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 sho w that the torus splits into three invariant sets on which the dynamics are quite different. These are: the orbit of the discontinuity set, the comple ment of this set in its closure, and the complement of the closure. There a re still some unsolved problems concerning the orbit of the discontinuity s et. However we do know that there are intervals of periodic orbits and at l east 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 period ic. Their orbits can be described in terms of a symbolism obtained from an equal length substitution rule or the triadic odometer.