points of $\varphi$, whose associated curve is short, are given by $$\min\mathcal A_H:=\inf_{\gamma\in{\mathrm{Crit\,}}\mathcal A_H}\mathcal A_H(\gamma),\qquad \max\mathcal A_H:=\sup_{\gamma\in{\mathrm{Crit\,}}\mathcal A_H}\mathcal A_H(\gamma).$$ The Calabi invariant of $H$ with respect to ${\mathrm{\omega}}$ is defined by $$\label{eq:calabi_intro}
{\mathrm{CAL}}_{{\mathrm{\omega}}}(H):=\int_{M\times[0,1]}H\,\omega^n\wedge{{\mathrm d}}t.$$ One can easily see that the minimal/maximal actions and the Calabi invariant are related in the following way. It is one particular case of Proposition \[p:qa1\].

\[p:hamsys\] Assume that a Hamiltonian diffeomorphism $\varphi:(M,{\mathrm{\omega}})\to (M,{\mathrm{\omega}})$ is generated by a quasi-autonomous Hamiltonian $H:M\times[0,1]\to{{\mathbb R}}$. For instance, $\varphi$ is the time-one map of a Hamiltonian isotopy $C^1$-close to the identity. Then, there holds $$\min\mathcal A_H\leq \frac{{\mathrm{CAL}}_{\mathrm{\omega}}(H)}{\int_M{\mathrm{\omega}}^n}\leq\max\mathcal A_H,$$ and any of the two equalities holds if and only if $H(q,t)=h(t)$ for all $(q,t)\in M\times[0,1]$ and some function $h:[0,1]\to{{\mathbb R}}$.

Proposition \[p:hamsys\] is false for general Hamiltonian diffeomorphisms. Indeed, adapting [@ABHS15 Proposition 2.28], one can construct a time-one map $\varphi$ of a Hamiltonian isotopy $C^0$-close to the identity not satisfying the inequality above. A proper way to go beyond the quasi-autonomous case might be to take into account not only the fixed points but all the periodic points of $\varphi$ as explored by Hutchings in [@Hut16 Theorem 1.2].

In [@BK19a] we saw how the use of a global surface of section 