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 ${\mathbf{L}}_{lim}(G,a)$ 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 ${\mathbf{L}}_{slip}(G,a)$, is also defined. In the case of the 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 $G$.

21 pages, 4 figures. Some minor corrections and a notation adjustment have been made