Output digitization of simple measure-preserving linear systems

We examine three simple linear systems from the viewpoint of ergodic theory.We digitize the output and record only the sign of the output at integer times.We show that even with this minimal output we can recover important information about the systems.In particular,for a two-dimensional system viewed as a fow on the circle,we can determine the rate of rotation.We then use these results to determine the slope of the trajectories for constant irrational fow on the two-dimensional torus.To achieve this,we randomize the system by partitioning the state space and only recording which partition the state is in at each integer time.We show directly that these systems have entropy zero.Finally,we examine two four-dimensional systems and reduce them to the study of linear fows on the two-dimensional torus.