  We performed extensive molecular dynamics (MD) simulations, supplemented by
Mode Coupling Theory (MCT) calculations, for the Square Shoulder (SS) model, a
purely repulsive potential where the hard-core is complemented by a finite
shoulder. For the one-component version of this model, MCT predicted [Sperl
{\it et al.} Phys. Rev. Lett. {\bf 104}, 145701 (2010)] the presence of
diffusion anomalies both upon cooling and upon compression and the occurrence
of glass-glass transitions. In the simulations, we focus on a non-crystallising
binary mixture, which, at the investigated shoulder width, shows a
non-monotonic behaviour of the diffusion upon cooling but not upon isothermal
compression. In addition, we find the presence of a disconnected glass-glass
line in the phase diagram, ending in two higher-order singularities. These
points generate a logarithmic dependence of the density correlators as well as
a subdiffusive behaviour of the mean squared displacement, although with the
interference of the nearby liquid-glass transition. We also perform novel MCT
calculations using as input the partial structure factors obtained within MD,
confirming the simulation results. The presence of two hard sphere glasses,
differing only in their hard core length, is revealed, showing that the simple
competition between the two is sufficient for creating a rather complex
dynamical behaviour.
