Linear zero mode spectra for quasicrystals

Abstract

A converse is given to the well-known fact that a hyperplane localised zero mode of a crystallographic bar-joint framework gives rise to a line or lines in the zero mode (RUM) spectrum. These connections motivate definitions of linear zero mode spectra for an aperiodic bar-joint framework G that are based on relatively dense sets of linearly localised flexes. For a Delone framework in the plane the limit spectrum L lim (G, a) is defined in this way, as a subset of the reciprocal space for a reference basis a of the ambient space. A smaller spectrum, the slippage spectrum L slip (G, a), is also defined. For quasicrystal parallelogram frameworks associated with regular multi-grids, in the sense of de Bruijn and Beenker, these spectra coincide and are determined in terms of the geometry of G.