  The goal of this paper is to analyze the effects of the matter fields in the
evolution of the Bianchi IX cosmology close to the singularity. Although the
dynamics of this model is very involved, asymptotically, as the singularity is
approached, it can be well approximated as a succession of Bianchi I periods
connected by quick bounces against potential walls. Moreover, in such limit,
matter fields (excluding stiff matter) are known to be subdominant with respect
to the anisotropies. Therefore, by performing an expansion around small
volumes, and assuming that the dominant matter contribution in this regime can
be described as a barotropic perfect fluid with a linear equation of state, we
obtain an approximate analytic solution of the dynamics. Then, we explicitly
compute the form of the transition (Kasner) map that relates the pre- and
post-bounce Bianchi I periods including the leading matter effects. As an
important conclusion, we observe that generically the presence of matter leads
to a post-bounce velocity with a lower deflection angle than in the vacuum
case, and thus effectively increases the convex curvature of the potential
walls. This effect may have important consequences in the chaotic nature of
general relativity near spacelike singularities.
