The study of dynamical systems is primarily concerned with single transformations or one-parameter flows (ie actions of N, Z, or R). The study of concrete and natural examples such as hyperbolic toral automorphisms, geodesic flows, and subshifts of finite type have contributed a lot to the development of the general theory.

Actions of other groups such as Z^d are less well understood, and it is quite difficult to find non-trivial smooth Z^d-actions on finite dimensional manifolds. Such actions also necessarily have zero entropy. I will talk about a large class of continuous Z^d-actions on compact metric spaces with diverse and interesting behavior: actions by automorphisms of compact abelian groups. Here the connections to commutative algebra allow a deep study of these actions. I will discuss some of these connections with a focus on explicit examples.