By means of left quasigroups L = (L, ·) and ternary systems, we construct dynamical Yang–Baxter maps associated with L, L, and (·) satisfying an invariance condition that the binary operation (·) of the left quasigroup L deﬁnes. Conversely, this construction characterize such dynamical Yang–Baxter maps. The unitary condition of the dynamical Yang–Baxter map is discussed. Moreover, we establish a correspondence between two dynamical Yang–Baxter maps constructed in this paper. This correspondence produces a version of the vertex-IRF correspondence.