GENERALIZED SHIFTS - UNPREDICTABILITY AND UNDECIDABILITY IN DYNAMIC-SYSTEMS

A class of shift-like dynamical systems is presented that displays a wide variety of behaviours. Three examples are presented along with some general definitions and results. A correspondence with Turing machines allows us to discuss issues of predictability and complexity. These systems possess a type of unpredictability qualitatively stronger than that which has been previously discussed in the study of low-dimensional chaos, and many simple questions about their dynamics are undecidable. We discuss the complexity of various sets they generate, including periodic points, basins of attraction, and time series. Finally, we show that they can be embedded in smooth maps in R2, or smooth flows in R3.