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

Attractor Solutions in Scalar-Field Cosmology