  A folklore result due to M.W. Hirsch states that most competitive maps admit
a carrying simplex, i.e., an invariant hypersurface which attracts all
nontrivial orbits. The common approach in the study of these maps is to focus
on the dynamical behavior on the carrying simplex. However, this manifold is
normally non-smooth. Therefore, not every tool coming from Differential
Geometry can be applied. In particular, the classical Grobman-Hartman theorem
can not be used on the carrying simplex. In this paper, we prove that the
restriction of the map to the carrying simplex in a neighborhood of an interior
fixed point is topologically conjugate to the restriction of the map to its
pseudo-unstable manifold by an invariant foliation. This implies that the
linearization techniques are applicable for studying the local dynamics of the
interior fixed points on the carrying simplex. We further construct the stable
and unstable manifolds on the carrying simplex. On the other hand, our results
also give partial responses to Hirsch's problem regarding the smoothness of the
carrying simplex. We discuss some applications in classical models of
population dynamics. Although we focus on monotone maps, many results of the
paper can be applied to maps that admit a non-smooth center-manifold.
