Abstract
The dynamical properties of spatially homogeneous and isotropic cosmological models containing a barotropic perfect fluid and multiple scalar fields with independent exponential potentials is investigated. It is shown that the assisted inflationary scaling solution is the global late-time attractor for the parameter values for which the model is inflationary, even when curvature and barotropic matter are included. For all other parameter values the multi-field curvature scaling solution is the global late-time attractor (in these solutions the curvature is not dynamically negligible asymptotically). Consequently, in general all of the scalar fields in multi-field models with exponential potentials are non-negligible in late-time behaviour, contrary to what is commonly believed.
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