role of self-intersection on manifolds in this new source of irreversibility (with a resulting “arrow of time”) is developed in Mackey (1992). As noted, the skeleton which configures attractors is composed of unstable, “saddle” fixed points, each of which attract (iron down) the trajectory along one dimension and repel or spread it out along another. Systems fulfilling the criteria for a chaotic dynamical system have the property of a countably infinite number of unstable periodic orbits composed of these unstable fixed points. Depending upon parameters, the orbital points can pull up their tails to be discrete with respect to each other or spread along the unstable direction to connect smoothly with others along a curve such as a saddle cycle. Parametric control of the strengths and structures of the saddle point skeleton of typical attractors can be used to change both the rate of generation of novel symbols as well as recurrances to old ones in the symbolic dynamics generating a brain dynamical system’s lexagraphic products (Bowen, 1978; Alexeev and Jacobson, 1981)). Using a variety of techniques to algorithmically register “return times,” experimental condition-sensitive “saddle orbits” composing unstable periodic orbits have been demonstrated in geometric reconstructions of real data series generated by a 40+ component chemical reaction (Lathrop and Kostelich, 1989), in response to natural stimuli in the time dependent behavior of the crayfish caudal photoreceptor (Pei and Moss, 1996) and in the interburst interval sequences recorded in hippocampal slices of the rat (So et al, 1997; So et al, 1998). If the reader uses the software listed above to simulate the time evolution of one of these attractors of abstract or real systems , she will learn that a remarkably small number of points, a very short time sample, will outline, “shadow” (Bowen, 1978), the complete array of unstable fixed points before filling in the attractor. It is tempting to speculate about t