Expansive Subdynamics

This paper provides a framework for studying the dynamics of commuting homeomorphisms. Let $\alpha$ be a continuous action of ${\mathbb Z}^d$ on an infinite compact metric space. For each subspace $V$ of ${\mathbb R}^d$ we introduce a notion of expansiveness for $\alpha$ along $V$, and show that there are nonexpansive subspaces in every dimension $\le d-1$. For each $k\le d$ the set ${\mathbb E} _k(\alpha )$ of expansive $k$-dimensional subspaces is open in the Grassmann manifold of all $k$-dimensional subspaces of ${\mathbb R}^d$. Various dynamical properties of $\alpha$ are constant, or vary nicely, within a connected component of ${\mathbb E} _k(\alpha )$, but change abruptly when passing from one expansive component to another. We give several examples of this sort of ``phase transition,'' including the topological and measure-theoretic directional entropies studied by Milnor, zeta functions, and dimension groups. For $d=2$ we show that, except for one unresolved case, every open set of directions whose complement is nonempty can arise as an ${\mathbb E} _1(\alpha )$. The unresolved case is that of the complement of a single irrational direction. Algebraic examples using commuting automorphisms of compact abelian groups are an important source of phenomena, and we study several instances in detail. We conclude with a set of problems and research directions suggested by our analysis.