function for $\varphi$ if it is a generating function for the Lagrangian submanifold $L_{\varphi_R}$ of $T^*_{-d\theta}(S^1 \times \mathbb{R}^{2n+1})$. By similar arguments as before, the Lagrangian submanifold $L_{\varphi_R}$ of $T^*_{-d\theta}(S^1 \times \mathbb{R}^{2n+1})$ extends to a Lagrangian submanifold of $T^*_{-d\theta}(S^1 \times S^{2n} \times \mathbb{R})$ lcs Hamiltonian isotopic to the zero section. Moreover, the last descends to a Lagrangian submanifold $\overline{L}_{\varphi}$ of $T^*_{-d\theta}(S^1 \times S^{2n} \times S^1)$ lcs Hamiltonian isotopic to the zero section, indeed

$$ \tau([[(\theta, x, y, z + 1, \Theta, X, Y, Z + 1)])] = \tau([[(\theta, x, y, z, \Theta, X, Y, Z)])] + (0, 0, 0, 1, 0, 0, 0, 0). $$

We say that $\overline{F} : S^1 \times S^{2n} \times S^1 \times \mathbb{R}^N \to \mathbb{R}$ is a generating function for $\varphi$ if it is a generating function for $\overline{L}_{\varphi}$. Applying Theorem 4.3 we obtain the following result.

**Proposition 5.5.** Let $\varphi$ be a compactly supported lcs Hamiltonian diffeomorphism of $S^1 \times \mathbb{R}^{2n+1}$ or $S^1 \times \mathbb{R}^{2n} \times S^1$. Then $\varphi$ has a generating function quadratic at infinity $\overline{F} : S^1 \times B \times \mathbb{R}^N \to \mathbb{R}$, where $B$ is either $S^{2n+1}$ or $S^{2n} \times S^1$, which is unique up to equivalence.

We define the action spectrum of a lcs Hamiltonian diffeomorphism of $S^1 \times \mathbb{R}^{2n+1}$ to be the set of actions (hence time-shifts) of its essential translated points. The translated points of a lcs Hamiltonian diffeomorphisms $\varphi$ of $S^1 \times \mathbb{R}^{2n} \times S^1$ do not have a well-defined time-shift, because the Lee flow is periodic and so for any translated point $p$ of $\varphi$ there is a $\mathbb{Z}$-family of Lee chords from $\varphi(p)$ to $p$. However, if $\varphi$ is compactly supported then it has a unique lift $\varphi_R$ that is the identity over the complement of the support of $\varphi$. Any point $\bar{p}$ of $S^1 \times \mathbb{R}^{2n+1}$ projecting to $p$ is a translated point of $\varphi_R$. Moreover, since $\varphi_R$ is equivariant by translation by 1 in the Lee direction, all the points projecting to $p$ have the same time-shift $T$ (as translated points of $\varphi_R$). We say that $T$ is the time-shift of the translated point $p$ of $\varphi$, and we define the action spectrum of $\varphi$ to be the set of actions (hence time-shifts) of its essential translated points (in other words, the action spectrum of $\varphi_R$).

**Proposition 5.6.** Let $\varphi$, $\varphi_1$ and $\varphi_2$ be compactly supported lcs Hamiltonian diffeomorphisms of $S^1 \times \mathbb{R}^{2n+1}$ or $S^1 \times \mathbb{R}^{2n} \times S^1$, and $\overline{F} : S^1 \times B \times \mathbb{R}^N \to \mathbb{R}$, $\overline{F}_1 : S^1 \times B \times \mathbb{R}^{N_1} \to \mathbb{R}$ and $\overline{F}_2 : S^1 \times B \times \mathbb{R}^{N_2} \to \mathbb{R}$ generating functions for $\varphi$, $\varphi_1$ and $\varphi_2$ respectively, where $B$ is either $S^{2n+1}$ or $S^{2n} \times S^1$. Then

(i) The action spectrum of $\varphi$ is equal to the set of critical values of $\overline{F}$.

(ii) The action spectrum of $\varphi_2^{-1} \circ \varphi_1$ is equal to the set of critical values of the difference function

$$ \overline{F}_1 - \overline{F}_2 : S^1 \times B \times \mathbb{R}^{N_1} \times \mathbb{R}^{N_2} \to \mathbb{R}, \quad (\overline{F}_1 - \overline{F}_2)(\theta, p, \zeta_1, \zeta_2) = \overline{F}_1(\theta, p, \zeta_1) - \overline{F}_2(\theta, p, \zeta_2). $$

*Proof.* If $\overline{F} : S^1 \times S^{2n+1} \times \mathbb{R}^N \to \mathbb{R}$ is a generating function for a compactly supported lcs Hamiltonian diffeomorphism $\varphi$ of $S^1 \times \mathbb{R}^{2n+1}$ then the induced function $F : S^1 \times \mathbb{R}^{2n+1} \times \mathbb{R}^N \to \mathbb{R}$ is a generating function for $L_{\varphi}$. Similarly, if $\overline{F} : S^1 \times S^{2n} \times S^1 \times \mathbb{R}^N \to \mathbb{R}$ is a generating function for a compactly supported lcs Hamiltonian diffeomorphism $\varphi$ of $S^1 \times \mathbb{R}^{2n} \times S^1$ then the induced function $F : S^1 \times \mathbb{R}^{2n+1} \times \mathbb{R}^N \to \mathbb{R}$ is a generating function for $L_{\varphi_R}$. In both cases, $F$ and $\overline{F}$ have the same critical values. Thus (i) and (ii) follow from Propositions 5.3 and 5.4. $\square$

For compactly supported lcs Hamiltonian diffeomorphisms $\varphi_1$ and $\varphi_2$ of $S^1 \times \mathbb{R}^{2n+1}$ or $S^1 \times \mathbb{R}^{2n} \times S^1$, we pose $\varphi_1 \le \varphi_2$ if $\varphi_2 \circ \varphi_1^{-1}$ is the time-1 map of a non-negative compactly supported lcs Hamiltonian isotopy. We will prove in Section 6 that $\le$ is a partial order. One of the main ingredients will be the next proposition.

**Proposition 5.7.** Let $\varphi_1$ and $\varphi_2$ be compactly supported lcs Hamiltonian diffeomorphisms of $S^1 \times \mathbb{R}^{2n+1}$ or $S^1 \times \mathbb{R}^{2n} \times S^1$ with $\varphi_1 \le \varphi_2$. Then there are generating functions quadratic at infinity $\overline{F}_1, \overline{F}_2 : S^1 \times B \times \mathbb{R}^N \to \mathbb{R}$ for $\varphi_1$ and $\varphi_2$ respectively, where $B$ is either $S^{2n+1}$ or $S^{2n} \times S^1$, with $\overline{F}_1 \le \overline{F}_2$.

Before proving Proposition 5.7 we recall from [Cha95, Theorem 3] and [The95, Section III.2] the following construction (see also [San11a, Proposition 3.4], [Gir10, Proposition 3.4] and [FSZ23, Lemma 2.10]). The transition function of a $C^1$-small contactomorphism $\phi$ of a 1-jet bundle $J^1\mathbb{R}^m$ is the function $G : J^1\mathbb{R}^m \to \mathbb{R}$ defined by

$$ G(q, p, z) = g_{p,z}(q) - f_{p,z}(q), $$

where, for every $p$ and $z$, $f_{p,z}(q) = z + pq$ and $g_{p,z}$ is the function such that $\mathrm{im}(j^1g_{p,z}) = \phi(\mathrm{im}(j^1f_{p,z}))$. In particular, $G = 0$ if and only if $\phi$ is the identity. Moreover, we have the following facts.
* If $\phi$ is a $C^1$-small contactomorphism of $J^1\mathbb{R}^m$ with transition function $G: J^1\mathbb{R}^m \to \mathbb{R}$ and $\Lambda$ a Legendrian submanifold of $J^1\mathbb{R}^m$ with generating function $F: \mathbb{R}^m \times \mathbb{R}^N \to \mathbb{R}$ then the function

$$G \sharp F: \mathbb{R}^m \times ((\mathbb{R}^m)^* \times \mathbb{R}^m \times \mathbb{R}^N) \to \mathbb{R}, \\ (q; p, q, \zeta) \mapsto G(q, p, F(q+q, \zeta) - p(q+q)) + F(q+q, \zeta) - pq$$

is a generating function for $\phi(\Lambda)$.

* Denote the coordinates in $J^1\mathbb{R}^m$ by $(q_1, \dots, q_m, p_1, \dots, p_m, z)$. If $\phi$ is a $C^1$-small contactomorphism of $J^1\mathbb{R}^m$ that is equivariant by translation by 1 in the $q_j$-direction then the transition function $G: J^1\mathbb{R}^m \to \mathbb{R}$ of $\phi$ satisfies

$$G(q_1, \dots, q_j + 1, \dots, q_m, p_1, \dots, p_m, z - p_j) = G(q_1, \dots, q_m, p_1, \dots, p_m, z).$$

Thus, if moreover $F: \mathbb{R}^m \times \mathbb{R}^N \to \mathbb{R}$ is a function that is invariant by translation by 1 in the $q_j$-direction then $G \sharp F$ is invariant by translation by 1 in the $q_j$-direction. This implies the following.

We say that $F: \mathbb{R}^{m_1+m_2} \times \mathbb{R}^N \to \mathbb{R}$ is a generating function for a Legendrian submanifold $\Lambda$ of $J^1(\mathbb{R}^{m_1} \times (S^1)^{m_2})$ if it is a generating function for the preimage of $\Lambda$ by the covering map $J^1\mathbb{R}^{m_1+m_2} \to J^1(\mathbb{R}^{m_1} \times (S^1)^{m_2})$. Let $\Lambda$ be a Legendrian submanifold of $J^1(\mathbb{R}^{m_1} \times (S^1)^{m_2})$ with generating function $F: \mathbb{R}^{m_1+m_2} \times \mathbb{R}^N \to \mathbb{R}$, $\{\phi_t\}$ a $C^1$-small contact isotopy of $J^1(\mathbb{R}^{m_1} \times (S^1)^{m_2})$ and $G_t: J^1\mathbb{R}^{m_1+m_2} \to \mathbb{R}$ the transition functions of the lift of $\{\phi_t\}$ to $J^1\mathbb{R}^{m_1+m_2}$. Then $G_t \sharp F$ is a 1-parameter family of generating functions for $\{\phi_t(\Lambda)\}$.

* Let $\{\phi_t\}$ be a $C^1$-small contact isotopy of $J^1\mathbb{R}^m$ with Hamiltonian function $H_t$ and transition functions $G_t$. For every $p$ and $z$, denote by $\{(\phi_t)_{p,z}\}$ the isotopy of $\mathbb{R}^m$ defined by

$$(\phi_t)_{p,z}(q) = \pi \circ \phi_t \circ j^1 f_{p,z}(q),$$

where $\pi: J^1\mathbb{R}^m \to \mathbb{R}^m$ is the projection. Then

$$\left. \frac{dG_t}{dt} \right|_{t=t_0} ((\phi_t)_{p,z}(q), p, z) = H_{t_0} \circ \phi_{t_0} \circ j^1 f_{p,z}(q).$$

* Let $\Lambda$ be a Legendrian submanifold of $J^1(\mathbb{R}^{m_1} \times (S^1)^{m_2})$ that coincides with the zero section outside a compact set of $\mathbb{R}^{m_1} \times (S^1)^{m_2}$, and $F: \mathbb{R}^{m_1+m_2} \times \mathbb{R}^N \to \mathbb{R}$ a generating function for $\Lambda$. We say that $F$ is a special generating function quadratic at infinity for $\Lambda$ if there is a non-degenerate quadratic form $F_\infty$ on $\mathbb{R}^N$ such that $F(x_1, \dots, x_{m_1}, \theta_1, \dots, \theta_{m_2}, \zeta) = F_\infty(\zeta)$ for $(x_1, \dots, x_{m_1})$ outside a compact set. If $F: \mathbb{R}^{m_1+m_2} \times \mathbb{R}^N \to \mathbb{R}$ is a special generating function quadratic at infinity for $\Lambda$ and $\{\phi_t\}_{t \in [0,1]}$ a $C^1$-small compactly supported contact isotopy of $J^1(\mathbb{R}^{m_1} \times (S^1)^{m_2})$ whose lift to $J^1\mathbb{R}^{m_1+m_2}$ has transition functions $G_t: J^1\mathbb{R}^{m_1+m_2} \to \mathbb{R}$, then there is a fibre preserving diffeomorphism $\Phi$ such that $F_t := (G_t \sharp F) \circ \Phi$ is a 1-parameter family of special generating functions quadratic at infinity for $\{\phi_t(\Lambda)\}$ with $F_0$ a stabilization of $F$. If $\|\phi_t\|_{C^1} \le \epsilon$ for all $t$ then $\|G_t\|_{C^0} \le \epsilon$ for all $t$ and so, since $G_0 = 0$, $F_1$ and $F_0$ are generating functions quadratic at infinity for $\phi_1(\Lambda)$ and $\Lambda$ respectively with $\|F_1 - F_0\|_{C^0} \le \epsilon$.

In particular the above facts imply the following results.

**Lemma 5.8.** Let $\Lambda_1$ and $\Lambda_2$ be Legendrian submanifolds of $J^1(\mathbb{R}^{m_1} \times (S^1)^{m_2})$ that are the image of the zero section by a compactly supported contact isotopy and such that $\Lambda_2$ is the image of $\Lambda_1$ by the time-1 map of a compactly supported non-negative contact isotopy. Let $\bar{\Lambda}_1$ and $\bar{\Lambda}_2$ be the induced Legendrian submanifolds of $J^1(S^{m_1} \times (S^1)^{m_2})$. Then there are generating functions quadratic at infinity $\bar{F}_1, \bar{F}_2: S^{m_1} \times (S^1)^{m_2} \times \mathbb{R}^N \to \mathbb{R}$ for $\bar{\Lambda}_1$ and $\bar{\Lambda}_2$ with $\bar{F}_1 \le \bar{F}_2$.

**Lemma 5.9.** Let $\phi$ be a compactly supported contactomorphism of $J^1(\mathbb{R}^{m_1} \times (S^1)^{m_2})$ with $\|\phi\|_{C^1} \le \epsilon$ sufficiently small, and $\Lambda$ a Legendrian submanifold of $J^1(\mathbb{R}^{m_1} \times (S^1)^{m_2})$ that is the image of the zero section by a compactly supported contact isotopy. Let $\bar{\phi}$ and $\bar{\Lambda}$ be the induced contactomorphism and Legendrian submanifold of $J^1(S^{m_1} \times (S^1)^{m_2})$. Then there are generating functions quadratic at infinity $\bar{F}_0, \bar{F}_1: S^{m_1} \times (S^1)^{m_2} \times \mathbb{R}^N \to \mathbb{R}$ for $\bar{\Lambda}$ and $\bar{\phi}(\bar{\Lambda})$ with $\|\bar{F}_1 - \bar{F}_0\|_{C^0} \le \epsilon$.

Using Lemma 5.8 we can now prove Proposition 5.7.
Proof of Proposition 5.7. Suppose first that $\varphi_1$ and $\varphi_2$ are compactly supported lcs Hamiltonian diffeomorphisms of $S^1 \times \mathbb{R}^{2n+1}$. Let $\{\psi_t\}$ be a compactly supported lcs Hamiltonian isotopy with $\psi_1 = \varphi_2 \circ \varphi_1^{-1}$ and with Hamiltonian function $H_t \ge 0$, and consider the compactly supported lcs Hamiltonian isotopy $\{\Psi_{\psi_t}\}$ of $T^*_{-d\theta}(S^1 \times \mathbb{R}^{2n+1})$ that is the conjugation by $\tau$ by the compactly supported lcs Hamiltonian isotopy $\{\mathrm{id}\boxtimes \psi_t\}$ of $(S^1 \times \mathbb{R}^{2n+1})\boxtimes (S^1 \times \mathbb{R}^{2n+1})$. Then $L_{\varphi_2} = \Psi_{\psi_1}(L_{\varphi_1})$. Moreover, $\{\Psi_{\psi_t}\}$ is a non-negative lcs Hamiltonian isotopy. Indeed, by Lemma 2.43 the Hamiltonian function of $\{\mathrm{id}\boxtimes \psi_t\}$ is

$$0 \boxplus H_t : [(\theta_1, p_1), (\theta_2, p_2)] \mapsto e^{\theta_2 - \theta_1} H_t(p_2)$$

and so, by Lemma 2.21, the Hamiltonian function of $\{\Psi_{\psi_t}\}$ is $(0\boxplus H_t)\circ\tau^{-1}$, which is non-negative because so is $H_t$. By Proposition 2.36, the lift of $\{\Psi_{\psi_t}\}$ to the contactization $J^1_{-d\theta}(S^1 \times \mathbb{R}^{2n+1})$ is a non-negative compactly supported contact isotopy $\{\widetilde{\Psi}_{\psi_t}\}$. The conjugation $\{U_{-d\theta} \circ \widetilde{\Psi}_{\psi_t} \circ (U_{-d\theta})^{-1}\}$ by the untwisting map (2) is thus a non-negative compactly supported contact isotopy of $J^1(S^1 \times \mathbb{R}^{2n+1})$, which sends $U_{-d\theta}(\widetilde{L}_{\varphi_1})$ to $U_{-d\theta}(\widetilde{L}_{\varphi_2})$. By Lemma 5.8, there are generating functions quadratic at infinity $\overline{F_1}, \overline{F_2} : S^1 \times S^{2n+1} \times \mathbb{R}^N \to \mathbb{R}$ for the compactifications in $J^1(S^1 \times S^{2n+1})$ of $U_{-d\theta}(\widetilde{L}_{\varphi_1})$ and $U_{-d\theta}(\widetilde{L}_{\varphi_2})$ with $\overline{F_1} \le \overline{F_2}$. Since these compactifications are $U_{-d\theta}(\widetilde{L}_{\varphi_1})$ and $U_{-d\theta}(\widetilde{L}_{\varphi_2})$, by Proposition 4.2 we conclude that $\overline{F_1}$ and $\overline{F_2}$ are generating functions quadratic at infinity for $\overline{L}_{\varphi_1}$ and $\overline{L}_{\varphi_2}$, hence for $\varphi_1$ and $\varphi_2$, with $\overline{F_1} \le \overline{F_2}$. If $\varphi_1$ and $\varphi_2$ are compactly supported lcs Hamiltonian diffeomorphisms of $S^1 \times \mathbb{R}^{2n} \times S^1$ a similar proof applies by considering their lifts to $S^1 \times \mathbb{R}^{2n+1}$. $\square$

Using Lemma 5.9 we also obtain the following result, which is needed later.

**Lemma 5.10.** Let $\varphi$ be a compactly supported lcs Hamiltonian diffeomorphism of $S^1 \times \mathbb{R}^{2n+1}$ or $S^1 \times \mathbb{R}^{2n} \times S^1$. For every $\epsilon > 0$ small enough there is $\delta > 0$ such that if $\psi$ is a compactly supported lcs Hamiltonian diffeomorphism with $\|\varphi \circ \psi^{-1}\|_{C^1} \le \delta$ then there are generating functions quadratic at infinity $\overline{F}_{\varphi}, \overline{F}_{\psi} : S^1 \times B \times \mathbb{R}^N \to \mathbb{R}$ for $\varphi$ and $\psi$ respectively, where $B$ is either $S^{2n+1}$ or $S^{2n} \times S^1$, with $\|\overline{F}_{\varphi} - \overline{F}_{\psi}\|_{C^0} \le \epsilon$.

*Proof.* Suppose first that $\varphi$ is a compactly supported lcs Hamiltonian diffeomorphism of $S^1 \times \mathbb{R}^{2n+1}$. For every $\psi$ we have $L_{\varphi} = \Psi_{\varphi \circ \psi^{-1}}(L_{\psi})$, where $\Psi_{\varphi \circ \psi^{-1}}$ is the conjugation by $\tau$ of $\mathrm{id}\boxtimes (\varphi \circ \psi^{-1})$, and so

$$U_{-d\theta}(\widetilde{L}_{\varphi}) = U_{-d\theta} \circ \widetilde{\Psi_{\varphi \circ \psi^{-1}}} \circ (U_{-d\theta})^{-1} (U_{-d\theta}(\widetilde{L}_{\psi})),$$

where $\widetilde{L}_{\varphi}, \widetilde{L}_{\psi}$ and $\widetilde{\Psi}_{\varphi \circ \psi^{-1}}$ are the lifts of $L_{\varphi}, L_{\psi}$ and $\Psi_{\varphi \circ \psi^{-1}}$ to the contactization $J^1_{-d\theta}(S^1 \times \mathbb{R}^{2n+1})$ and $U_{-d\theta}$ is the untwisting map (2). Take $\delta > 0$ such that if $\|\phi\|_{C^1} \le \delta$ then $\|U_{-d\theta} \circ \widetilde{\Psi}_{\varphi} \circ (U_{-d\theta})^{-1}\|_{C^1} \le \epsilon$. Let $\psi$ be a compactly supported lcs Hamiltonian diffeomorphism of $S^1 \times \mathbb{R}^{2n+1}$ with $\|\varphi \circ \psi^{-1}\|_{C^1} \le \delta$, and let $\{\phi_t\}_{t \in [0,1]}$ be a compactly supported lcs Hamiltonian isotopy of $S^1 \times \mathbb{R}^{2n+1}$ with $\phi_1 = \varphi \circ \psi^{-1}$ and $\|\phi_t\|_{C^1} \le \delta$ for all $t$. Then $\|U_{-d\theta} \circ \widetilde{\Psi}_{\phi_t} \circ (U_{-d\theta})^{-1}\|_{C^1} \le \epsilon$ for all $t$. By Lemma 5.9, there are generating functions quadratic at infinity $\overline{F}_{\varphi}$ and $\overline{F}_{\psi}$ for the compactifications of $U_{-d\theta}(\widetilde{L}_{\varphi})$ and $U_{-d\theta}(\widetilde{L}_{\psi})$ with $\|\overline{F}_{\varphi} - \overline{F}_{\psi}\|_{C^0} \le \epsilon$. Since these compactifications are $U_{-d\theta}(\widetilde{L}_{\varphi})$ and $U_{-d\theta}(\widetilde{L}_{\psi})$, by Proposition 4.2 we deduce that $\overline{F}_{\varphi}$ and $\overline{F}_{\psi}$ are generating functions quadratic at infinity for $\overline{L}_{\varphi}$ and $\overline{L}_{\psi}$, hence of $\varphi$ and $\psi$. If $\varphi$ is a compactly supported lcs Hamiltonian diffeomorphism of $S^1 \times \mathbb{R}^{2n} \times S^1$ the proof is similar. $\square$

We end this section by discussing the relation with generating functions for contactomorphisms. We start with the following lemma.

**Lemma 5.11.** Let $\phi$ be a contactomorphism of $\mathbb{R}^{2n+1}$, and $\tilde{\phi}$ its lift to $S^1 \times \mathbb{R}^{2n+1}$. Consider the Legendrian submanifold $\Lambda_{\phi}$ of $J^1\mathbb{R}^{2n+1}$, and the exact Lagrangian submanifold $L_{\tilde{\phi}}$ of $T^*_{-d\theta}(S^1 \times \mathbb{R}^{2n+1})$. Let $\tilde{L}_{\tilde{\phi}}$ be the lift of $L_{\tilde{\phi}}$ to the contactization $J^1_{-d\theta}(S^1 \times \mathbb{R}^{2n+1})$. Then

$$U_{-d\theta}(\widetilde{L}_{\tilde{\phi}}) = 0_{S^1} \times \Lambda_{\phi},$$

where $U_{-d\theta}$ is the untwisting map (2) from $J^1_{-d\theta}(S^1 \times \mathbb{R}^{2n+1})$ to $J^1(S^1 \times \mathbb{R}^{2n+1})$ and where we identify $J^1(S^1 \times \mathbb{R}^{2n+1})$ with $T^*S^1 \times J^1\mathbb{R}^{2n+1}$.
Proof. Identify $T^*_{-d\theta}(S^1 \times \mathbb{R}^{2n+1})$ with $S^1 \times J^1\mathbb{R}^{2n+1}$ by the strict lcs diffeomorphism of Lemma 2.13. By Proposition 2.44 we then have $L_{\tilde{\phi}} = S^1 \times \Lambda_{\phi}$. We thus conclude using Lemma 2.38. $\square$

Using Lemma 5.11 we now obtain the following relation between generating functions.

**Proposition 5.12.** Let $\phi$ be a compactly supported contactomorphism of $\mathbb{R}^{2n+1}$ or $\mathbb{R}^{2n} \times S^1$ contact isotopic to the identity, and $\tilde{\phi}$ its lift to $S^1 \times \mathbb{R}^{2n+1}$ or $S^1 \times \mathbb{R}^{2n} \times S^1$ respectively. Let $\bar{F} : B \times \mathbb{R}^N \to \mathbb{R}$, where $B$ is either $S^{2n+1}$ or $S^{2n} \times S^1$, be a generating function for $\phi$. Then the function $\bar{F} \circ \text{pr}_2 : S^1 \times (B \times \mathbb{R}^N) \to \mathbb{R}$, where $\text{pr}_2$ is the projection to the second factor, is a generating function for $\tilde{\phi}$.

Proof. Suppose first that $\phi$ is a contactomorphism of $\mathbb{R}^{2n+1}$. By definition, $\bar{F} : S^{2n+1} \times \mathbb{R}^N \to \mathbb{R}$ is a generating function for the Legendrian submanifold $\bar{\Lambda}_{\phi}$ of $J^1S^{2n+1}$. This implies that $\bar{F} \circ \text{pr}_2$ is a generating function for the Legendrian submanifold $0_{S^1} \times \bar{\Lambda}_{\phi}$ of $J^1(S^1 \times S^{2n+1}) \equiv T^*S^1 \times J^1S^{2n+1}$. Lemma 5.11 implies that $0_{S^1} \times \bar{\Lambda}_{\phi} = U_{-d\theta}(\bar{L}_{\tilde{\phi}})$. By Proposition 4.2 we thus conclude that $\bar{F} \circ \text{pr}_2$ is a generating function for $\bar{L}_{\tilde{\phi}}$, hence for $\tilde{\phi}$. If $\phi$ is a contactomorphism of $\mathbb{R}^{2n} \times S^1$ the proof is similar, by applying Lemma 5.11 to its lift to $\mathbb{R}^{2n+1}$. $\square$

## 6. SPECTRAL SELECTORS FOR COMPACTLY SUPPORTED LCS HAMILTONIAN DIFFEOMORPHISMS OF $S^1 \times \mathbb{R}^{2n+1}$ AND $S^1 \times \mathbb{R}^{2n} \times S^1$

In this section we define spectral selectors for compactly supported lcs Hamiltonian diffeomorphisms of $S^1 \times \mathbb{R}^{2n+1}$ and $S^1 \times \mathbb{R}^{2n} \times S^1$. Our construction is analogous to those in [Vit92, Bhu01, San10, San11a] of spectral selectors for compactly supported Hamiltonian diffeomorphisms of $\mathbb{R}^{2n}$ and for compactly supported contactomorphisms of $\mathbb{R}^{2n+1}$ and $\mathbb{R}^{2n} \times S^1$ contact isotopic to the identity.

We first recall from [Vit92] the definition of spectral selectors for functions quadratic at infinity. Let $B$ be a compact manifold and $F : E = B \times \mathbb{R}^N \to \mathbb{R}$ a function quadratic at infinity. Let $E^- \to B$ be the subbundle of $E \to B$ on which the quadratic form $F_\infty$ is negative definite, and consider the Thom isomorphism

$$T : H^*(B) \to H^{*+\text{ind}(F_\infty)}(D(E^-), S(E^-)),$$

where we denote by $D(E^-)$ and $S(E^-)$ the disc and sphere bundles of the vector bundle $E^- \to B$. We consider the sublevel sets

$$E^a = \{ e \in E \mid F(e) \le a \},$$

and we denote $E^{-\infty} = E^a$ for $a$ smaller than all the critical values of $F$. We then consider the inclusion

$$i_a : (E^a, E^{-\infty}) \hookrightarrow (E, E^{-\infty}),$$

and the induced homomorphism in cohomology

$$i_a^* : H^*(E, E^{-\infty}) \to H^*(E^a, E^{-\infty}).$$

Finally we consider the composition

$$i_a^* \circ \text{ex} \circ T : H^*(B) \to H^{*+\text{ind}(F_\infty)}(E^a, E^{-\infty}),$$

where

$$\text{ex} : H^*(D(E^-), S(E^-)) \to H^*(E, E^{-\infty})$$

is the excision isomorphism. For $a$ smaller than all the critical values of $F$ the homomorphism $i_a^* \circ \text{ex} \circ T$ vanishes, while for $a$ bigger than all the critical values of $F$ it is an isomorphism. For any non-zero class $u$ in $H^*(B)$ we define

$$c(u, F) = \inf \{ a \in \mathbb{R} \mid i_a^* \circ \text{ex} \circ T(u) \neq 0 \}.$$

The properties in the following proposition are proved in [Vit92]. In (iv) and (v), $[B]$ and $1$ denote respectively the fundamental and unit classes in $H^*(B)$. For (vii), recall that two functions $F_1$ and $F_2$ quadratic at infinity are said to be equivalent if there are non-degenerate quadratic forms $Q_1$ and $Q_2$ and a fibre preserving diffeomorphism $\Phi$ such that $F_1 \oplus Q_1 = (F_2 \oplus Q_2) \circ \Phi$.

**Proposition 6.1.** The spectral selectors $c(u, F)$ satisfy the following properties:
(i) Spectrality: $c(u, F)$ is a critical value of $F$.

(ii) Continuity: if $|F_1 - F_2|_{C^0} \le \epsilon$ then

$$|c(u, F_1) - c(u, F_2)| \le \epsilon.$$

(iii) Monotonicity: if $F_1, F_2: B \times \mathbb{R}^N \to \mathbb{R}$ satisfy $F_1 \le F_2$ then $c(u, F_1) \le c(u, F_2)$.

(iv) Poincaré duality: $c([B], -F) = -c(1, F)$.

(v) Non-degeneracy: if $F: B \times \mathbb{R}^N \to \mathbb{R}$ is a generating function quadratic at infinity for a Lagrangian submanifold $L$ of $T^*B$ and if $c([B], F) = c(1, F)$ then $L$ is the zero section.

(vi) Triangle inequality: let $F_1: B \times \mathbb{R}^{N_1} \to \mathbb{R}$ and $F_2: B \times \mathbb{R}^{N_2} \to \mathbb{R}$ be functions quadratic at infinity, and consider the function quadratic at infinity $F_1 + F_2: B \times \mathbb{R}^{N_1} \times \mathbb{R}^{N_2} \to \mathbb{R}$ defined by

$$(F_1 + F_2)(q, \zeta_1, \zeta_2) = F_1(q, \zeta_1) + F_2(q, \zeta_2).$$

Then

$$c(u \cup v, F_1 + F_2) \ge c(u, F_1) + c(v, F_2).$$

(vii) Equivalence: if $F_1$ and $F_2$ are equivalent then $c(u, F_1) = c(u, F_2)$ for any $u$.

Let now $\varphi$ be a compactly supported lcs Hamiltonian diffeomorphism of $S^1 \times \mathbb{R}^{2n+1}$ or $S^1 \times \mathbb{R}^{2n} \times S^1$, and $\bar{F}: S^1 \times B \times \mathbb{R}^N \to \mathbb{R}$, where $B$ is either $S^{2n+1}$ or $S^{2n} \times S^1$, a generating function quadratic at infinity for $\varphi$. We define

$$c_+(\varphi) = c([S^1 \times B], \bar{F}) \quad \text{and} \quad c_-(\varphi) = c(1, \bar{F}),$$

where $[S^1 \times B]$ and $1$ are the fundamental and unit classes in $H^*(S^1 \times B)$. By Proposition 6.1 (vii) and the uniqueness statement in Proposition 5.5, the quantities $c_+(\varphi)$ and $c_-(\varphi)$ are well-defined, in the sense that they do not depend on the choice of the generating function quadratic at infinity $\bar{F}$ for $\varphi$. Moreover, by Proposition 6.1 (i) and Proposition 5.6 (i) they belong to the action spectrum of $\varphi$. In particular, $c_\pm(\mathrm{id}) = 0$.

Before discussing the other properties of $c_+$ and $c_-$ we prove the following lemma, which will be the key ingredient to obtain the invariance by conjugation property.

**Lemma 6.2.**

(i) Let $\varphi$ and $\psi$ be compactly supported lcs Hamiltonian diffeomorphisms of $S^1 \times \mathbb{R}^{2n+1}$. Suppose that $p$ is an essential translated point of $\varphi$ of time-shift equal to zero. Then $\psi(p)$ is an essential translated points of $\psi \circ \varphi \circ \psi^{-1}$ of time-shift equal to zero. Moreover, $\psi(p)$ is a non-degenerate translated point of $\psi \circ \varphi \circ \psi^{-1}$ if and only if $p$ is a non-degenerate translated point of $\varphi$.

(ii) Let $\varphi$ and $\psi$ be compactly supported lcs Hamiltonian diffeomorphisms of $S^1 \times \mathbb{R}^{2n} \times S^1$. Suppose that $p$ is an essential translated point of $\varphi$ of time-shift equal to an integer $k$. Then $\psi(p)$ is an essential translated points of $\psi \circ \varphi \circ \psi^{-1}$ of time-shift equal to $k$. Moreover, $\psi(p)$ is a non-degenerate translated point of $\psi \circ \varphi \circ \psi^{-1}$ if and only if $p$ is a non-degenerate translated point of $\varphi$.

*Proof.* We prove (ii), the proof of (i) being similar. Since the time-shift of $p$ is an integer, we have $\varphi(p) = p$. Let $g$ and $f$ be the conformal factors of $\varphi$ and $\psi$ respectively. Then the conformal factor of $\psi \circ \varphi \circ \psi^{-1}$ is $(f \circ \varphi + g - f) \circ \psi^{-1}$. In particular, since $g(p) = 0$ and $\varphi(p) = p$ we have

$$(f \circ \varphi + g - f) \circ \psi^{-1}(\psi(p)) = 0.$$

Let $\varphi_R$ and $\psi_R$ be the lifts of $\varphi$ and $\psi$ to $S^1 \times \mathbb{R}^{2n+1}$, and $\bar{p}$ a point of $S^1 \times \mathbb{R}^{2n+1}$ projecting to $p$. Then $\bar{p}$ is a translated point of $\varphi_R$ of time-shift $k$, in particular there is a Lee chord of time-shift $k$ from $\varphi_R(\bar{p})$ to $\bar{p}$. Since $k$ is an integer and $\psi_R$ is equivariant by translation by 1 in the Lee direction, this implies that there is a Lee chord of time-shift $k$ between $\psi_R(\varphi_R(\bar{p})) = (\psi_R \circ \varphi_R \circ \psi_R^{-1})(\psi_R(\bar{p}))$ and $\psi_R(\bar{p})$. Since $\psi_R(\bar{p})$ projects to $\psi(p)$ and $\psi_R \circ \varphi_R \circ \psi_R^{-1} = (\psi \circ \varphi \circ \psi^{-1})_R$, we conclude that $\psi(p)$ is a translated point of $\psi \circ \varphi \circ \psi^{-1}$ of time-shift $k$. Let now $S_\varphi$ be the action of $\varphi$ and $S_\psi$ the action of $\psi$. Then the action of $\psi \circ \varphi \circ \psi^{-1}$ is

$$S_{\psi \circ \varphi \circ \psi^{-1}} = e^{f \circ \psi^{-1}} (S_{\psi} - S_{\varphi} - e^{-g} S_{\psi} \circ \varphi) \circ \psi^{-1}.$$

Since $p$ is an essential translated point of $\varphi$, by Example 3.11 we have $S_\varphi(p) = 0$. Since moreover $g(p) = 0$ and $\varphi(p) = p$, we conclude that $S_{\psi \circ \varphi \circ \psi^{-1}}(\psi(p)) = 0$ and so, by Example 3.11, $\psi(p)$ is an essential translated point of $\psi \circ \varphi \circ \psi^{-1}$. We now show that $\psi(p)$ is non-degenerate if and only if so is $p$. Suppose that $p$ is a
degenerate translated point of $\varphi$, thus there is a non-zero tangent vector $Y$ at $p$ such that $\varphi_*(Y) = Y$ and $dg(Y) = 0$. Then

$$ (\psi \circ \varphi \circ \psi^{-1})_*(\psi_*(Y)) = \psi_*(Y) $$

and

$$ d((f \circ \varphi + g - f) \circ \psi^{-1})(\psi_*(Y)) = df(\varphi_*(Y)) + dg(Y) - df(Y) = 0, $$

thus $\psi(p)$ is a degenerate translated point of $\psi \circ \varphi \circ \psi^{-1}$. The other implication is similar. $\square$

We can now prove the following properties of the spectral selectors $c_+$ and $c_-$.

**Proposition 6.3.** The spectral selectors $c_+$ and $c_-$ on Ham$^c(S^1 \times \mathbb{R}^{2n+1})$ and Ham$^c(S^1 \times \mathbb{R}^{2n} \times S^1)$ satisfy the following properties:

(i) Continuity: $c_+$ and $c_-$ are continuous with respect to the $C^1$-topology.

(ii) Monotonicity: if $\varphi_1 \le \varphi_2$ then $c_+(\varphi_1) \le c_+(\varphi_2)$ and $c_-(\varphi_1) \le c_-(\varphi_2)$.

(iii) Poincaré duality: $c_-(\varphi) = -c_+(\varphi^{-1})$.

(iv) Positivity: $c_+(\varphi) \ge 0$ and $c_-(\varphi) \le 0$.

(v) Non-degeneracy: $c_+(\varphi) = c_-(\varphi) = 0$ if and only if $\varphi$ is the identity.

(vi) Triangle inequality: $c_+(\varphi_1 \circ \varphi_2) \le c_+(\varphi_1) + c_+(\varphi_2)$ and $c_-(\varphi_1 \circ \varphi_2) \ge c_-(\varphi_1) + c_-(\varphi_2)$.

(vii) Invariance by conjugation: if $\varphi$ and $\psi$ are compactly supported lcs Hamiltonian diffeomorphisms of $S^1 \times \mathbb{R}^{2n+1}$ then $c_+(\varphi) = 0$ if and only if $c_+(\psi \circ \varphi \circ \psi^{-1}) = 0$, and similarly for $c_-$. If $\varphi$ and $\psi$ are compactly supported lcs Hamiltonian diffeomorphisms of $S^1 \times \mathbb{R}^{2n} \times S^1$ then $[c_+(\psi \circ \varphi \circ \psi^{-1})] = [c_+(\varphi)]$ and $[c_+(\psi \circ \varphi \circ \psi^{-1})] = [c_+(\varphi)]$, and similarly for $c_-$, where $[\cdot]$ and $[\cdot]$ denote the ceiling and floor functions.

*Proof.* Continuity (i) follows from Proposition 6.1 (ii) and Lemma 5.10, and monotonicity (ii) from Proposition 6.1 (iii) and Proposition 5.7.

Poincaré duality (iii) can be seen as follows. Let $\overline{F}_\varphi$ and $\overline{F}_{\varphi^{-1}}$ be generating functions quadratic at infinity for $\varphi$ and $\varphi^{-1}$ respectively. We first show that, for any non-zero class $u$,

$$ (14) \qquad c(u, \overline{F}_{\varphi^{-1}}) = c(u, -\overline{F}_\varphi). $$

To see this, let $\{\varphi_t\}$ be a compactly supported lcs Hamiltonian isotopy with $\varphi_1 = \varphi$, let $\overline{F}_t$ and $\overline{G}_t$ be 1-parameter families of generating functions quadratic at infinity for $\{\varphi_t\}$ and $\{\varphi_t \circ \varphi^{-1}\}$ respectively, and set

$$ c_t = c(u, \overline{G}_t - \overline{F}_t). $$

By Proposition 5.6 (ii), for every $t$ the set of critical values of $\overline{G}_t - \overline{F}_t$ is equal to the action spectrum of $\varphi^{-1}$. The functions $\overline{G}_t - \overline{F}_t$ have thus all the same set $S$ of critical values. By Proposition 6.1 (i), $c_t$ is in $S$ for all $t$. By Proposition 6.1 (ii), the map $t \mapsto c_t$ is continuous. Since $S$ is nowhere dense we conclude that $t \mapsto c_t$ is constant, and so $c_0 = c(u, \overline{G}_0 - \overline{F}_0)$ is equal to $c_1 = c(u, \overline{G}_1 - \overline{F}_1)$. But $\overline{G}_0$ is a generating function quadratic at infinity for $\varphi^{-1}$, hence equivalent to $\overline{F}_{\varphi^{-1}}$, and $\overline{F}_0$ is a generating function quadratic at infinity for the identity, hence equivalent to the zero function. Thus $\overline{G}_0 - \overline{F}_0$ is equivalent to $\overline{F}_{\varphi^{-1}}$. Similarly, $\overline{G}_1 - \overline{F}_1$ is equivalent to $-\overline{F}_\varphi$. By Proposition 6.1 (vii), we thus obtain (14). Using (14) and Proposition 6.1 (iv) we then conclude that

$$ c_-(\varphi) = c(1, \overline{F}_\varphi) = -c([S^1 \times B], -\overline{F}_\varphi) = -c([S^1 \times B], \overline{F}_{\varphi^{-1}}) = -c_+(\varphi^{-1}). $$

By the Poincaré duality property (iii), positivity (iv) follows if we prove that $c_-(\varphi) \le 0$. This can be seen as follows. Let $\overline{F}: E = S^1 \times B \times \mathbb{R}^N \to \mathbb{R}$, where $B$ is either $S^{2n+1}$ or $S^{2n} \times S^1$, be a generating function quadratic at infinity for $\varphi$. Since

$$ c_-(\varphi) = c(1, \overline{F}) = \inf\{a \in \mathbb{R} \mid i_a^* \circ \mathrm{ex} \circ T(1) \neq 0\}, $$

we need to show that $i_0^* \circ \mathrm{ex} \circ T(1) \neq 0$. Suppose first that $B$ is $S^{2n+1}$, and denote by $p_\infty \in S^{2n+1}$ the point at infinity when seeing $S^{2n+1}$ as the 1-point compactification of $\mathbb{R}^{2n+1}$. Denote by $E_{S^1 \times \{p_\infty\}}$, $E_{S^1 \times \{p_\infty\}}^0$ and
$E_{S^1 \times \{p_\infty\}}^{-\infty}$ the preimages of $S^1 \times \{p_\infty\}$ by the projections of $E$, $E^0$ and $E^{-\infty}$ to $S^1 \times S^{2n+1}$. Consider the commutative diagram

$$
\begin{tikzcd}
& & H^*(S^1 \times S^{2n+1}) \arrow[r, "\text{ex} \circ T"'] & H^{*+\text{ind}(\bar{F}_\infty)}(E, E^{-\infty}) \arrow[r, "i_0"'] & H^{*+\text{ind}(\bar{F}_\infty)}(E^0, E^{-\infty}) \\
& & \arrow[uur] \arrow[dl] & \arrow[uur] \arrow[dl] & \arrow[dl] \\
& & H^*(S^1 \times \{p_\infty\}) \arrow[r, "\text{ex} \circ T"'] & H^{*+\text{ind}(\bar{F}_\infty)}(E_{S^1 \times \{p_\infty\}}, E_{S^1 \times \{p_\infty\}}^{-\infty}) \arrow[r, "i_0"'] & H^{*+\text{ind}(\bar{F}_\infty)}(E_{S^1 \times \{p_\infty\}}^0, E_{S^1 \times \{p_\infty\}}^{-\infty})
\end{tikzcd}
$$

where the vertical arrows are the homomorphisms induced by the inclusions. Since the image of

$$
i_{\bar{F}, -d\theta} : \Sigma_{\bar{F}} \to T^*_{-d\theta}(S^1 \times S^{2n+1}), \quad (\theta, p, \zeta) \mapsto \left( \theta, p, \frac{\partial \bar{F}}{\partial \theta}(\theta, p, \zeta) + \bar{F}(\theta, p, \zeta), \frac{\partial \bar{F}}{\partial p}(\theta, p, \zeta) \right)
$$

is $\overline{L}_{\varphi}$, which by construction is equal to the zero section on a neighborhood of $S^1 \times \{p_\infty\}$, for every $\theta \in S^1$
and $\zeta \in \mathbb{R}^N$ the point $i_{\overline{F}, -d\theta}(\theta, p_\infty, \zeta)$ belongs to the zero section. If $(\theta, p_\infty, \zeta)$ is a critical point of $\overline{F}$ it thus
has critical value zero. We conclude that zero is the only critical value of $\overline{F}|_{E_{S^1 \times \{p_\infty\}}}$, and so that the map

$$
i_0 : H^{*+\mathrm{ind}(\bar{F}_\infty)}(E_{S^1 \times \{p_\infty\}}, E_{S^1 \times \{p_\infty\}}^{-\infty}) \to H^{*+\mathrm{ind}(\bar{F}_\infty)}(E_{S^1 \times \{p_\infty\}}^0, E_{S^1 \times \{p_\infty\}}^{-\infty})
$$

is an isomorphism. In particular, the image of the unit class of $H^*(S^1 \times \{p_\infty\})$ by

$$
i_0^* \circ \text{ex} \circ T : H^*(S^1 \times \{p_\infty\}) \to H^{*+\text{ind}(\bar{F}_\infty)}(E_{S^1 \times \{p_\infty\}}^0, E_{S^1 \times \{p_\infty\}}^{-\infty})
$$

is different than zero. Since the first vertical arrow sends the unit class of $H^*(S^1 \times S^{2n+1})$ to the unit class of $H^*(S^1 \times \{p_\infty\})$, we conclude that the image of the unit class of $H^*(S^1 \times S^{2n+1})$ by

$$
i_0^* \circ \text{ex} \circ T : H^*(S^1 \times S^{2n+1}) \to H^{*+\text{ind}(\bar{F}_\infty)}(E^0, E^{-\infty})
$$

is different than zero, as we wanted. If $B$ is $S^{2n} \times S^1$ the proof is similar, considering the restriction of $E$ to $S^1 \times \{p_\infty\} \times S^1$, where $p_\infty$ is the point at infinity of $S^{2n}$.

We now prove non-degeneracy (v). Suppose that $c_+(\varphi) = c_-(\varphi) = 0$, thus

$$
c([S^1 \times B], \bar{F}) = c(1, \bar{F}) = 0
$$

for any generating function quadratic at infinity $\bar{F}: S^1 \times B \times \mathbb{R}^N \to \mathbb{R}$ for $\varphi$. By Proposition 6.1 (v), $\bar{F}$ generates the zero section of $T^*(S^1 \times B)$, hence the zero section of $J^1(S^1 \times B)$. But, by definition, $\bar{F}$ generates the Lagrangian submanifold $\bar{L}_\varphi$ of $T^*_{-d\theta}(S^1 \times B)$ and so, by Proposition 4.2, the Legendrian submanifold $U_{-d\theta}(\widetilde{\bar{L}_\varphi})$ of $J^1(S^1 \times B)$. We thus conclude that $U_{-d\theta}(\widetilde{\bar{L}_\varphi})$ is the zero section. But this implies that $\widetilde{\bar{L}_\varphi}$ is the zero section, thus $\bar{L}_\varphi$ is the zero section and so $\varphi$ is the identity.

By the Poincaré duality property (iii), it is enough to prove the triangle inequality (vi) for $c_+$. This can be
done as follows. Let $\overline{F}_{\varphi_1}$, $\overline{F}_{\varphi_2}$ and $\overline{F}_{\varphi_1 \circ \varphi_2}$ be generating functions quadratic at infinity for $\varphi_1$, $\varphi_2$ and $\varphi_1 \circ \varphi_2$
respectively. We first show that, for any non-zero class $u$,

$$
(15) \qquad c(u, \overline{F}_{\varphi_2}) = c(u, \overline{F_{\varphi_1 \circ \varphi_2}} - \overline{F_{\varphi_1}}).
$$

To see this, let {$\varphi_t$} be a lcs Hamiltonian isotopy with time-1 map $\varphi_1$, let $\overline{F}_t$ and $\overline{G}_t$ be 1-parameter families of generating functions quadratic at infinity for {$\varphi_t$} and {$\varphi_t \circ \varphi_2$} respectively, and set

$$
c_t = c(u, \overline{G}_t - \overline{F}_t).
$$

By Proposition 5.6 (ii), for every $t$ the set of critical values of $\overline{G}_t - \overline{F}_t$ is equal to the action spectrum of $\varphi_2$.
The functions $\overline{G}_t - \overline{F}_t$ have thus all the same set $S$ of critical values. By Proposition 6.1 (i), $c_t$ is in $S$ for all
$t$. By Proposition 6.1 (ii) the map $t \mapsto c_t$ is continuous. Since $S$ is nowhere dense we conclude that $t \mapsto c_t$
is constant, and so $c_0 = c(u, \overline{G}_0 - \overline{F}_0)$ is equal to $c_1 = c(u, \overline{G}_1 - \overline{F}_1)$. Since $\overline{G}_0 - \overline{F}_0$ is equivalent to $\overline{F}_{\varphi_2}$ and
$\overline{G}_1 - \overline{F}_1$ is equivalent to $\overline{F}_{\varphi_1 \circ \varphi_2} - \overline{F}_{\varphi_1}$, by Proposition 6.1 (vii) we obtain (15). Using (15) and Proposition
6.1 (iv) and (vi) we thus conclude that

$$
\begin{align*}
c_+(\varphi_2) &= c([S^1 \times B], \overline{F}_{\varphi_2}) = c([S^1 \times B], \overline{F_{\varphi_1 \circ \varphi_2}} - \overline{F_{\varphi_1}}) \ge c([S^1 \times B], \overline{F_{\varphi_1 \circ \varphi_2}}) + c(1, -\overline{F_{\varphi_1}}) \\
&= c([S^1 \times B], \overline{F_{\varphi_1 \circ \varphi_2}}) - c([S^1 \times B], \overline{F_{\varphi_1}}) = c_+(\varphi_1 \circ \varphi_2) - c_+(\varphi_1).
\end{align*}
$$
The proof of the invariance by conjugation property (vii) is analogous to the proof of [Bhu01, Theorem 1.1 (i)] and [San11a, Lemma 3.16]. Suppose first that $\varphi$ and $\psi$ are compactly supported lcs Hamiltonian diffeomorphisms of $S^1 \times \mathbb{R}^{2n} \times S^1$. Let $\{\psi_t\}$ be a compactly supported lcs Hamiltonian isotopy with $\psi_1 = \psi$, let $\overline{F}_t : E \to \mathbb{R}$ a 1-parameter family of generating functions quadratic at infinity for $\{\psi_t \circ \varphi \circ \psi_t^{-1}\}$, and set $c_t = c(u, \overline{F}_t)$, where $u$ is either the fundamental or the unit class of $S^1 \times S^{2n} \times S^1$. The result follows if we prove that if $c_{t_0} = k \in \mathbb{Z}$ for some $t_0$ then $c_t = k$ for all $t$. Let $x_t$ be a path in $E$, for $t$ in some subinterval of $[0, 1]$ containing $t_0$, such that each $x_t$ is a critical point of $\overline{F}_t$ with critical value $c_t$. Let $q_t$ be the corresponding path of essential translated points of $\psi_t \circ \varphi \circ \psi_t^{-1}$. Assume first that $q_{t_0}$ is a non-degenerate translated point of $\psi_{t_0} \circ \varphi \circ \psi_{t_0}^{-1}$. Since $q_0$ has time-shift $c_{t_0} = k \in \mathbb{Z}$, by Lemma 6.2 for every $t$ the point $\psi_t \circ \psi_{t_0}^{-1}(q_{t_0})$ is an essential non-degenerate translated point of $\psi_t \circ \varphi \circ \psi_t^{-1}$ of time-shift $k$. Let $y_t$ be the corresponding path of non-degenerate critical points of $\overline{F}_t$ of critical value $k$. Since $y_{t_0} = x_{t_0}$ we deduce that $x_t = y_t$ for all $t$, and so $c_t = k$ for all $t$. This ends the proof for $S^1 \times \mathbb{R}^{2n} \times S^1$ under the assumption that $q_{t_0}$ is a non-degenerate translated point of $\psi_{t_0} \circ \varphi \circ \psi_{t_0}^{-1}$. If $q_{t_0}$ is degenerate, we obtain the result by an approximation argument as in [Bhu01]. For every $\epsilon > 0$ there is a generating function quadratic at infinity $G : E \to \mathbb{R}$ such that $\|G - \overline{F}_{t_0}\|_{C^\infty} < \epsilon$, $G \equiv \overline{F}_{t_0}$ on an neighborhood of $S^1 \times \{p_\infty\} \times S^1 \times \mathbb{R}^N$, all the critical points of $G$ outside this neighborhood are non-degenerate and $c(u, G) = c(u, \overline{F}_{t_0})$. Since $\|G - \overline{F}_{t_0}\|_{C^\infty} < \epsilon$, $G$ is a generating function quadratic at infinity for a compactly supported lcs Hamiltonian diffeomorphism $\phi_{t_0}$ of $S^1 \times \mathbb{R}^{2n} \times S^1$ with $\|\phi_{t_0} \circ (\psi_{t_0} \circ \varphi \circ \psi_{t_0})^{-1}\|_{C^\infty} < \epsilon$. Let $\phi_t = (\psi_t \circ \psi_{t_0}) \circ \phi_{t_0} \circ (\psi_t \circ \psi_{t_0})^{-1}$. Then $\|\phi_t \circ (\psi_t \circ \varphi \circ \psi_t)^{-1}\|_{C^\infty} < \epsilon$. Let $G_t$ be a 1-parameter family of generating functions quadratic at infinity for $\{\phi_t\}$. By the first part of the proof, $c(u, G_t) = k$ for all $t$. But $c(u, G_t)$ is equal to $c_+(\phi_t)$ or $c_-(\phi_t)$. Since $\epsilon$ is arbitrary, by continuity (i) we thus obtain that $c_t = k$ for all $t$. For compactly supported lcs Hamiltonian diffeomorphisms of $S^1 \times \mathbb{R}^{2n+1}$ a similar argument works for $k = 0$, showing that $c_+(\varphi) = 0$ if and only if $c_+(\psi \circ \varphi \circ \psi^{-1}) = 0$ and similarly for $c_-$. $\square$

Let $\phi$ be a compactly supported contactomorphism of $\mathbb{R}^{2n+1}$ or $\mathbb{R}^{2n} \times S^1$ contact isotopic to the identity, and $\overline{F} : B \times \mathbb{R}^N \to \mathbb{R}$, where $B$ is either $S^{2n+1}$ or $S^{2n} \times S^1$, a generating function quadratic at infinity for $\phi$. As in [Bhu01, San10, San11a], we define $c_+(\phi) = c([B], \overline{F})$ and $c_-(\phi) = c(1, \overline{F})$. The spectral selectors $c_+$ and $c_-$ on $\text{Cont}_0^c(\mathbb{R}^{2n+1})$ and $\text{Cont}_0^c(\mathbb{R}^{2n} \times S^1)$ enjoy properties similar to those discussed above for the spectral selectors $c_+$ and $c_-$ on $\text{Ham}^c(S^1 \times \mathbb{R}^{2n+1})$ and $\text{Ham}^c(S^1 \times \mathbb{R}^{2n} \times S^1)$. Except for the analogues of (iii) and (vi), these properties are proved in [Bhu01, San10, San11a]. As discussed in the next remark, the contact analogues of (iii) and (vi) are in fact stronger than the corresponding properties proved in [Bhu01, San10, San11a].

**Remark 6.4.** Arguments as in the proof of Proposition 6.3 (iii) and (vi) also apply to the spectral selectors $c_\pm$ of compactly supported contactomorphisms of $\mathbb{R}^{2n+1}$ and $\mathbb{R}^{2n} \times S^1$, proving in particular that $c_-(\phi) = -c_+(\phi^{-1})$, $c_+(\phi_1 \circ \phi_2) \le c_+(\phi_1) + c_+(\phi_2)$ and $c_-(\phi_1 \circ \phi_2) \ge c_-(\phi_1) + c_-(\phi_2)$. In [San11a] and [San10] only weaker versions of these properties are proved: $\lfloor c_-(\phi) \rfloor = -\lfloor c_+(\phi^{-1}) \rfloor$, $\lfloor c_+(\phi_1 \circ \phi_2) \rfloor \le \lfloor c_+(\phi_1) \rfloor + \lfloor c_+(\phi_2) \rfloor$ and $\lfloor c_-(\phi_1 \circ \phi_2) \rfloor \ge \lfloor c_-(\phi_1) \rfloor + \lfloor c_-(\phi_2) \rfloor$. It turns out that this difference is due to the fact that in the present article we use a different formula for the identification $\tau_{\text{cont}}$ of the contact product of $\mathbb{R}^{2n+1}$ with $J^1\mathbb{R}^{2n+1}$. If we would use the identification of [San11a, San10], in the contact analogue of Proposition 5.6 (ii) we would obtain that for any two compactly supported contactomorphisms $\phi_1$ and $\phi_2$ of $\mathbb{R}^{2n+1}$ or $\mathbb{R}^{2n} \times S^1$ contact isotopic to the identity and generating functions quadratic at infinity $\overline{F}_1$ and $\overline{F}_2$ for $\phi_1$ and $\phi_2$, the set of critical values of $\overline{F}_1 - \overline{F}_2$ is equal to the action spectrum of $\phi_1 \circ \phi_2^{-1}$, not of $\phi_2^{-1} \circ \phi_1$. Thus in the contact analogue of the proof of Proposition 6.3 (iii) the set of critical values of $\overline{G}_t - \overline{F}_t$ would not be equal to the action spectrum of $\varphi^{-1}$ but to the action spectrum of $\varphi_t \circ \varphi^{-1} \circ \varphi_t^{-1}$ and so we would need to integrate in the proof also arguments as in the proof of Proposition 6.3 (vii), where the floor and ceiling functions are necessary. Something similar also happens for the triangle inequality.

We now discuss the relation between the spectral selectors of lcs Hamiltonian diffeomorphisms and those of contactomorphisms.

**Proposition 6.5.** Let $\phi$ be a compactly supported contactomorphism of $\mathbb{R}^{2n+1}$ or $\mathbb{R}^{2n} \times S^1$ contact isotopic to the identity, and $\tilde{\phi}$ its lift to $S^1 \times \mathbb{R}^{2n+1}$ or $S^1 \times \mathbb{R}^{2n} \times S^1$. Then $c_+(\tilde{\phi}) = c_+(\phi)$ and $c_-(\tilde{\phi}) = c_-(\phi)$.
*Proof.* By the Poincaré duality property, Proposition 6.3 (iii), and the analogous property for the spectral selectors of contactomorphisms (cf. Remark 6.4), it is enough to prove that $c_-(\tilde{\phi}) = c_-(\phi)$ for any compactly supported contactomorphism $\phi$ of $\mathbb{R}^{2n+1}$ or $\mathbb{R}^{2n} \times S^1$ contact isotopic to the identity. Let $\bar{F} : B \times \mathbb{R}^N \to \mathbb{R}$ be a generating function quadratic at infinity for $\phi$, where $B$ is either $S^{2n+1}$ or $S^{2n} \times S^1$. By Proposition 5.12, the function $\bar{F} \circ \text{pr}_2 : S^1 \times (B \times \mathbb{R}^N) \to \mathbb{R}$, where $\text{pr}_2$ is the projection to the second factor, is a generating function quadratic at infinity for $\tilde{\phi}$. Denote $E_B = B \times \mathbb{R}^N$ and $E = S^1 \times B \times \mathbb{R}^N$, and similarly $E_B^a = \{\bar{F} \le a\}$ and $E^a = \{\bar{F} \circ \text{pr}_2 \le a\}$. Then $E = S^1 \times E_B$ and $E^a = S^1 \times E_B^a$. Consider the commutative diagram

$$ \begin{tikzcd} H^*(B) \arrow[r, "\text{ex} \circ T"] \arrow[d] & H^{*+\text{ind}(\bar{F}_\infty)}(E_B, E_B^{-\infty}) \arrow[r, "i_a"] \arrow[d] & H^{*+\text{ind}(\bar{F}_\infty)}(E_B^a, E_B^{-\infty}) \arrow[d] \\ H^*(S^1 \times B) \arrow[r, "\text{ex} \circ T"] & H^{*+\text{ind}(\bar{F}_\infty)}(E, E^{-\infty}) \arrow[r, "i_a"] & H^{*+\text{ind}(\bar{F}_\infty)}(E^a, E^{-\infty}) \end{tikzcd} $$

where the vertical arrows are the homomorphisms induced by the projection $S^1 \times B \to B$ and the induced projections on fibre bundles. Since the first vertical arrow sends the unit class of $H^*(B)$ to the unit class of $H^*(S^1 \times B)$ and since the last vertical arrow is injective, we deduce that the image of the unit class of $H^*(B)$ by $i_a^* \circ \text{ex} \circ T$ is non-zero if and only if the image of the unit class of $H^*(S^1 \times B)$ by $i_a^* \circ \text{ex} \circ T$ is non-zero. This implies that $c(1, \bar{F} \circ \text{pr}_2) = c(1, \bar{F})$, and so $c_-(\tilde{\phi}) = c_-(\phi)$. $\square$

Recall that the spectral selectors for compactly supported Hamiltonian symplectomorphisms of $\mathbb{R}^{2n}$ are defined in [Vit92] by

$$ c_+(\varphi) = c([S^{2n}], \bar{F}) \quad \text{and} \quad c_-(\varphi) = c(1, \bar{F}) $$

for any generating function quadratic at infinity $\bar{F} : S^{2n} \times \mathbb{R}^N \to \mathbb{R}$ for $\varphi$. For any $\varphi$ we have

$$ c_{\pm}(\tilde{\varphi}) = c_{\pm}(\varphi), $$

where $\tilde{\varphi}$ denotes the lift of $\varphi$ to $\mathbb{R}^{2n+1}$ or $\mathbb{R}^{2n} \times S^1$. Indeed, this follows from the fact that the Legendrian submanifold $\Lambda_{\tilde{\varphi}}$ of $J^1\mathbb{R}^{2n+1} \equiv T^*\mathbb{R} \times J^1\mathbb{R}^{2n}$ is equal to $0_\mathbb{R} \times \tilde{L}_\varphi$, where $\tilde{L}_\varphi$ is the lift of $L_\varphi \subset T^*\mathbb{R}^{2n}$ to $J^1\mathbb{R}^{2n}$ (cf. [San11a, Proposition 3.18]).

As a first application of our spectral selectors we obtain a proof of Theorem 1.2 (ii).

*Proof of Theorem 1.2 (ii).* Let $\varphi$ be a compactly supported lcs Hamiltonian diffeomorphisms of $S^1 \times \mathbb{R}^{2n+1}$ or $S^1 \times \mathbb{R}^{2n} \times S^1$ with non-empty support. We have to show that $\varphi$ has at least one essential translated point in the interior of its support. By Proposition 6.3 (v), either $c_+(\varphi) \neq 0$ or $c_-(\varphi) \neq 0$. But, $c_+(\varphi)$ and $c_-(\varphi)$ belong to the action spectrum of $\varphi$. We thus conclude that $\varphi$ has at least one essential translated point that has action (and time-shift) different from zero, and thus belongs to the interior of the support of $\varphi$. $\square$

## 7. BI-INVARIANT PARTIAL ORDERS ON THE HAMILTONIAN GROUPS OF $S^1 \times \mathbb{R}^{2n+1}$ AND $S^1 \times \mathbb{R}^{2n} \times S^1$, AND AN INTEGER-VALUED BI-INVARIANT METRIC ON THE HAMILTONIAN GROUP OF $S^1 \times \mathbb{R}^{2n} \times S^1$

Recall that for compactly supported lcs Hamiltonian diffeomorphisms $\varphi_1$ and $\varphi_2$ of $S^1 \times \mathbb{R}^{2n+1}$ or $S^1 \times \mathbb{R}^{2n} \times S^1$ we have posed $\varphi_1 \le \varphi_2$ if $\varphi_2 \circ \varphi_1^{-1}$ is the time-1 map of a non-negative compactly supported lcs Hamiltonian isotopy. We now show that $\le$ is a bi-invariant partial order, and so $S^1 \times \mathbb{R}^{2n+1}$ and $S^1 \times \mathbb{R}^{2n} \times S^1$ are strongly orderable. We obtain this by comparing $\le$ with a bi-invariant partial order $\leq$ defined using the spectral selectors of the previous section, which is the lcs analogue of the partial orders on $\text{Ham}^c(\mathbb{R}^{2n})$, $\text{Cont}_0^c(\mathbb{R}^{2n+1})$ and $\text{Cont}_0^c(\mathbb{R}^{2n} \times S^1)$ defined in [Vit92], [Bhu01] and [San11a] respectively.

For compactly supported lcs Hamiltonian diffeomorphisms $\varphi_1$ and $\varphi_2$ of $S^1 \times \mathbb{R}^{2n+1}$ or $S^1 \times \mathbb{R}^{2n} \times S^1$ we pose

$$ \varphi_1 \le \varphi_2 \quad \text{if} \quad c_+(\varphi_1 \circ \varphi_2^{-1}) = 0. $$

**Theorem 7.1.** The relation $\leq$ on $\text{Ham}^c(S^1 \times \mathbb{R}^{2n+1})$ or $\text{Ham}^c(S^1 \times \mathbb{R}^{2n} \times S^1)$ is a bi-invariant partial order.
*Proof.* Reflexivity follows from the fact that $c_+(\mathrm{id}) = 0$. Suppose that $\varphi_1 \preceq \varphi_2$ and $\varphi_2 \preceq \varphi_3$, thus $c_+(\varphi_1 \circ \varphi_2^{-1}) = c_+(\varphi_2 \circ \varphi_3^{-1}) = 0$. By Proposition 6.3 (iv) and (vi) we then have

$$0 \le c_+(\varphi_1 \circ \varphi_3^{-1}) \le c_+(\varphi_1 \circ \varphi_2^{-1}) + c_+(\varphi_2 \circ \varphi_3^{-1}) = 0,$$

thus $c_+(\varphi_1 \circ \varphi_3^{-1}) = 0$ and so $\varphi_1 \preceq \varphi_3$, showing transitivity. Suppose that $\varphi_1 \preceq \varphi_2$ and $\varphi_2 \preceq \varphi_1$, thus $c_+(\varphi_1 \circ \varphi_2^{-1}) = c_+(\varphi_2 \circ \varphi_1^{-1}) = 0$. By Proposition 6.3 (iii) we then have

$$c_-(\varphi_1 \circ \varphi_2^{-1}) = -c_+(\varphi_2 \circ \varphi_1^{-1}) = 0.$$

It thus follows from Proposition 6.3 (v) that $\varphi_1 \circ \varphi_2^{-1}$ is the identity, and so $\varphi_1 = \varphi_2$. This proves anti-symmetry. As for bi-invariance, if $\varphi_1 \preceq \varphi_2$, thus $c_+(\varphi_1 \circ \varphi_2^{-1}) = 0$, then for every $\psi$ we have

$$c_+((\varphi_1 \circ \psi) \circ (\varphi_2 \circ \psi)^{-1}) = c_+(\varphi_1 \circ \varphi_2^{-1}) = 0,$$

thus $\varphi_1 \circ \psi \preceq \varphi_2 \circ \psi$, and, by Proposition 6.3 (vii),

$$c_+((\psi \circ \varphi_1) \circ (\psi \circ \varphi_2)^{-1}) = c_+(\psi \circ \varphi_1 \circ \varphi_2^{-1} \circ \psi^{-1}) = 0,$$

thus $\psi \circ \varphi_1 \preceq \psi \circ \varphi_2$.

□

In order to deduce that the relation $\leq$ is also a partial order, and so that $S^1 \times \mathbb{R}^{2n+1}$ and $S^1 \times \mathbb{R}^{2n} \times S^1$ are strongly orderable (Theorem 1.3 (ii)), we use the following comparison.

**Proposition 7.2.** Let $\varphi_1$ and $\varphi_2$ be compactly supported lcs Hamiltonian diffeomorphisms of $S^1 \times \mathbb{R}^{2n+1}$ or $S^1 \times \mathbb{R}^{2n} \times S^1$ with $\varphi_1 \leq \varphi_2$. Then $\varphi_1 \preceq \varphi_2$.

*Proof.* Since $\varphi_1 \leq \varphi_2$, $\varphi_2 \circ \varphi_1^{-1}$ is the time-1 map of a compactly supported non-negative lcs Hamiltonian isotopy. By definition this implies that $\mathrm{id} \leq \varphi_2 \circ \varphi_1^{-1}$, and so by Proposition 6.3 (ii) and (iv) we have

$$0 = c_-(\mathrm{id}) \le c_-(\varphi_2 \circ \varphi_1^{-1}) \le 0,$$

thus $c_-(\varphi_2 \circ \varphi_1^{-1}) = 0$. By Proposition 6.3 (iii) this implies that $c_+(\varphi_1 \circ \varphi_2^{-1}) = 0$, and so $\varphi_1 \preceq \varphi_2$. □

□

We can now prove that $S^1 \times \mathbb{R}^{2n+1}$ and $S^1 \times \mathbb{R}^{2n} \times S^1$ are strongly orderable.

*Proof of Theorem 1.3 (ii).* We have to show that the relation $\leq$ on $\mathrm{Ham}^c(S^1 \times \mathbb{R}^{2n+1})$ or $\mathrm{Ham}^c(S^1 \times \mathbb{R}^{2n} \times S^1)$ is a bi-invariant partial order. Reflexivity follows from the fact that $\{\mathrm{id}\}$ has Hamiltonian function $H_t \equiv 0$. Transitivity and bi-invariance follow from Lemmas 2.20 and 2.21. As for anti-symmetry, if $\varphi_1 \leq \varphi_2$ and $\varphi_2 \leq \varphi_1$ then by Proposition 7.2 we also have $\varphi_1 \preceq \varphi_2$ and $\varphi_2 \preceq \varphi_1$. But, by Theorem 7.1, $\preceq$ is anti-symmetric. We thus conclude that $\varphi_1 = \varphi_2$. □

□

Recall that a bi-invariant metric on a group $G$ is a map $d: G \times G \to \mathbb{R}$ that satisfies the following properties:

(i) Positivity: $d(g_1, g_2) \geq 0$, and $d(g, g) = 0$.

(ii) Non-degeneracy: $d(g_1, g_2) = 0$ if and only if $g_1 = g_2$.

(iii) Symmetry: $d(g_1, g_2) = d(g_2, g_1)$.

(iv) Triangle inequality: $d(g_1, g_3) \leq d(g_1, g_2) + d(g_2, g_3)$.

(v) Bi-invariance: $d(hg_1, hg_2) = d(g_1h, g_2h) = d(g_1, g_2)$.

In [San10] the third author defined an integer valued bi-invariant metric $d_{\mathrm{cont}}$ on $\mathrm{Cont}_0^c(\mathbb{R}^{2n} \times S^1)$ by posing

$$d_{\mathrm{cont}}(\phi_1, \phi_2) = [c_+(\phi_1 \circ \phi_2^{-1})] - [c_-(\phi_1 \circ \phi_2^{-1})],$$

and proved that this metric is unbounded. This metric is the contact analogue of the Viterbo metric $d_{\mathrm{symp}}$ on $\mathrm{Ham}^c(\mathbb{R}^{2n})$, which is defined by

$$d_{\mathrm{symp}}(\varphi_1, \varphi_2) = c_+(\varphi_1 \circ \varphi_2^{-1}) - c_-(\varphi_1 \circ \varphi_2^{-1})$$

and is also unbounded. Similarly, for compactly supported lcs Hamiltonian diffeomorphisms $\varphi_1$ and $\varphi_2$ of $S^1 \times \mathbb{R}^{2n} \times S^1$ we define

$$d(\varphi_1, \varphi_2) = [c_+(\varphi_1 \circ \varphi_2^{-1})] - [c_-(\varphi_1 \circ \varphi_2^{-1})].$$
**Theorem 7.3.** The map $d$ is an unbounded bi-invariant metric on $\text{Ham}^c(S^1 \times \mathbb{R}^{2n} \times S^1)$.

*Proof.* Positivity follows from Proposition 6.3 (iv) and the fact that $c_+(id) = 0$. Suppose that

$$d(\varphi_1, \varphi_2) = [c_+(\varphi_1 \circ \varphi_2^{-1})] - [c_-(\varphi_1 \circ \varphi_2^{-1})] = 0.$$

Since $c_+(\varphi_1 \circ \varphi_2^{-1}) \ge 0$ and $c_-(\varphi_1 \circ \varphi_2^{-1}) \le 0$, this implies that $c_+(\varphi_1 \circ \varphi_2^{-1}) = c_-(\varphi_1 \circ \varphi_2^{-1}) = 0$ and, by Proposition 6.3 (v), that $\varphi_1 \circ \varphi_2^{-1}$ is the identity, or, equivalently, that $\varphi_1 = \varphi_2$. This proves non-degeneracy. Symmetry, the triangle inequality and bi-invariance follow from Proposition 6.3 (iii), (vi) and (vii) respectively. Observe now that, by Proposition 6.5, if $\phi_1$ and $\phi_2$ are compactly supported contactomorphisms of $\mathbb{R}^{2n} \times S^1$ contact isotopic to the identity and $\phi_1$ and $\phi_2$ their lifts to $S^1 \times \mathbb{R}^{2n} \times S^1$ then $d(\phi_1, \phi_2) = d_{\text{cont}}(\phi_1, \phi_2)$. Since $d_{\text{cont}}$ is unbounded, we conclude $d$ is also unbounded. $\square$

Recall that a partially ordered metric space is a metric space $(Z, d)$ endowed with a partial order $\le$ that is compatible with $d$, in the sense that for all $a, b, c \in Z$ with $a \le b \le c$ we have $d(a, b) \le d(a, c)$.

**Theorem 7.4.** $(\text{Ham}^c(S^1 \times \mathbb{R}^{2n} \times S^1), d, \le)$ and $(\text{Ham}^c(S^1 \times \mathbb{R}^{2n} \times S^1), d, \le)$ are partially ordered metric spaces.

*Proof.* By Proposition 7.2, it is enough to prove that $\le$ is compatible with $d$. For this, suppose that $\varphi_1 \preceq \varphi_2 \preceq \varphi_3$. Then $c_+(\varphi_1 \circ \varphi_2^{-1}) = c_+(\varphi_2 \circ \varphi_3^{-1}) = c_+(\varphi_1 \circ \varphi_3^{-1}) = 0$. Thus, using Proposition 6.3 (iii) and (vi) we get

$$ \begin{aligned} d(\varphi_1, \varphi_2) &= [c_+(\varphi_1 \circ \varphi_2^{-1})] - [c_-(\varphi_1 \circ \varphi_2^{-1})] = -[c_-(\varphi_1 \circ \varphi_2^{-1})] = [c_+(\varphi_2 \circ \varphi_1^{-1})] \\ &\le [c_+(\varphi_2 \circ \varphi_3^{-1})] + [c_+(\varphi_3 \circ \varphi_1^{-1})] = [c_+(\varphi_3 \circ \varphi_1^{-1})] \\ &= [c_+(\varphi_1 \circ \varphi_3^{-1})] + [c_+(\varphi_3 \circ \varphi_1^{-1})] = [c_+(\varphi_1 \circ \varphi_3^{-1})] - [c_-(\varphi_1 \circ \varphi_3^{-1})] \\ &= d(\varphi_1, \varphi_3). \end{aligned} $$

$\square$

**Remark 7.5.** By an argument of Polterovich, which is written down in [San15, Proposition 5.1], the identity component of the group of compactly supported contactomorphisms of any contact manifold does not admit any bi-invariant metric taking values arbitrarily close to zero. A similar argument also works in the lcs case to show that the group of compactly supported lcs diffeomorphisms of any lcs manifold $(M, [(\eta, \omega)])$ does not admit any bi-invariant metric that takes values arbitrarily close to zero. Indeed, let $B$ be a Darboux ball in $M$ and let $\psi_1$ and $\psi_2$ be two lcs diffeomorphisms that are supported in $B$ and do not commute, i.e. $[\psi_1, \psi_2] := \psi_1 \circ \psi_2 \circ \psi_1^{-1} \circ \psi_2^{-1} \ne \text{id}$. Suppose by contradiction that $d$ is bi-invariant metric on the group of compactly supported lcs diffeomorphisms of $(M, [(\eta, \omega)])$ that takes values arbitrarily close to zero, and consider the associated conjugation-invariant norm $\|\cdot\| = d(\cdot, \text{id})$. Let $\varphi$ be a compactly supported lcs diffeomorphism with $\|\varphi\| = \epsilon$ for $\epsilon$ arbitrarily small. Since $\varphi$ is not the identity, there is a small ball $B_\varphi$ in $M$ that is displaced by $\varphi$. Since it is possible to squeeze any given domain of $(\mathbb{R}^{2n}, [(0, \omega_0)])$ into an arbitrarily small one by the Liouville flow (which is a lcs isotopy), we can find a compactly supported lcs diffeomorphism $\theta$ of $M$ such that $\theta(B) \subset B_\phi$. By Lemma 2.21, $\theta \circ \psi_1 \circ \theta^{-1}$ and $\theta \circ \psi_2 \circ \theta^{-1}$ are lcs diffeomorphisms supported in $B_\phi$. Since we assume that $\|\cdot\|$ is conjugation-invariant and $[\psi_1, \psi_2] \ne \text{id}$, we have

$$ \| [\theta \circ \psi_1 \circ \theta^{-1}, \theta \circ \psi_2 \circ \theta^{-1}] \| = \| \theta \circ [\psi_1, \psi_2] \circ \theta^{-1} \| = \| [\psi_1, \psi_2] \| \ne 0. $$

On the other hand, we also have

$$ \| [\theta \circ \psi_1 \circ \theta^{-1}, \theta \circ \psi_2 \circ \theta^{-1}] \| \le 4 \|\varphi\| = 4\epsilon. $$

Indeed, as in [San15, Proposition 5.1], this follows from the fact that if $a_1$ and $a_2$ are supported in a domain $U$ and $b$ displaces $U$ then for every conjugation-invariant norm $\|\cdot\|$ we have $\|[a_1, a_2]\| \le 4\|b\|$. Since $\epsilon$ was chosen arbitrarily small, we thus obtain a contradiction. As the Liouville flow is not Hamiltonian, it is unclear to us whether the group of compactly supported lcs Hamiltonian diffeomorphisms admits a bi-invariant metric that takes values arbitrarily close to zero.
8. A LCS CAPACITY FOR DOMAINS OF $S^1 \times \mathbb{R}^{2n} \times S^1$, AND PROOF OF LCS NON-SQUEEZING

We define the *lcs capacity* of an open and bounded domain $\mathcal{U}$ of $S^1 \times \mathbb{R}^{2n} \times S^1$ by

$$c(\mathcal{U}) = \sup\{\lceil c_+(\varphi) \rceil | \varphi \in \mathrm{Ham}(\mathcal{U})\},$$

where $\mathrm{Ham}(\mathcal{U})$ denotes the group of lcs diffeomorphisms of $S^1 \times \mathbb{R}^{2n} \times S^1$ that are the time-1 map of a lcs Hamiltonian isotopy supported in $\mathcal{U}$. By the following lemma, $c(\mathcal{U})$ is a well-defined integer number.

**Lemma 8.1.** For every compactly supported lcs Hamiltonian diffeomorphism $\psi$ of $S^1 \times \mathbb{R}^{2n} \times S^1$ such that $\psi(\mathcal{U}) \cap \mathcal{U} = \emptyset$ we have

$$c(\mathcal{U}) \le d(\mathrm{id}, \psi).$$

*Proof.* We first prove that

$$(16) \qquad \lceil c_+(\psi \circ \varphi) \rceil = \lceil c_+(\psi) \rceil$$

for every $\varphi \in \mathrm{Ham}(\mathcal{U})$. Let $\{\varphi_t\}$ be a lcs Hamiltonian isotopy supported in $\mathcal{U}$ with $\varphi_1 = \varphi$, let $\overline{F}_t : E \to \mathbb{R}$ be a 1-parameter family of generating functions quadratic at infinity for $\{\psi \circ \varphi_t\}$, and set

$$c_t = c_+(\psi \circ \varphi_t) = c([S^1 \times S^{2n} \times S^1], \overline{F}_t).$$

The results follows if we prove that if $c_{t_0} = k \in \mathbb{Z}$ for some $t_0$ then $c_t = k$ for all $t$. Let $x_t$ be a path in $E$, for $t$ in a subinterval of $[0, 1]$ containing $t_0$, such that each $x_t$ is a critical point of $\overline{F}_t$ with critical value $c_t$. Let $q_t$ be the corresponding path of essential translated points of $\psi \circ \varphi_t$. As in the proof of Proposition 6.3 (vii), without loss of generality we can assume that $q_{t_0}$ is a non-degenerate essential translated point of $\psi \circ \varphi_{t_0}$. Since $q_{t_0}$ has time-shift $k \in \mathbb{Z}$, we have $\psi \circ \varphi_{t_0}(q_{t_0}) = q_{t_0}$. Since $\varphi_{t_0}$ is supported in $\mathcal{U}$ and $\psi(\mathcal{U}) \cap \mathcal{U} = \emptyset$, this implies that $q_{t_0} \notin \mathcal{U}$ and so $\varphi_t(q_{t_0}) = q_{t_0}$ for all $t$ and $\psi(q_{t_0}) = q_{t_0}$, thus $\psi \circ \varphi_t(q_{t_0}) = q_{t_0}$ for all $t$. Let $g_t$ and $f$ be the conformal factors of $\varphi_t$ and $\psi$ respectively. Then the conformal factor of $\psi \circ \varphi_t$ is $f \circ \varphi_t + g_t$. Since $q_{t_0}$ is a translated point of $\psi \circ \varphi_{t_0}$ and since $g_t(q_{t_0}) = 0$ for all $t$ (because $q_{t_0}$ does not belong to the support of $\varphi_t$), we have

$$0 = (f \circ \varphi_{t_0} + g_{t_0})(q_{t_0}) = f(q_{t_0}),$$

and so

$$(f \circ \varphi_t + g_t)(q_{t_0}) = f(q_{t_0}) = 0$$

for all $t$. We conclude that $q_{t_0}$ is a translated point of $\psi \circ \varphi_t$ of time-shift $k$, for all $t$. The action of $\psi \circ \varphi_t$ is $S_{\varphi_t} - e^{-g_t} S_{\psi} \circ \varphi_t$. Since $q_{t_0}$ is an essential translated point of $\psi \circ \varphi_{t_0}$, by Example 3.11 we have

$$(S_{\varphi_{t_0}} - e^{-g_{t_0}} S_{\psi} \circ \varphi_{t_0})(q_{t_0}) = S_{\varphi_{t_0}}(q_{t_0}) - S_{\psi}(q_{t_0}).$$

Since moreover $S_{\varphi_t}(q_{t_0}) = 0$ for all $t$ (by Remark 2.1, since $q_{t_0}$ does not belong to the support of $\varphi_t$ and the Lee form $-d\theta$ is not exact on the complement of the support of $\varphi_t$), we thus have

$$(S_{\varphi_t} - e^{-g_t} S_{\psi} \circ \varphi_t)(q_{t_0}) = -S_{\psi}(q_{t_0}) = 0$$

for all $t$. By Example 3.11, we conclude that $q_{t_0}$ is an essential translated point of $\psi \circ \varphi_t$ for all $t$. Moreover, the fact that $q_{t_0}$ is a non-degenerate translated point of $\psi \circ \varphi_{t_0}$ implies that it is a non-degenerate translated point of $\psi \circ \varphi_t$ for all $t$ (using again that $q_{t_0}$ does not belong to the support of $\varphi_t$). Let $y_t$ be the corresponding path of non-degenerate critical points of $\overline{F}_t$ of critical value $k$. Since $y_{t_0} = x_{t_0}$, we deduce that $y_t = x_t$ for all $t$, and so $c_t = k$ for all $t$. This proves (16). Using (16) and Proposition 6.3 (iii) and (vi) we then obtain

$$\lceil c_+(\varphi) \rceil \le \lceil c_+(\psi \circ \varphi) \rceil + \lceil c_+(\psi^{-1}) \rceil = \lceil c_+(\psi) \rceil - \lfloor c_-(\psi) \rfloor = d(\mathrm{id}, \psi)$$

for every $\varphi \in \mathrm{Ham}(\mathcal{U})$, and so $c(\mathcal{U}) \le d(\mathrm{id}, \psi)$. $\square$

We define the *displacement energy* of an open and bounded domain $\mathcal{U}$ of $S^1 \times \mathbb{R}^{2n} \times S^1$ by

$$E(\mathcal{U}) = \inf\{d(\mathrm{id}, \psi) | \psi(\mathcal{U}) \cap \mathcal{U} \neq \emptyset\}.$$

Lemma 8.1 then also implies the energy-capacity inequality

$$c(\mathcal{U}) \le E(\mathcal{U}).$$

**Proposition 8.2.** The lcs capacity $c$ satisfies the following properties:
(i) Monotonicity: if $\mathcal{U}_1 \subset \mathcal{U}_2$ then $c(\mathcal{U}_1) \le c(\mathcal{U}_2)$.

(ii) Lcs invariance: for any compactly supported lcs Hamiltonian diffeomorphism $\psi$ of $S^1 \times \mathbb{R}^{2n} \times S^1$, we have $c(\mathcal{U}) = c(\psi(\mathcal{U}))$.

*Proof.* Monotonicity holds by definition. Lcs invariance follows from Proposition 6.3 (vii), because $\varphi \in \mathrm{Ham}(\mathcal{U})$ if and only if $\psi \circ \varphi \circ \psi^{-1} \in \mathrm{Ham}(\psi(\mathcal{U}))$. $\square$

As in [Vit92], for any open and bounded domain $\mathcal{U}$ of $\mathbb{R}^{2n}$ we define

$$c_{\mathrm{symp}}(\mathcal{U}) = \sup\{c_+(\varphi) \mid \varphi \in \mathrm{Ham}(\mathcal{U})\},$$

where $\mathrm{Ham}(\mathcal{U})$ denotes the group of diffeomorphisms of $\mathbb{R}^{2n}$ that are the time-1 map of a Hamiltonian isotopy supported in $\mathcal{U}$. Similarly, as in [San11a], for any open and bounded domain $\mathcal{U}$ of $\mathbb{R}^{2n} \times S^1$ we define

$$c_{\mathrm{cont}}(\mathcal{U}) = \sup\{\lceil c_+(\varphi) \rceil \mid \varphi \in \mathrm{Cont}(\mathcal{U})\},$$

where $\mathrm{Cont}(\mathcal{U})$ denotes the group of diffeomorphisms of $\mathbb{R}^{2n} \times S^1$ that are the time-1 map of a contact isotopy supported in $\mathcal{U}$. For every open and bounded domain $\mathcal{U}$ of $\mathbb{R}^{2n}$ we then have

$$(17) \qquad c_{\mathrm{cont}}(\mathcal{U} \times S^1) = \lceil c_{\mathrm{symp}}(\mathcal{U}) \rceil.$$

We also have the following result.

**Proposition 8.3.** For every open and bounded domain $\mathcal{U}$ of $\mathbb{R}^{2n} \times S^1$ we have

$$c(S^1 \times \mathcal{U}) = c_{\mathrm{cont}}(\mathcal{U}).$$

*Proof.* The result follows from Proposition 6.3 (ii) and Proposition 6.5 if we prove that for every $\varphi \in \mathrm{Ham}(S^1 \times \mathcal{U})$ there is $\phi \in \mathrm{Cont}(\mathcal{U})$ such that $\varphi \le \tilde{\phi}$. This can be seen as follows. Let $\{\varphi_t\}$ be a lcs Hamiltonian isotopy of $S^1 \times \mathbb{R}^{2n} \times S^1$ with $\varphi_1 = \varphi$ and with Hamiltonian function $H_t$ supported in $S^1 \times \mathcal{U}$. Let $h_t$ be the conformal factors of $\{\varphi_t\}$, and $G$ a function on $\mathbb{R}^{2n} \times S^1$ supported in $\mathcal{U}$ such that

$$G(p) \le -e^{-h_t(\theta,p)} H_t \circ \varphi_t(\theta,p) \quad \text{for all } p, \theta \text{ and } t.$$

Let $\{\chi_t\}$ be the contact isotopy with Hamiltonian function $G$, and $\{\tilde{\chi}_t\}$ its lift to $S^1 \times \mathbb{R}^{2n} \times S^1$. By Example 2.12 and Example 2.19, $\{\tilde{\chi}_t\}$ has conformal factor $\tilde{g}_t(\theta, p) = g_t(p)$ and Hamiltonian function $\tilde{G}(\theta, p) = G(p)$. Let $\{\phi_t\}$ be the contact isotopy defined by $\phi_t = \chi_t^{-1}$. Then $\{\tilde{\phi}_t\}$ has conformal factor $-\tilde{g}_t \circ \tilde{\chi}_t^{-1}$ and, by Lemma 2.20 (i), Hamiltonian function $-e^{-\tilde{g}_t} \tilde{G} \circ \tilde{\chi}_t$. By Lemma 2.20, $\{\tilde{\phi}_t \circ \phi_t^{-1}\}$ has thus Hamiltonian function

$$-e^{-\tilde{g}_t} \tilde{G} \circ \tilde{\chi}_t + e^{-\tilde{g}_t} (-e^{-h_t} H_t \circ \varphi_t) \circ \tilde{\chi}_t = -e^{-\tilde{g}_t} (\tilde{G} + e^{-h_t} H_t \circ \varphi_t) \circ \tilde{\chi}_t \ge 0.$$

This shows that $\varphi_1 \le \tilde{\phi}_1$, as we wanted. $\square$

In particular, by Proposition 8.3 and (17) we have

$$(18) \qquad c(S^1 \times B^{2n}(R) \times S^1) = c_{\mathrm{cont}}(B^{2n}(R) \times S^1) = \lceil c_{\mathrm{symp}}(B^{2n}(R)) \rceil = \lceil \pi R^2 \rceil.$$

Using this we now obtain a proof of the lcs non-squeezing theorem for integers.

*Proof of Theorem 1.4.* Suppose that $\pi R_2^2 \le k \le \pi R_1^2$ for $k \in \mathbb{N}_0$. We have to show that there is no lcs squeezing of $S^1 \times B^{2n}(R_1) \times S^1$ into $S^1 \times B^{2n}(R_2) \times S^1$, i.e. no lcs Hamiltonian isotopy $\{\varphi_t\}_{t \in [0,1]}$ of $S^1 \times \mathbb{R}^{2n} \times S^1$ such that

$$\varphi_1(\overline{S^1 \times B^{2n}(R_1) \times S^1}) \subset S^1 \times B^{2n}(R_2) \times S^1.$$

Suppose by contradiction that such a lcs squeezing exists. We then also have a lcs squeezing of a neighborhood of $S^1 \times B^{2n}(R_1) \times S^1$ into $S^1 \times B^{2n}(R_2) \times S^1$, so without loss of generality we can assume that $\pi R_2^2 \le k < \pi R_1^2$. By (18) and Proposition 8.2 we have

$$c(\varphi_1(S^1 \times B^{2n}(R_1) \times S^1)) = c(S^1 \times B^{2n}(R_1) \times S^1) = \lceil \pi R_1^2 \rceil > k$$

and

$$c(\varphi_1(S^1 \times B^{2n}(R_1) \times S^1)) \le c(S^1 \times B^{2n}(R_2) \times S^1) = \lceil \pi R_2^2 \rceil \le k.$$
This contradiction finishes the proof.

☐

## 9. THE HAMILTONIAN GROUP OF $S^1 \times \mathbb{R}^{2n+1} \times S^1$ DOES NOT HAVE THE ROKHLIN PROPERTY

Recall that a topological group $G$ is said to have the Rokhlin property if there exists $g \in G$ whose conjugacy class $\{gh^{-1} | h \in G\}$ is dense in $G$. This notion has been introduced by Glasner and Weiss [GW01, GW08]. Serraille and Stojisavljević [SS24] discovered a connection between the Rokhlin property for groups of contactomorphisms or contact homeomorphisms and contact non-squeezing. More precisely, they proved that the closure of the identity component of the group of compactly supported contactomorphisms of $\mathbb{R}^{2n+1}$ inside the group of compactly supported homeomorphisms endowed with the $C^0$-topology (with respect to the Euclidean distance) has the Rokhlin property, while the closure of the identity component of the group of compactly supported contactomorphisms of $\mathbb{R}^{2n} \times S^1$ inside the group of compactly supported homeomorphisms endowed with the $C^0$-topology (with respect to the distance on $\mathbb{R}^{2n} \times S^1 = \mathbb{R}^{2n} \times \mathbb{R}/\mathbb{Z}$ induced by the Euclidean distance on $\mathbb{R}^{2n+1}$) does not have the Rokhlin property. The proof of the Rokhlin property for $\mathbb{R}^{2n+1}$ uses the fact that any ball in $\mathbb{R}^{2n+1}$ can be squeezed by a compactly supported contact isotopy into an arbitrarily small ball. On the other hand, the proof of the fact that the Rokhlin property does not hold in the case of $\mathbb{R}^{2n} \times S^1$ is based on the contact non-squeezing theorem of Eliashberg, Kim and Polterovich. In fact, the same proof also shows that the identity component of the group of compactly supported contactomorphisms of $\mathbb{R}^{2n} \times S^1$, endowed with the $C^0$-topology, does not have the Rokhlin property. Similarly, we obtain the following corollary of Theorem 1.4, where $\mathrm{Ham}^c(S^1 \times \mathbb{R}^{2n} \times S^1)$ is endowed with the $C^0$-topology with respect to the distance induced by the Euclidean distance on $\mathbb{R}^{2n+2}$.

**Corollary 9.1.** *The topological group $\mathrm{Ham}^c(S^1 \times \mathbb{R}^{2n} \times S^1)$ does not have the Rokhlin property.*

*Proof.* We proceed exactly as in [SS24]. Suppose by contradiction that there exists $g \in \mathrm{Ham}^c(S^1 \times \mathbb{R}^{2n} \times S^1)$ whose conjugacy class is dense in $\mathrm{Ham}^c(S^1 \times \mathbb{R}^{2n} \times S^1)$. Since $g$ has compact support, we can assume that its support is contained in $S^1 \times B(R) \times S^1$ for some $R$. Choose $f \in \mathrm{Ham}^c(S^1 \times \mathbb{R}^{2n} \times S^1)$ such that $f(x) \neq x$ for all $x \in S^1 \times B(R+1) \times S^1$. Such $f$ can be found by considering the time-1 map of a lcs Hamiltonian isotopy with Hamiltonian function that is equal to $\frac{1}{2}$ on $S^1 \times B(R+1) \times S^1$ and vanishes outside a larger set. Since we assume that the conjugacy class of $g$ is dense in $\mathrm{Ham}^c(S^1 \times \mathbb{R}^{2n} \times S^1)$, there exists a sequence $(h_n)$ in $\mathrm{Ham}^c(S^1 \times \mathbb{R}^{2n} \times S^1)$ such that $(h_n \circ g \circ h_n^{-1})$ converges to $f$. This implies that

$$ (19) \qquad S^1 \times B(R+1) \times S^1 \not\subseteq \mathrm{supp}(h_n \circ g \circ h_n^{-1}) = h_n(\mathrm{supp}(g)) $$

for all $n$. Indeed, if $S^1 \times B(R+1) \times S^1 \subset h_n(\mathrm{supp}(g))$ then $S^1 \times B(R+1) \times S^1 \subset h_n(S^1 \times B(R) \times S^1)$, and so $h_n^{-1}(S^1 \times B(R+1) \times S^1) \subset S^1 \times B(R) \times S^1$, contradicting Theorem 1.4. By taking $n$ large enough in (19) we obtain

$$ S^1 \times B(R+1) \times S^1 \not\subseteq \mathrm{supp}(f), $$

and thus there exists $x \in S^1 \times B(R+1) \times S^1$ such that $f(x) = x$, contradicting the construction of $f$. ☐

## REFERENCES

[AK18] M. Abouzaid and T. Kragh, Simple homotopy equivalence of nearby Lagrangians. *Acta Math.* **220** (2018), 207–237.

[AM13] P. Albers and W. Merry, Translated points and Rabinowitz Floer homology. *J. Fixed Point Theory Appl.* **13** (2013), 201–214.

[AFM15] P. Albers, U. Fuchs and W. Merry, Orderability and the Weinstein conjecture. *Compos. Math.* **151** (2015), 2251–2272.

[All22a] S. Allais, On the minimal number of translated points in contact lens spaces. *Proc. Amer. Math. Soc.* **150** (2022), 2685–2693.

[All22b] S. Allais, Morse estimates for translated points on unit tangent bundles. Preprint, arXiv:2205.13946. To appear in *Annales de l’Institut Fourier*.

[AAR23] S. Allais and P.-A. Arlove, Spectral selectors and contact orderability. Preprint, arXiv:2309.10578. To appear in *Communications in Contemporary Mathematics*.

[AAR24] S. Allais and M.-C. Arnaud, The dynamics of conformal Hamiltonian flows: dissipativity and conservativity. *Rev. Mat. Iberoam.* **40** (2024), 987–1021.

[AD15] V. Apostolov and G. Dloussky, Locally conformally symplectic structures on compact non-Kähler complex surfaces. *Int. Math. Res. Not.* 2016, no. 9, 2717–2747.
[AD18] V. Apostolov and G. Dloussky, On the Lee classes of locally conformally symplectic complex surfaces. *J. Symplectic Geom.* **16** (2018), 931–958.

[Ban78] A. Banyaga, Sur la structure du groupe des difféomorphismes qui préservent une forme symplectique. *Comment. Math. Helv.* **53** (1978), 174–227.

[Ban02] A. Banyaga, Some properties of locally conformal symplectic structures. *Comment. Math. Helv.* **77** (2002), 383–398.

[BaMa18] G. Bazzoni and J. C. Marrero, On locally conformal symplectic manifolds of the first kind. *Bull. Sci. Math.* **143** (2018), 1–57.

[BeMe24] M. Bertelson and G. Meigniez, Conformal symplectic structures, foliations and contact structures. *J. Symplectic Geom.* **22** (2024), 393–439.

[Bhu01] M. Bhupal, A partial order on the group of contactomorphisms of $\mathbb{R}^{2n+1}$ via generating functions. *Turkish J. Math.* **25** (2001), 125–135.

[BZ15] M. S. Borman and F. Zapolsky, Quasimorphisms on contactomorphism groups and contact rigidity. *Geom. Topol.* **19** (2015), 365–411.

[Can24] D. Cant, Contactomorphisms of the sphere without translated points. *J. Symplectic Geom.* **22** (2024), 1–10.

[Cha95] M. Chaperon, On generating families. In *The Floer memorial volume* (H. Hofer et al. eds), (Progr. Math., vol. 133), Birkhäuser, Basel, 1995, pp. 283–296.

[CM19] B. Chantraine and E. Murphy, Conformal symplectic geometry of cotangent bundles. *J. Symplectic Geom.* **17** (2019), 639–661.

[CS24] B. Chantraine and K. Sackel, Products of locally conformal symplectic manifolds. Preprint, arXiv: 2401.14918.

[Che96] Y. Chekanov, Critical points of quasi-functions and generating families of Legendrian manifolds. *Funct. Anal. Appl.* **30** (1996), 118–128.

[CN10] V. Chernov and S. Nemirovski, Legendrian links, causality, and the Low conjecture. *Geom. Funct. Anal.* **19** (2010), 1320–1333.

[Chi17] S. Chiu, Non-squeezing property of contact balls. *Duke Math. J.* **166** (2017), 605–655.

[CFP17] V. Colin, E. Ferrand and P. Pushkar, Positive isotopies of Legendrian submanifolds and applications. *Int. Math. Res. Not.* **2017**, 6231–6254.

[Cur24] A. Currier, On the projection of exact Lagrangians in locally conformally symplectic geometry. Preprint, arXiv:2408.07760.

[Cur25] A. Currier, Morse–Novikov homology and $\beta$-critical points. Preprint, arXiv:2501.17809.

[EKP06] Y. Eliashberg, S. S. Kim and L. Polterovich, Geometry of contact transformations and domains: orderability versus squeezing. *Geom. Topol.* **10** (2006), 1635–1747.

[EM23] Y. Eliashberg and E. Murphy, Making cobordisms symplectic. *J. Amer. Math. Soc.* **36** (2023), 1–29.

[EP00] Y. Eliashberg and L. Polterovich, Partially ordered groups and geometry of contact transformations. *Geom. Funct. Anal.* **10** (2000), 1448–1476.

[Fra16] M. Fraser, Contact non-squeezing at large scale in $\mathbb{R}^{2n} \times S^1$. *Internat. J. Math.* **27** (2016), 1650107.

[FPR18] M. Fraser, L. Polterovich and D. Rosen, On Sandon-type metrics for contactomorphism groups. *Ann. Math. Qué.* **42** (2018), 191–214.

[FSZ23] M. Fraser, S. Sandon and B. Zhang, Contact non-squeezing at large scale via generating functions. *J. Fixed Point Theory Appl.* **27** (2025), Paper No. 46, 57 pp.

[Gir10] E. Giroux, Sur la géométrie et la dynamique des transformations de contact (d'après Y. Eliashberg, L. Polterovich et al.). *Séminaire Bourbaki. Volume 2008/2009. Exposés 997–1011. Astérisque* (2010), no.332, Exp. No. 1004, viii, 183–220.

[Giv90] A.B. Givental, Nonlinear generalization of the Maslov index. *Theory of singularities and its applications*, 71–103, *Adv. Soviet Math.* **1**, Amer. Math. Soc., Providence, RI, 1990.

[GW01] E. Glasner and B. Weiss, The Topological Rohlin Property and Topological Entropy. *Am. J. Math.*, **123** (2001), 1055–1070.

[GW08] E. Glasner and B. Weiss, Topological groups with Rokhlin properties. *Colloq. Math.* **110** (2008), 51–80.

[GKPS20] G. Granja, Y. Karshon, M. Pabiniak and S. Sandon, Givental’s non-linear Maslov index on lens spaces. *Int. Math. Res. Not.* **2021**(2020), 18225–18299.

[HS24] J. Hedicke and E. Shelukhin, Non-orderability and the contact Hofer norm. Preprint, arXiv:2411.19887.

[LS85] F. Laudenbach and J.-C. Sikorav, Persistence d'intersection avec la section nulle au cours d'une isotopie hamiltonienne dans un fibré cotangent. *Invent. Math.* **82** (1985), 349–357.

[Lee43] H.-C. Lee, A kind of even-dimensional differential geometry and its application to exterior calculus. *Amer. J. Math.* **65** (1943), 433–438.

[Lib55] P. Libermann, Sur les structures presque complexes et autres structures infinitésimales régulières. *Bull. Soc. Math. France* **83** (1955), 195–224.

[Lic83] A. Lichnerowicz, Sur les algèbres de Lie locales de Kirillov–Shiga. *C. R. Acad. Sci. Paris Sér. I Math.* **296** (1983), 915–920.

[McDS17] D. McDuff and D. Salamon, Introduction to symplectic topology. Third edition. *Oxf. Grad. Texts Math. Oxford University Press*, Oxford (2017).

[MN18] M. Meiwes and K. Naef, Translated points on hypertight contact manifolds. *J. Topol. Anal.* **10** (2018), 289–322.

[MU19] W. Merry and I. Uljarevic, Maximum principles in symplectic homology. *Israel J. Math.* **229** (2019), 39–65.

[OS17] A. Otiman and M. Stanciu, Darboux–Weinstein theorem for locally conformally symplectic manifolds. *J. Geom. Phys.* **111** (2017), 1–5.
[San10] S. Sandon, An integer valued bi-invariant metric on the group of contactomorphisms of $R^{2n} \times S^1$. *J. Topol. Anal.* **2** (2010), 327–339.

[San11a] S. Sandon, Contact homology, capacity and non-squeezing in $R^{2n} \times S^1$ via generating functions. *Ann. Inst. Fourier (Grenoble)* **61** (2011), 145–185.

[San11b] S. Sandon, A Morse estimate for translated points of contactomorphisms of spheres and projective spaces. *Geom. Dedicata* **165** (2013), 95–110.

[San12] S. Sandon, On iterated translated points for contactomorphisms of $R^{2n+1}$ and $R^{2n} \times S^1$. *Internat. J. Math.* **23** (2012), 1250042, 14 pp.

[San15] S. Sandon, Bi-invariant metrics on the contactomorphism groups. *São Paulo J. Math. Sci.* **9** (2015), 195–228.

[SS24] B. Serraille and V. Stojisavljević, On certain $C^0$-aspects of contactomorphism groups. Preprint, arXiv: 2411.11422.

[She17] E. Shelukhin, The Hofer norm of a contactomorphism. *J. Symplectic Geom.* **15** (2017), 1173–1208.

[Sik87] J.C. Sikorav, Problemes d'intersections et de points fixes en géométrie hamiltonienne. *Comment. Math. Helv.* **62** (1987), 62–73.

[Ter21] B. Tervil, Translated points for contactomorphisms of prequantization spaces over monotone symplectic toric manifolds. *J. Symplectic Geom.* **19** (2021), 1421–1493.

[The95] D. Théret, Utilisation des fonctions génératrices en géométrie symplectique globale. *Ph.D. Thesis, Université Denis Diderot* (Paris 7), 1995.

[The99] D. Théret, A complete proof of Viterbo's uniqueness theorem on generating functions. *Topology Appl.* **96** (1999), 249–266.

[Vai76] I. Vaisman, On locally conformal almost Kähler manifolds. *Israel J. Math.* **24** (1976), 338–351.

[Vai85] I. Vaisman, Locally conformal symplectic manifolds. *Internat. J. Math. Math. Sci.* **8** (1985), 521–536.

[Vit92] C. Viterbo, Symplectic topology as the geometry of generating functions. *Math. Ann.* **292** (1992), 685–710.

DÉPARTEMENT DE MATHÉMATIQUES, C.P. 218 UNIVERSITÉ LIBRE DE BRUXELLES, 1050 BRUXELLES, BELGIUM

Email address: melanie.bertelson@ulb.be

DÉPARTEMENT DE MATHÉMATIQUES, C.P. 218 UNIVERSITÉ LIBRE DE BRUXELLES, 1050 BRUXELLES, BELGIUM

Email address: pranav.vijay.chakravarthy@ulb.be

UNIVERSITÉ DE STRASBOURG, CNRS, IRMA UMR 7501, F-67000 STRASBOURG, FRANCE

Email address: sandon@math.unistra.fr