Dynamics of non-ergodic piecewise affine maps of the torus

We discuss the dynamics of a class of non-ergodic piecewise affine maps of the torus. These maps exhibit highly complex and little understood behavior. We present computer graphics of some examples and analyses of some with a decreasing degree of completeness. For the best understood example, we show that the torus splits into three invariant sets on which the dynamics are quite different. These are: the orbit of the discontinuity set, the complement of this set in its closure, and the complement of the closure. There are still some unsolved problems concerning the orbit of the discontinuity set. However we do know that there are intervals of periodic orbits and at least one infinite orbit. The map on the second invariant set is minimal and uniquely ergodic. The third invariant set is one of full Lebesgue measure and consists of a countable number of open octagons whose points are periodic. Their orbits can be described in terms of a symbolism obtained from an equal length substitution rule or the triadic odometer.