with period $T(\alpha_*)$. In this case, the quotient is a closed symplectic manifold $(M,\omega)$ and the quotient map $\mathfrak p:\Sigma\to M$ is a non-trivial circle bundle. More precisely, we have $\mathfrak p^*\omega={{\mathrm d}}\alpha_*$ so that $-\tfrac{1}{T(\alpha_*)}\omega$ represent the Euler class of the bundle. Vice versa, for every closed symplectic manifold $(M,\omega)$ such that $\frac1T\omega$ represents an integer class for some $T>0$, there exists a manifold $\Sigma$ and a Zoll contact form with period $T$ yielding $(M,\omega)$ according to the construction above, see [@BW58].

By [@Bot80], if $\mathfrak h$ denotes the free homotopy class of prime periodic orbits of $\Phi_{\alpha_*}$, then $\Phi_\alpha$ carries a periodic orbit in the class $\mathfrak h$, if $\alpha$ is sufficiently close to $\alpha_*$ . We denote by $T_{\min}(\alpha,\mathfrak h)$, respectively $T_{\max}(\alpha,\mathfrak h)$ the minimal, respectively maximal, period of prime periodic orbits of $\Phi_\alpha$ in the class $\mathfrak h$. If $\Sigma$ has dimension three, $T_{\min}(\alpha,\mathfrak h)$ and $T_{\max}(\alpha,\mathfrak h)$ satisfy the following contact systolic-diastolic inequality, which was originally conjectured by Álvarez-Paiva and Balacheff [@APB14].

\[thm:contact\_sys\][@BK19a Theorem 1.4] Let $\Sigma$ be a closed manifold of dimension three, and let $|H_1^\mathrm{tor}(\Sigma;{{\mathbb Z}})|$ be the order of the torsion part of $H_1(\Sigma,{{\mathbb Z}})$. If $\alpha_*$ is a Zoll contact form 