  We construct moduli spaces of linear self-maps of projective space with
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 $(\mathbb{P}^N)^n$, and we take the geometric invariant theory
(GIT) quotient. These moduli spaces arise in algebraic dynamics and integrable
systems. Our main result is a dynamical characterization of the GIT semistable
and stable loci in the space of linear maps $T$ with marked points. We show
that GIT stability can be checked by counting the marked points on flags with
certain Hessenberg functions relative to $T$. 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 $A_N$ lattice.
