A flow on a compact Hausdorff space X is given by a map t : X --> X. The general goal of this paper is to find the "cyclic parts" of such a flow. To do this, we approximate (X,t) by a flow on a Stone space (that is, a totally disconnected, compact Hausdorff space). Such a flow can be examined by analyzing the resulting flow on the Boolean algebra of clopen subsets, using the spectrum defined in our previous TAC paper, The cyclic spectrum of a Boolean flow. In this paper, we describe the cyclic spectrum in terms that do not rely on topos theory. We then compute the cyclic spectrum of any finitely generated Boolean flow. We define when a sheaf of Boolean flows can be regarded as cyclic and find necessary conditions for representing a Boolean flow using the global sections of such a sheaf. In the final section, we define and explore a related spectrum based on minimal subflows of Stone spaces.
Keywords: Boolean flow, dynamical systems, spectrum, sheaf
2000 MSC: 06D22, 18B99, 37B99
Theory and Applications of Categories,
Vol. 16, 2006,
No. 17, pp 434-459.