  We come back on the dynamical properties of k-essential cosmological models
and show how the interesting phenomenological features of those models are
related to the existence of boundaries in the phase surface. We focus our
attention to the branching points where the energy density has an extremum and
the effective speed of sound diverges. We discuss the behaviour of solutions of
a general class of cosmological models exhibiting such curves and give two
possible interpretations; the most interesting possibility regards the arrow of
time that is reversed in trespassing the branching curve. This study teaches to
us something new about general FLRW cosmologies where the fluids driving the
cosmic evolution have equations of state that are multivalued functions of the
energy density and other thermodynamical quantities.
