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

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