EXPANSIVE SUBDYNAMICS

This paper provides a framework for studying the dynamics of commuting homeomorphisms. Let alpha be a continuous action of Z(d) on an infini te compact metric space. For each subspace V of R(d) we introduce a no tion of expansiveness for alpha along V, and show that there are nonex pansive subspaces in every dimension less than or equal to d - 1. For each k less than or equal to d the set E(k)(alpha) of expansive k-dime nsional subspaces is open in the Grassmann manifold of all k-dimension al subspaces of R(d). Various dynamical properties of alpha are consta nt, or vary nicely, within a connected component of E(k)(alpha), but c hange abruptly when passing from one expansive component to another. W e give several examples of this sort of ''phase transition,'' includin g the topological and measure-theoretic directional entropies studied by Milnor, zeta functions, and dimension groups. For d = 2 we show tha t, except for one unresolved case, every open set of directions whose complement is nonempty can arise as an E(1)(alpha). The unresolved cas e is that of the complement of a single irrational direction. Algebrai c examples using commuting automorphisms of compact abelian groups are an important source of phenomena, and we study several instances in d etail. We conclude with a set of problems and research directions sugg ested by our analysis.