  Discrete dynamical systems defined by the iteration of a polynomial map of
the unit simplex to itself appear in the context of population genetic systems
evolving under mutation, recombination and weak selection. Although exceptional
progress has been made in finding particular solutions to these systems, our
knowledge of the general properties of the space of all possible dynamical
systems of this kind is still limited. We prove that the space of
bounded-degree polynomial maps of the unit simplex to itself is a compact and
convex subset of a Euclidean space. We provide an explicit characterization of
such a space and of its boundary. A special class of maps in the boundary, the
folding maps, which generalize the logistic map for any dimension and degree
are defined and constructed. Finally, we use numerical methods to study the
ergodic and mixing properties of maps in the neighborhood of several of these
folding maps.
