on $\Sigma$, there exists a $C^2$-neighbourhood $\mathcal U$ of ${{\mathrm d}}\alpha_*$ such that for every contact form $\alpha$ on $\Sigma$ with ${{\mathrm d}}\alpha\in\mathcal U$, there holds $$T_{\min}(\alpha,\mathfrak h)^2\leq \frac{1}{|H_1^\mathrm{tor}(\Sigma;{{\mathbb Z}})|}\mathrm{Volume}(\alpha)\leq T_{\max}(\alpha,\mathfrak h)^2,$$ where any of the two equalities holds if and only if $\alpha$ is Zoll.

When $\Sigma=S^3$ the result above is due to Abbondandolo, Bramham, Hryniewicz and Salomão, see [@ABHS15]. In that paper, forms $\alpha$ are also constructed, which are $C^0$-close to $\alpha_*$ and violate the systolic inequality. Furthermore, the same authors showed in [@ABHSnew] that, if $(\Sigma,\xi)$ is an arbitrary contact closed three-manifold, then, for every $C>0$, there exists a contact form $\alpha_C$ with $\ker\alpha_C=\xi$ such that $T_{\min}(\alpha_C)^2>C\cdot\mathrm{Volume}(\alpha_C)$.

### Symplectic systolic-diastolic inequality {#symplectic-systolic-diastolic-inequality .unnumbered}

Let $(M,{\mathrm{\omega}})$ be a connected closed symplectic manifold of dimension $2n$, and let $\varphi:M\to M$ be a Hamiltonian diffeomorphism generated by a Hamiltonian $H:M\times[0,1]\to{{\mathbb R}}$. We orient $M$ so that ${\mathrm{\omega}}^n$ is positive. One can define the Hamiltonian action $\mathcal A_H:\Lambda_{\mathrm{short}}(M)\to{{\mathbb R}}$ on the space of “short” one-periodic curves $q:S^1\to M$ by $$\label{e:hamact}
\mathcal A_H(\gamma):=\int_{D^2}{\widehat q}^{\hspace{2pt}*}{\mathrm{\omega}}+\int_0^1 H(q(t),t){{\mathrm d}}t,$$ where $\widehat q:D^2\to M$ is a “small” capping disc for $q\in\Lambda_{\mathrm{short}}(M)$ (see Section \[sec:conjsys\] for a precise definition). The minimal and maximal Hamiltonian actions of fixed 