Predecessors and starting point of this article on multidimensional discrete time-autonomy (ta) and time-controllability (tc) were three papers from 2002 to 2004 by Sasane, Controneo, Thomas and Willems on the continuous case, a paper from 2011 by Napp, Rapisarda and Rocha on time-relevant two-dimensional (2D)-behaviours and a submitted paper by Bisiacco and Valcher on 2D-tc and dead-beat control. We also acknowledge interesting talks on this subject by Bisiacco and Valcher in Innsbruck. We develop a general framework for discrete multidimensional behaviours in which the discrete time variable in N plays a special part among all independent variables. Following the models of our predecessors we define ta and tc of a behaviour by properties of its trajectories. The goal of this article is the characterisation of these properties by constructive algebraic conditions which is fully achieved for ta and for tc in dimension two and partially for tc in dimensions higher than two.