  We study equivariant families of discrete Hamiltonians on amenable geometries
and their integrated density of states (IDS). We prove that the eigenspace of a
fixed energy is spanned by eigenfunctions with compact support. The size of a
jump of the IDS is consequently given by the equivariant dimension of the
subspace spanned by such eigenfunctions. From this we deduce uniform
convergence (w.r.t. the spectral parameter) of the finite volume approximants
of the IDS. Our framework includes quasiperiodic operators on Delone sets,
periodic and random operators on quasi-transitive graphs, and operators on
percolation graphs.
