  Models of cosmological scalar fields often feature "attractor solutions" to
which the system evolves for a wide range of initial conditions. There is some
tension between this well-known fact and another well-known fact: Liouville's
theorem forbids true attractor behavior in a Hamiltonian system. In universes
with vanishing spatial curvature, the field variables (\phi,\dot\phi) specify
the system completely, defining an effective phase space. We investigate
whether one can define a unique conserved measure on this effective phase
space, showing that it exists for m^2\phi^2 potentials and deriving conditions
for its existence in more general theories. We show that apparent attractors
are places where this conserved measure diverges in the (\phi,\dot\phi)
variables and suggest a physical understanding of attractor behavior that is
compatible with Liouville's theorem.
