Mapping tori of $A_{\mathrm{\infty }}$-autoequivalences and Legendrian lifts of exact Lagrangians in circular contactizations

We study mapping tori of quasi-autoequivalences $\tau :\mathcal{A}\to \mathcal{A}$ which induce a free action of $\mathbf{Z}$ on objects. More precisely, we compute the mapping torus of $\tau$ when it is strict and acts bijectively on hom-sets, or when the $A_{\mathrm{\infty }}$-category $\mathcal{A}$ is directed and there is a bimodule map $\mathcal{A}(-,-)\to \mathcal{A}(-,\tau (-))$ satisfying some hypotheses. Then we apply these results in order to link together the Fukaya $A_{\mathrm{\infty }}$-category of a family of exact Lagrangians, and the Chekanov-Eliashberg DG-category of Legendrian lifts in the circular contactization.