

$$\begin{aligned} \text{ vol }(M,\alpha )= & {} \int _{M\backslash B}\alpha \wedge {\text {d}}\alpha ^n=\int _{{\mathbb {R}}/{\mathbb {Z}}\times \text{ int }(\Sigma )}\psi ^*(\alpha \wedge d\alpha ^n) \\= & {} \int _{{\mathbb {R}}/{\mathbb {Z}}\times \text{ int }(\Sigma )}T {\text {d}}s \wedge \omega ^n=\int _{\text{ int }(\Sigma )}T \omega ^n=\int _{\Sigma }T \omega ^n, \end{aligned}$$

where we have used that $\omega ^n\vert _B\equiv 0$, and the claim follows. In case where $\tau$ is Hamiltonian, we want to relate the Hamiltonian action of a periodic orbit of $\tau$ to the Reeb action of the corresponding Reeb orbit in the ambient contact manifold.

Let $H: S^1 \times \Sigma \rightarrow {\mathbb {R}}^+$ be a Hamiltonian generating $\tau$, i.e., the isotopy $\phi _t$ defined by $\phi _0=id$, $\frac{{\text {d}}}{{\text {d}}t}\phi _t=X_{H_t}\circ \phi _t$ satisfies $\phi _1=\tau$. The sign convention for the Hamiltonian vector field is $i_{X_{H_t}}\omega =-dH_t$. We usually view this Hamiltonian isotopy as defining an element $\phi =\phi _H=[\{\phi _t\}]$ in the universal cover $\widetilde{\text{ Diff }}(\Sigma ,\omega )$ of the space of symplectomorphisms $\text{ Diff }(\Sigma ,\omega )$. By Cartan’s formula, we have

$$\begin{aligned} \partial _t \phi _t^*\lambda =\phi _t^*{\mathcal {L}}_{X_{H_t}}\lambda =\phi _t^*(i_{X_{H_t}}\omega + d(i_{X_{H_t}}\lambda ))=\phi _t^*d(i_{X_{H_t}}\lambda -H_t), \end{aligned}$$

and so integrating, we obtain

$$\begin{aligned} \tau ^*\lambda -\lambda =dF_H, \end{aligned}$$

(C.10)

where

$$\begin{aligned} F_H=\int _0^1 (i_{X_{H_t}}\lambda - H_t) \circ \phi _t \;{\text {d}}t. \end{aligned}$$

(C.11)

Combining (C.8) and (C.10), we deduce that

$$\begin{aligned} \tau =F_H+C \end{aligned}$$

(C.12)

for some constant C (assuming $\Sigma$ is connected).

We determine the constant C under a suitable assumption, which we assume holds in all what follows. Namely, assume that $\tau$ extends to $\Sigma$ with the same formula, i.e., via an extension of the return time function T to $\Sigma$. Assume also that $H_t\vert _B\equiv const:=C_t>0$ for some H generating $\tau$. Equivalently, $X_{H_t}\vert _B=h_tR_B$ for some (not necessarily positive) smooth function $h_t$ on B, satisfying $h_t=dH_t(V_\lambda )\vert _B$ where $V_\lambda$ is the Liouville vector field associated with $\lambda$. In this case, denoting $\gamma _x(t)=\phi _t(x)$ for $x \in B$ and $t\in [0,1]$, we get

$$\begin{aligned} F_H(x)=\int _{\gamma _x} \lambda _B-\int _0^1C_t{\text {d}}t=\int _0^1 (h_t(\phi _t(x))-C_t){\text {d}}t, \end{aligned}$$

(C.13)

On the other hand, let $\beta _x(t)=\varphi _t(x)$ be the Reeb orbit through x ending at $\beta _x(1)=\tau (x)$, for $t \in [0,1]$, which we assume parametrized, so that ${\dot{\beta }}_x=T(x)R_B(\beta _x)$. Note that $\beta _x$ is a reparametrization of $\gamma _x$, and so we obtain

$$\begin{aligned} \tau (x)=\int _{\beta _x} \lambda _B=\int _{\gamma _x}\lambda _B. \end{aligned}$$

This means that T is the unique primitive of $\tau ^*\lambda -\lambda$ satisfying $T(x)=\int _{\gamma _x}\lambda _B$ for $x \in B$. Combining (C.12) and (C.13), we conclude that

$$\begin{aligned} C=\int _0^1C_t{\text {d}}t>0, \end{aligned}$$

a positive constant.

By the above computation, T is what is usually called the action of $\phi =\phi _H$ with respect to $\lambda$, and is independent of the isotopy class (with fixed endpoints) of the path $\phi _H$. The Calabi invariant is then by definition the average action $CAL(\phi _H,\omega )=\int T \omega ^n$, which is independent of $\lambda$; cf. [2, 28]. Combining with (C.9), we obtain

$$\begin{aligned} CAL(\phi _H,\omega )=\text{ vol }(M,\alpha ). \end{aligned}$$

Let $\gamma :S^1={\mathbb {R}}/k{\mathbb {Z}}\rightarrow \Sigma$, defined by $\gamma (t)=\phi _t(x)$, be a k-periodic Hamiltonian orbit associated to the k-periodic point x of $\tau$. That is, we have $x=\gamma (0)$, $\gamma (1)=\tau (x),\dots , \gamma (k)=\tau ^k(x)=x$, and assume that k is the minimal period of x. We then get

$$\begin{aligned} \sum _{i=1}^k F_H(\tau ^i(x))={\mathcal {A}}_{H^{\#k}}(\gamma ) \end{aligned}$$

is precisely the Hamiltonian action of $\gamma$ with respect to the Hamiltonian

$$\begin{aligned} H_t^{\#k}=\sum _{i=1}^k H_t \circ \phi _t^{-i} \end{aligned}$$

generating $\tau ^k$. If $\beta : S^1={\mathbb {R}}/{\mathbb {Z}} \rightarrow M$ is the Reeb orbit corresponding to $\gamma$, (C.12) implies that its period is

$$\begin{aligned} \int _{S^1}\beta ^*\alpha =\sum _{i=1}^k T(\tau ^i(x))= \sum _{i=1}^k F_H(\tau ^i(x))+kC={\mathcal {A}}_{H^{\#k}}(\gamma )+kC. \end{aligned}$$

Since $C>0$, this implies the following: if the Hamiltonian action of every k-periodic orbit $\gamma$ grows to infinity with k, then the period of the associated Reeb orbits $\beta$ also. In other words, long Hamiltonian periodic orbits in the global hypersurface of section give long Reeb orbits in the ambient contact manifold.

We summarize the above discussion in the following:

Lemma C.1

Let $(M^{2n+1},\alpha )$ be a contact manifold, $(\Sigma ^{2n},\omega =d\alpha \vert _\Sigma )$ a Liouville domain which is a global hypersurface of section for the Reeb flow, $(B^{2n-1},\alpha _B)=(\partial \Sigma ,\alpha \vert _B)$, $\tau :\text{ int }(\Sigma ) \rightarrow \text{ int }(\Sigma )$ the Poincaré return map, and $T: \text{ int }(\Sigma )\rightarrow \mathbb R^+$ the first return time. Then:

• (1) $\text{ vol }(\Sigma ,\omega )=\text{ vol }(B,\alpha _B)$.

• (2) $\text{ vol }(M,\alpha )=\int _{\Sigma }T \omega ^n$.

• (3) $\tau$ is an exact symplectomorphism.

• (4) If $\tau$ is Hamiltonian with generating isotopy $\phi _H=[\{\phi _t\}]\in \widetilde{\text{ Diff }}(\Sigma ,\omega )$, and extends to $\Sigma$ as a (not necessarily positive) reparametrization of the Reeb flow at B, then:

  • (i) $CAL(\phi _H,\omega )=\text{ vol }(M,\alpha )$.

  • (ii) The period of a Reeb orbit $\beta$ on M corresponding to a k-periodic Hamiltonian orbit $\gamma$ on $\Sigma$ is

    $$\begin{aligned} \int _{S^1}\beta ^*\alpha ={\mathcal {A}}_{H^{\#k}}(\gamma )+kC \end{aligned}$$

    for some positive constant $C>0$, where

    $$\begin{aligned} {\mathcal {A}}_{H^{\#k}}(\gamma )=\int _{S^1}\gamma ^*\lambda -\int _0^1H^{\#k}_t(\gamma (t)){\text {d}}t \end{aligned}$$

    is the Hamiltonian action of $\gamma$ with respect to the Hamiltonian

    $$\begin{aligned} H_t^{\#k}=\sum _{i=1}^k H_t \circ \phi _t^{-i} \end{aligned}$$

    generating $\tau ^k$. In particular, if $\gamma$ has large action, then $\beta$ has large period.

Appendix D: Strong convexity implies strong index-positivity

In this appendix, we give a general condition for index-positivity to hold, which is also relevant for the restricted three-body problem. A connected compact hypersurface $\Sigma \subset {\mathbb {R}}^{4}$ is said to bound a strongly convex domain $W \subset {\mathbb {R}}^{4}$ whenever there exists a smooth function $\phi : {\mathbb {R}}^{4}\rightarrow {\mathbb {R}}$ satisfying:

• (i) (Regularity) $\Sigma =\{\phi =0\}$ is a regular level set;

• (ii) (Bounded domain) $W = \{z\in {\mathbb {R}}^{4}: \phi (z) \le 0\}$ is bounded and contains the origin; and

• (iii) (Positive-definite Hessian) $\nabla ^2\phi _{z}(h, h) > 0$ for $z\in W$ and for each non-zero tangent vector $h\in T\Sigma$.

In this case, the radial vector field is transverse to $\Sigma$, and so $\Sigma$ is a contact-type 3-sphere, inheriting a contact form $\alpha$ induced by the standard Liouville form in ${\mathbb {R}}^{4}$.

Lemma D.1

Suppose that $\Sigma$ bounds a strongly convex domain. Then, $\Sigma$ is strongly index-positive.

Remark D.2

In the planar restricted three-body problem, the values of energy/mass ratio $(c,\mu )$ for which the Levi–Civita regularization bounds a strictly convex domain is called the convexity range, which in particular implies that the dynamics is dynamically convex (cf. [5, 6, 24]). It follows that index-positivity holds in the convexity range for the quotient ${\mathbb {R}}P^3$, which is part of the assumptions of Theorem A.

Proof

Write $\Sigma = \phi ^{-1}(0)$ as in the definition above. Denote the contact form on $\Sigma$ by $\alpha :=\lambda |_{\Sigma }$. We will use the standard quaternions I, J, K, where I is chosen to coincide with the standard complex structure.

The tangent space of $\Sigma$ is spanned by the vectors

$$\begin{aligned} R=X_\phi /\alpha (X_\phi ) =I \nabla \phi /\alpha (X_\phi ) =Iw,\; U=Jw -\alpha (Jv)R,\; V=Kw -\alpha (Kv)R. \end{aligned}$$

We note that U and V give a symplectic trivialization $\epsilon$ of $(\xi =\ker \alpha ,d\alpha )$. To see this, we compute

$$\begin{aligned} d\alpha (U,V)= d \alpha ( Jw, Kw) =w^t J^t I^t K w=w^t K^t K w=w^t w=1. \end{aligned}$$

To prove the claim, we investigate the rate of change of a version of the rotation number. See Chapter 10.6 in [16] for a detailed standard description of the Robbin–Salamon index in terms of the rotation number. We will detail the version that we will use below.

We look at the linearization of the Hamiltonian flow:

$$\begin{aligned} \dot{X} =\nabla _X X_\phi = I \nabla ^2\phi \cdot X. \end{aligned}$$

(D.14)

Starting with $X(0) \in \xi$, we compute how quickly the vector X rotates with respect to the frame. Define the angular form

$$\begin{aligned} \Theta = \frac{u{\text {d}}v-v{\text {d}}u}{u^2+v^2} = \frac{u d\alpha (U, \cdot ) +v d\alpha (V,\cdot ) }{u^2+v^2} =\frac{d\alpha (u U+v V,\cdot )}{u^2+v^2} , \end{aligned}$$

where (u, v) are cartesian coordinates on the plane spanned by the frame (U, V), so we may write $X=u U+ v V$. We plug in $\dot{X}$ and find

$$\begin{aligned} \Theta (\dot{X})= & {} \frac{d\alpha (X,\dot{X})}{u^2+v^2}= \frac{(u U+v V)^t I^t I \nabla ^2\phi \cdot (u U+v V)}{u^2+v^2} \nonumber \\= & {} \frac{\nabla ^2\phi (uU+vV,uU+vV)}{u^2+v^2} \ge \lambda _{\min }>0, \end{aligned}$$

(D.15)

where $\lambda _{\min }$ is the minimal eigenvalue of $\nabla ^2\phi$ over the compact hypersurface $\Sigma$. After we have set up some notation, we will see that this is enough to get a lower bound on the growth rate of the Robbin–Salamon index. With our global trivialization $\epsilon$, we can define the matrix

$$\begin{aligned} \psi (t)=\epsilon \circ dFl^R_t \circ \epsilon ^{-1}. \end{aligned}$$

By applying Eq. (D.14) to the initial vectors $\epsilon ^{-1}(1,0)$ and $\epsilon ^{-1}(0,1)$, we get a linear evolution equation for the matrix $\psi (t)$

$$\begin{aligned} {{\dot{\psi }}} =A(t) \psi , \end{aligned}$$

(D.16)

where A is a time-dependent matrix. We will view this ODE as a vector field on Sp(2): the linearized Reeb flow along each Reeb orbit will give rise to such a vector field.

To relate the above angle to the Conley–Zehnder index, we also need to recall the Iwasawa decomposition, also known as KAN decomposition, of Sp(2). Write

$$\begin{aligned} KAN:=\left\{ \left( \left( \begin{array}{cc} \cos (\phi ) &{} -\sin (\phi ) \\ \sin (\phi ) &{} \cos (\phi ) \end{array} \right) , \left( \begin{array}{cc} a &{} 0 \\ 0 &{} a^{-1} \end{array} \right) , \left( \begin{array}{cc} 1 &{} t \\ 0 &{} 1 \end{array} \right) \right) ~\Bigg |~ \phi \in [0,2\pi ), a \in {\mathbb {R}}_{>0},t \in {\mathbb {R}}\right\} . \end{aligned}$$

And put

$$\begin{aligned} \text{ kan }: KAN \longrightarrow Sp(2), (\phi ,a, t) \longmapsto \left( \begin{array}{cc} \cos (\phi ) &{} -\sin (\phi ) \\ \sin (\phi ) &{} \cos (\phi ) \end{array} \right) \left( \begin{array}{cc} a &{} 0 \\ 0 &{} a^{-1} \end{array} \right) \left( \begin{array}{cc} 1 &{} t \\ 0 &{} 1 \end{array} \right) . \end{aligned}$$

This map has the inverse

$$\begin{aligned} \begin{aligned} \text{ kan}^{-1}: Sp(2)&\longrightarrow KAN \\ \left( \begin{array}{cc} a &{} b \\ c &{} d \end{array} \right)&\longmapsto \left( \frac{1}{\sqrt{a^2+c^2}} \left( \begin{array}{cc} a &{} -c \\ c &{} a \end{array} \right) , \left( \begin{array}{cc} \sqrt{a^2+c^2} &{} 0 \\ 0 &{} \frac{1}{\sqrt{a^2+c^2}} \end{array} \right) , \left( \begin{array}{cc} 1 &{} \frac{ab+cd}{\sqrt{a^2+c^2}} \\ 0 &{} 1 \end{array} \right) \right) . \end{aligned} \end{aligned}$$

The KAN angle can locally be determined as

$$\begin{aligned} \arg ( \text{ kan}^{-1}(\psi )) ={{\,\mathrm{atan}\,}}(c/a), \end{aligned}$$

so we see that the change in angle equals

$$\begin{aligned} \frac{{\text {d}}}{{\text {d}}t}\arg ( \text{ kan}^{-1}(\psi )) =\frac{{\text {d}}}{{\text {d}}t}{{\,\mathrm{atan}\,}}(c/a)=\frac{a\dot{c} -c\dot{a}}{a^2+c^2}. \end{aligned}$$

On the other hand, the rate of change of the KAN angle equals $\Theta (\dot{X})$. Indeed, the first column of $\psi (t)$ is the vector $Z(t):=\epsilon (X(t))=\left( \begin{array}{c} u \\ v \end{array} \right)$ if we put $X(0)=\epsilon ^{-1}(1,0)$.

By Eq. (D.15), this rate of change is at least $\lambda _{\min }$, where $\lambda _{\min }$ is the minimal eigenvalue of $-IA$ (which we assume to be positive-definite). This means that each slice

$$\begin{aligned} S_\phi =\text{ kan }\left( \left\{ \left( \left( \begin{array}{cc} \cos (\phi ) &{} -\sin (\phi ) \\ \sin (\phi ) &{} \cos (\phi ) \end{array} \right) , \left( \begin{array}{cc} a &{} 0 \\ 0 &{} a^{-1} \end{array} \right) , \left( \begin{array}{cc} 1 &{} t \\ 0 &{} 1 \end{array} \right) \right) ~\Bigg |~ a \in {\mathbb {R}}_{>0},t \in {\mathbb {R}}\right\} \right) \end{aligned}$$

is a global surface of section for the vector field associated with Eq. (D.16): the maximal return time is $\frac{2\pi }{\lambda _{\min }}$. Now, take a matrix $\psi (0)$ in the slice $S_0$, and let $\psi (t)$ denote the solution to Eq. (D.16).

Claim: Each crossing is regular and contributes positively.

To see this, recall that the crossing form of a path $\psi$ in Sp(2) at a crossing t is defined as the bilinear form

$$\begin{aligned} \omega _0(\cdot , {{\dot{\psi }}}(t) \cdot )|_{\ker (\psi (t)-{{\,\mathrm{id}\,}})}. \end{aligned}$$

Since U, V is a symplectic frame, we have with $Z=(u,v)$ (i.e., $X=uU+vV$), the following inequality:

$$\begin{aligned} \omega _0(Z,{{\dot{\psi }}} Z)=d\alpha (X, \dot{X}) \ge \lambda _{\min } (u^2+v^2), \end{aligned}$$

by Eq. (D.15). This establishes the claim.

Let $t_\ell$ denote the $\ell$-th return time to $S_0$ of the path $\psi$. The symplectic path $\psi |_{[0,t_\ell ]}$ is not necessarily a loop, but we can make it into a loop by connecting $\psi (t_\ell )$ to $\psi (0)$ while staying in the slice $S_0$. We can and will do this by adding at most one crossing, which we make regular. Call the extension to a loop ${{\tilde{\psi }}}$. The additional crossing that we may have inserted can contribute negatively.

Now, use the loop axiom for the Robbin–Salamon index. This tells us that

$$\begin{aligned} \mu _{{\text {RS}}}({{\tilde{\psi }}})=2\mu _\ell ({{\tilde{\psi }}})=2 \ell . \end{aligned}$$

By the catenation property of the Robbin–Salamon index and positivity of all but the last (potential) crossing, we have

$$\begin{aligned} \mu _{{\text {RS}}}(\psi )\ge 2\ell -2. \end{aligned}$$

Now, consider a symplectic path $\psi$ of length T. We can bound the winding number as

$$\begin{aligned} \ell \ge \left\lfloor \frac{\lambda _{\min }}{2\pi } T \right\rfloor . \end{aligned}$$

With this in mind, we obtain for a Hamiltonian arc $\gamma$ of length T

$$\begin{aligned} \mu _{{\text {RS}}}(\gamma ;\epsilon ) \ge \frac{2\lambda _{\min }}{2\pi } T-4. \end{aligned}$$

When this Hamiltonian arc $\gamma$ is viewed as a Reeb arc $\gamma _R$ with Reeb action $T_R$, we can rewrite this bound as follows, using:

$$\begin{aligned} T_R =\int _0^T \alpha (X_\phi ) {\text {d}}t \le T \cdot \max \alpha (X_\phi ). \end{aligned}$$

We find

$$\begin{aligned} \mu _{{\text {RS}}}(\gamma _R;\epsilon ) \ge \frac{\lambda _{\min }}{\pi \max \alpha (X_\phi ) }T_R -4. \end{aligned}$$

$\square$

Remark D.3

Observe that the proof actually shows the stronger claim that index-positivity holds when the Hessian of $\phi$ restricted to the contact structure is positive-definite. Note also that the latter condition is not enough for dynamical convexity.

Finally, we note that the bound obtained can be sharpened, since the index is necessarily positive by observing that $\psi (0)={{\,\mathrm{id}\,}}$, so it is a crossing and using that each crossing of the path $\psi$ contributes positively.

Appendix E: Strongly index-definite symplectic paths

In this appendix, we prove a crucial index growth estimate needed to rule out non-relevant boundary orbits via index considerations (needed in Lemma 4.5 in the main body of the paper).

Definition E.1

Consider the linear ODE ${{\dot{\psi }}}(t)=A(t)\psi (t)$, where $A:{\mathbb {R}}_{\ge 0}\rightarrow \mathfrak {sp}(2n)$ and $A(0)=0$. Its solution is a path of symplectic matrices with $\psi (0)=\mathbb {1}$. We say that the ODE is strongly index-definite if there exist constants $c>0,d \in {\mathbb {R}}$, such that

$$\begin{aligned} \vert \mu _{{\text {RS}}}(\psi \vert _{[0,t]})\vert \ge ct+d, \end{aligned}$$

where $\mu _{{\text {RS}}}$ is the Robbin–Salamon index [33].

Note that we make no non-degeneracy assumptions on the symplectic paths in the above definition.

We now consider the specific family of linear ODEs $\dot{\psi }(t)=A(t)\psi (t)$, where the matrix A has the special form

$$\begin{aligned} A(t)=\left( \begin{array}{@{}c|c@{}} R(t) &{} \begin{matrix} {\overline{X}}(t) &{} 0 \\ {\overline{Y}}(t) &{} 0 \\ \end{matrix} \\ \hline \begin{matrix} 0 &{} 0 \\ {\overline{Y}}(t) &{} -{\overline{X}}(t) \end{matrix} &{} \begin{matrix} a(t) &{} 0 \\ b(t) &{} -a(t) \end{matrix} \end{array}\right) \in \mathfrak {sp}(2n). \end{aligned}$$

Here, we use the notation $({\overline{X}},{\overline{Y}})=(X_1,Y_1,\dots ,X_{n-1},Y_{n-1})$, and we assume $R(t) \in \mathfrak {sp}(2n-2)$, $A(0)=0$.

Lemma E.2

Assume that the linear ODE ${\dot{M}}(t)=R(t)M(t)$ is strongly index-definite as an ODE in dimension $2n-2$. Then, the same holds for the linear ODE ${{\dot{\psi }}}(t)=A(t)\psi (t)$.

Proof

One may check that

$$\begin{aligned} {\mathfrak {g}}=\left\{ \left( \begin{array}{@{}c|c@{}} R &{} \begin{matrix} {\overline{X}} &{} 0 \\ {\overline{Y}} &{} 0 \\ \end{matrix} \\ \hline \begin{matrix} 0 &{} 0 \\ {\overline{Y}} &{} -{\overline{X}} \end{matrix} &{} \begin{matrix} a &{} 0 \\ b &{} -a \end{matrix} \end{array}\right) : R \in \mathfrak {sp}(2n-2)\right\} \end{aligned}$$

is a Lie subalgebra of $\mathfrak {sp}(2n)$. The corresponding Lie subgroup of Sp(2n) is

$$\begin{aligned} G=\left\{ \left( \begin{array}{@{}c|c@{}} M &{} \begin{matrix} {\overline{x}} &{} 0 \\ {\overline{y}} &{} 0 \\ \end{matrix} \\ \hline \begin{matrix} 0 &{} 0 \\ {\overline{u}} &{} {\overline{v}} \end{matrix} &{} \begin{matrix} \alpha &{} 0 \\ \beta &{} \alpha ^{-1} \end{matrix} \end{array}\right) : M \in Sp(2n-2),\; \alpha >0,\; (-{\overline{y}},{\overline{x}})\cdot M +\alpha \cdot ( {\overline{u}},{\overline{v}})=0 \right\} . \end{aligned}$$

We deduce that $\psi \in G$. We then write

$$\begin{aligned} \psi =\left( \begin{array}{@{}c|c@{}} M &{} \begin{matrix} {\overline{x}} &{} 0 \\ {\overline{y}} &{} 0 \\ \end{matrix} \\ \hline \begin{matrix} 0 &{} 0 \\ {\overline{u}} &{} {\overline{v}} \end{matrix} &{} \begin{matrix} \alpha &{} 0 \\ \beta &{} \alpha ^{-1} \end{matrix} \end{array}\right) \in G, \end{aligned}$$

where M is a solution to $\dot{M}=R M$, and consider the following homotopy of paths:

$$\begin{aligned} \psi _s=\left( \begin{array}{@{}c|c@{}} M &{} \begin{matrix} s{\overline{x}} &{} 0 \\ s{\overline{y}} &{} 0 \\ \end{matrix} \\ \hline \begin{matrix} 0 &{} 0 \\ s{\overline{u}} &{} s{\overline{v}} \end{matrix} &{} \begin{matrix} \alpha &{} 0 \\ \beta &{} \alpha ^{-1} \end{matrix} \end{array}\right) . \end{aligned}$$

Note that $\psi _s$ is a path in $G\subset Sp(2n)$ for every s, and $\psi _0$ has no off-diagonal terms. For any given t, this gives a homotopy in G relative endpoints of $\psi \vert _{[0,t]}$ to a concatenated path of the form $\psi _0\vert _{[0,t]} \# \phi _t$, where $\phi _t(s)=\psi _s(t)$. We therefore have

$$\begin{aligned} \mu _{{\text {RS}}}(\psi \vert _{[0,t]})=\mu _{{\text {RS}}}(\psi _0\vert _{[0,t]})+\mu _{{\text {RS}}}(\phi _t). \end{aligned}$$

(E.17)

On the other hand, from the block decomposition of $\psi _0$ and the fact that the lower block can be homotoped to a symplectic shear by joining $\alpha (t)$ to 1, we have

$$\begin{aligned} \mu _{{\text {RS}}}(\psi _0\vert _{[0,t]})=\mu _{{\text {RS}}}(M\vert _{[0,t]})\pm \frac{1}{2}\text{ sign }(\beta (t)), \end{aligned}$$

(E.18)

where the sign depends on conventions. Moreover, one may easily check that the characteristic polynomial of an element in G is completely independent of the off-diagonal terms. In particular, we obtain that

$$\begin{aligned} \det (\psi _s-\mathbb {1})=\det (\psi _0-\mathbb {1})=\det (M-\mathbb {1})(\alpha -1)(\alpha ^{-1}-1) \end{aligned}$$

is independent of s. In other words, $\psi (t)$ is an intersection point with the Maslov cycle if and only if $\psi _0(t)$ is, and the eigenvalue 1 has the same algebraic multiplicity for both such intersections. Moreover, if $\psi (t)$ is not an intersection, then $\phi _t$ does not intersect the Maslov cycle at all.

One may check that if $\alpha (t)\ne 1$, then the geometric multiplicity of 1 as an eigenvalue of $\phi _t(s)$ is independent of s (and, therefore, $\mu _{{\text {RS}}}(\phi _t)=0$ for such t). If $\alpha (t)=1$, this may not necessarily still hold. However, we may appeal to the following general fact, whose proof was provided to the authors by Alberto Abbondandolo:

Lemma E.3

There exists a universal bound $C=C(n)$ (depending only on dimension), such that, if $\phi :[0,1] \rightarrow Sp(2n)$ is a continuous path of symplectic matrices for which the algebraic multiplicity of the eigenvalue 1 of the matrix $\phi (t)$ is independent of t, then

$$\begin{aligned} |\mu _{{\text {RS}}}(\phi )|\le C. \end{aligned}$$

Proof of Lemma E.3

Step 1. We first reduce to the case where $\phi$ has 1 as the only eigenvalue. We have a continuous symplectic splitting ${\mathbb {R}}^{2n}=V(t)\oplus W(t)$ where V(t) is the generalized eigenspace of $\phi (t)$ corresponding to 1, and W(t) is the direct sum of the generalized eigenspaces of $\phi (t)$ corresponding to the other eigenvalues (here, the dimensions of V(t) and W(t) are t-independent by assumption), for which $\phi (t)=\phi _V(t) \oplus \phi _W(t)$ splits symplectically. Since $\phi _W$ does not intersect the Maslov cycle by construction, we have $\mu _{{\text {RS}}}(\phi )=\mu _{{\text {RS}}}(\phi _V)+\mu _{{\text {RS}}}(\phi _W)=\mu _{{\text {RS}}}(\phi _V).$

Step 2. A loop $\phi$ of symplectic matrices having 1 as the only eigenvalue is nullhomotopic in Sp(2n), and hence, $\mu _{{\text {RS}}}(\phi )=0$. This follows for instance by the interpretation of the Robbin–Salamon index as the total winding number of the Krein-positive eigenvalues on the unit circle (see, e.g., [1, Lemma 1.3.7]).

Step 3. The identity matrix may be joined to any symplectic matrix M satisfying spec$(M)=\{1\}$ via a path M(t) satisfying spec$(M(t))=\{1\}$, and for which $|\mu _{{\text {RS}}}(M(t))|\le C$ for some universal bound C. Indeed, we may write $M=e^{JS}$ where S is a symmetric matrix having 0 as the only eigenvalue, and consider the path $M(t)=e^{tJS}$. This satisfies the required properties, since M(t) changes strata of the Maslov cycle only at $t=0$, the geometric multiplicity of 1 jumping from 2n at $t=0$ to perhaps a lower one at $t>0$, and so the contribution of this wall-crossing to $\mu _{{\text {RS}}}(M)$ is universally bounded.

The proof finishes by combining the previous steps, where we join the endpoints of a path $\phi$ as in Step 1 to the identity as in Step 3, use the concatenation property of $\mu _{{\text {RS}}}$, and appeal to Step 2. $\square$

Combining Eqs. (E.17) and (E.18) with Lemma E.3, we conclude that

$$\begin{aligned} |\mu _{{\text {RS}}}(\psi \vert _{[0,t]})-\mu _{{\text {RS}}}(M\vert _{[0,t]})|\le C \end{aligned}$$

for some universal constant $C=C(n)$, from which the conclusion of Lemma E.2 is immediate. $\square$