Topological entropy and periodic points for Zd actions on compact abelian groups with the Descending Chain Condition
This thesis deals with the ergodic theory of actions of Zd on compact abelian groups satisfying a regularity condition: the descending chain condition. Using results of Klaus Schmidt, Bruce Kitchens and Doug Lind, we show that the descending chain condition guarantees that the global topological entropy of such an action coincides with the growth rate of periodic points whenever this exists (§II.3). This links the global entropy to the dynamics of such systems and allows the global entropy to be computed for some important examples (§II4). The algebraic entropy of a Zd action on a discrete abelian group is defined. Following Justin Peters, we show that this entropy coincides with the topological entropy of the adjoint action on the compact dual group (§III.1). Two possible zeta functions for actions are introduced, and their poles are found for expansive systems with the descending chain condition (§IV.2). We comparethese functions with the one-dimensional case of a single automorphism, where the least real pole of the zeta function lies at exp(-entropy). We relate an instance of convergence at a pole with the Mahler measure of a polynomial described as a limit of one dimensional Mahler measures (§IV.3). Appendix A is a paper written with Doug Lind in which the basic onedimensional entropy formula (Yuzvinskii's formula) is computed using adelic methods. Appendix B reproduces for the sake of completeness the higher-dimensional entropy formula due to Doug Lind and Klaus Schmidt. Appendix C contains some examples of calculations of numbers of periodic points for simple higher-dimensional systems.