Attractor solutions in scalar-field cosmology

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 and (phi) over dot 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-(phi) over dot variables and suggest a physical understanding of attractor behavior that is compatible with Liouville's theorem.