Expansive subdynamics for algebraic $Z^d$-actions

Abstract

A general framework for investigating topological actions of $Z^d$ on compact metric spaces was proposed by Boyle and Lind in terms of expansive behavior along lower-dimensional subspaces of $R^d$. Here we completely describe this expansive behavior for the class of algebraic $Z^d$-actions given by commuting automorphisms of compact abelian groups. The description uses the logarithmic image of an algebraic variety together with a directional version of Noetherian modules over the ring of Laurent polynomials in several commuting variables. We introduce two notions of rank for topological $Z^d$-actions, and for algebraic $Z^d$-actions describe how they are related to each other and to Krull dimension. For a linear subspace of $R^d$ we define the group of points homoclinic to zero along the subspace, and prove that this group is constant within an expansive component.