Characterisation of spectra and wave functions of aperiodic crystal structures

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Abstract
The scale invariance of quasiperiodic and other aperiodic well ordered structures is discussed in connection with scale invariance of electrons and phonons in such systems. The scale properties are found from a multifractal analysis of spectra and wave functions. Multifractal behaviour is found also in systems without self-similarity in the structure. Discussed are quasiperiodic systems, and two other chains which are obtained with a substitution rule. Finally a discussion is given of the lattice dynamics of models for quasi-crystals. The density of states is calculated for a model of AlMn. There are indications that there are vibration eigenstates that are not extended.