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.