Abstract : The standard cellular automata (CA) model is based on three main features: locality, uniformity and synchronicity. Recently, some variants have been introduced, most of them consist in relaxing one of those three properties. In this paper, we study the dynamical behavior of m-ACA (using fair measures), a variant of cellular automata in which the synchronicity property has been relaxed. Inspired by literature about α-asynchronous CA (a special case of m-ACA), the paper focuses on doubly quiescent elementary rules i.e., rules with radius 1, boolean alphabet and such that homogeneuous configurations are fixed points. We show that for many of these rules, the limit behavior is fully characterized by a subshift of finite type.