Rational contact instantons and Legendrian Fukaya category

This is the first of a series of papers in preparation on the Fukaya-type $A_\infty$ category generated by tame Legendrian submanifolds, called the Legendrian contact instanton Fukaya category (abbreviated as the Legendrian CI Fukaya category) and its applications to contact dynamics and topology. In the present paper, we give the construction of an $A_\infty$ category whose objects are Legendrian links and whose structure maps are defined by the moduli spaces of finite energy contact instantons on tame contact manifolds in the sense of [Oh21b]. In a sequel [KO], jointed by Jongmyeong Kim, we will explain the relationships with various previous results in the literature concerning Rabinowitz Fukaya categories [CF09, CFO10], [GGV], [BJK] on the Liouville manifolds with ideal boundary of contact manifolds, and the Floer theory of Lagrangian cobordism [CDRGG20], [EES05].