Our EPSRC-funded research (£348,847) investigates the spectral properties of inflation tiling dynamical systems, using a novel renormalisation-based approach. The project team, led by Professor Uwe Grimm, comprises post-doctoral researchers Dr Yasushi Nagai (funded by EPSRC) and Dr Neil Mañibo (funded by a PRIME fellowship from the German Academic Exchange Service DAAD), as well as post-graduate student Ibai Aedo who is jointly supervised with Dr Ian Short. We collaborate closely with groups from the University of Bielefeld (Michael Baake, Franz Gähler), George Washington University (E. Arthur Robinson jr) and Vassar College (Natalie Priebe Frank).
Spectral properties provide a measure of the order in the system. In particular, many well-understood examples of Pisot type show pure point spectra, irrespective of whether you consider the dynamical spectrum or the closely related mathematical diffraction spectrum. The latter encodes the Fourier transform of the correlations in the structure, which due to the self-similarity of inflation tilings satisfies a set of recursive relations.
These relations give rise to Fourier matrix cocycles which can be analysed by means of their associated Lyapunov exponents, providing criteria for the presence or absence of spectral components in the diffraction spectrum, and hence in the dynamical spectrum. It is an interesting observation that Lyapunov exponents also play a key role in the analysis of spectral properties of aperiodic Schrödinger operators, and may thus provide a link between these different, and apparently unrelated, spectral properties.