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 φ and ̇ φ 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 m²φ² 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 φ- ̇ φ variables and suggest a physical understanding of attractor behavior that is compatible with Liouville's theorem.