Expansive Subdynamics

Abstract

This paper provides a framework for studying the dynamics of commuting homeomorphisms. Let α \alpha be a continuous action of Z d \mathbb {Z}^d on an infinite compact metric space. For each subspace V V of R d \mathbb {R}^d we introduce a notion of expansiveness for α \alpha along V V , and show that there are nonexpansive subspaces in every dimension ≤ d − 1 \le d-1 . For each k ≤ d k\le d the set E k ( α ) \mathbb {E}_k(\alpha ) of expansive k k -dimensional subspaces is open in the Grassmann manifold of all k k -dimensional subspaces of R d \mathbb {R}^d . Various dynamical properties of α \alpha are constant, or vary nicely, within a connected component of E k ( α ) \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 d=2 we show that, except for one unresolved case, every open set of directions whose complement is nonempty can arise as an E 1 ( α ) \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.