The localization properties of non-interacting linear excitations in one-dimensional aperiodically ordered structures are investigated from a theoretical point of view. The models used have various relevance for real systems, like quasicrystals, photonic crystals, and deterministic aperiodic superlattices. The main objective is to gain a conceptual understanding of the localization phenomenon in different lattice models, especially with respect to their correlation measures.

The localization properties of electronic wavefunctions in various nearest neighbor tight -binding models are studied in the framework of the dynamical systems induced by the trace maps of their corresponding transfer matrices. With a unit hopping and an on-site potential modulated by the Rudin-Shapiro sequence, which in analogy with a random potential has an absolutely continuous correlation measure, the electronic spectrum is proved to be purely singular continuous and of zero Lebesgue measure. The absence of localization is also confirmed by numerical simulations of the dynamics of electronic wavepackets showing weakly anomalous diffusion and an algebraic decay of the temporal autocorrelation function. These results are also found to be invariant under the introduction of correlated hopping integrals.

The nature of localization of elastic vibrations in harmonic lattices is also studied. The generalized eigenvalue problem arising from classical interactions in diatomic chains can be mapped to mixed tight-binding models, which enables the use of the spectral theory of discrete Schrödinger operators. Like for the Rudin-Shapiro model, it is found that the vibrational spectra of harmonic chains with masses distributed according to the Thue-Morse sequence and the period-doubling sequence are purely singular continuous. These results are obtained by transforming the lattices to on-site models by the use of certain renormalization procedures.

Remembering that the correlation measure of the T hue-Morse sequence is purely singular continuous, while that of the period-doubling sequence is pure point, these results strongly suggest that the criticality of localization in deterministic aperiodic lattices is generic and quite independent of the character of the correlation measure associated to the modeling sequence.