Lyapunov Exponents in the Spectral Theory of Primitive Inflation Systems
In this work, we consider primitive inflation rules as generators of aperiodic tilings, and subsequently, of aperiodic point sets (which are toy models for quasicrystals) deemed adequate for diffraction analysis. We harvest the combinatorial-geometric properties of these systems to obtain renormalisation relations for the pair correlation functions, which carry over to measures that generate the diffraction measure. This yields a measure-valued renormalisation satisfied by each of the components of the diffraction. Using tools from the theory of Lyapunov exponents, we provide a sufficient criterion to rule out the presence of absolutely continuous components in the diffraction and a necessary condition to have a non-trivial absolutely continuous part. Moreover, we provide a computable bound which one can use to use invoke this criterion. We show that this holds for large classes of systems, and, as a sanity check, show that the necessary criterion for existence is satisfied by systems which are a priori known to have absolutely continuous diffraction. Furthermore, we present the recovery of known singularity results and point out connections to number-theoretic quantities which naturally arise from these objects, such as logarithmic Mahler measures.
Universität Bielefeld
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