  We study the spectral location of strongly pattern equivariant Hamiltonians
arising through configurations on a colored lattice. Roughly speaking, two
configurations are "close to each other" if, up to a translation, they "almost
coincide" on a large fixed ball. The larger this ball is, the more similar they
are, and this induces a metric on the space of the corresponding dynamical
systems. Our main result states that the map which sends a given configuration
into the spectrum of its associated Hamiltonian, is H\"older (even Lipschitz)
continuous in the usual Hausdorff metric. Specifically, the spectral distance
of two Hamiltonians is estimated by the distance of the corresponding dynamical
systems.
