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

H\"older Continuity of the Spectra for Aperiodic Hamiltonians