GIT stability of linear maps on projective space with marked points

We construct moduli spaces of linear self-maps of N-dimensional projective space with n marked points, up to projective equivalence. That is, we let the special linear group act simultaneously by conjugation on projective linear maps and diagonally on (PN) ⁿ, and we take the geometric invariant theory (GIT) quotient. These moduli spaces arise in algebraic dynamics in two ways: first, as ambient varieties of degree 1 portrait spaces; second, as the domains of discrete integrable systems such as the pentagram map. Our main result is a dynamical characterization of the GIT semistable and stable loci in the space of linear maps with marked points. The proof is combinatorial: to describe the weight polytopes for this action, we compute the vertices and facets of certain convex polyhedra generated by roots of the AN lattice.