Mapping tori of A∞-autoequivalences andLegendrian lifts of exact Lagrangians in circular contactizations

We study mapping tori of quasi-autoequivalences (Formula Presented) which induce a free action of Z on objects. More precisely, we compute the mapping torus of when it is strict and acts bijectively on hom-sets, or when the A1-category A is directed and there is a bimodule map (Formula Presented) satisfying some hypotheses. Then we apply these results in order to link together the Fukaya A1-category of a family of exact Lagrangians, and the Chekanov–Eliashberg DG-category of Legendrian lifts in the circular contactization.