We present a numerical simulation study of a simple monatomic Lennard-Jones liquid under shear flow, as a function of both temperature T and shear rate .gamma. By investigating different observables we find that (i) there exists a line, T(.gamma), in the (T-(.gamma)) plane that sharply marks the border between an "equilibrium" and a "shear-controlled" region for both the dynamic and the thermodynamic quantities; and (ii) along this line the structural relaxation time, tau(alpha)(T(.gamma)), is proportional to .gamma(-1), i.e., to the typical time scale introduced by the shear flow. Above T(.gamma), the liquid dynamics is unaffected by the shear flow, while below T(.gamma) both T and .gamma control the particle motion.