Dynamics of Nonergodic Piecewise Affine Maps of the Torus

We discuss the dynamics of a class of nonergodic 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 case we understand the best 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 compliment of the closure. About all we can say about the behavior of this map on the first set is that there are intervals of periodic orbits and at least one infinite orbit. The map on the second invariant set is measure-theoretically conjugate to the triadic odometer. The third invariant set is one of full Lebesque measure and consists of a countable number of open octagons whose points are periodic. Their orbits can be described in terms of a symbolism gotten from an equal length substitution rule or the triadic odometer.