The pair correlations of primitive inflation rules are analysed via their exact renormalisation relations. We introduce the inflation displacement algebra that is generated by the Fourier matrix of the inflation and deduce various consequences of its structure. Moreover, we derive a sufficient criterion for the absence of absolutely continuous diffraction components, as well as a necessary criterion for its presence. This is achieved via estimates for the Lyapunov exponents of the Fourier matrix cocycle of the inflation rule. While we develop the theory first for the classic setting in one dimension, we also present its extension to primitive inflation rules in higher dimensions with finitely many prototiles up to translations.

Abstract. The family of primitive binary substitutions defined by 1 → 0 → 01 m with m ∈ N is investigated. The spectral type of the corresponding diffraction measure is analysed for its geometric realisation with prototiles (intervals) of natural length. Apart from the wellknown Fibonacci inflation (m = 1), the inflation rules either have integer inflation factors, but non-constant length, or are of non-Pisot type. We show that all of them have singular diffraction, either of pure point type or essentially singular continuous.

A method of confirming the absence of absolutely continuous diffraction via the positivity of Lyapunov exponents derived from the corresponding Fourier matrices is presented, which provides an approach that is independent of previous results on the basis of Dekking's criterion. This yields a positive result for all constant length substitutions on a binary alphabet which are primitive and aperiodic.Comment: 12 page

We use generalised Zeckendorf representations of natural numbers to investigate mixing properties of symbolic dynamical systems. The systems we consider consist of bi-infinite sequences associated with so-called random substitutions. We focus on random substitutions associated with the Fibonacci, tribonacci and metallic mean numbers and take advantage of their respective numeration schemes.

We develop a general theory of continuous substitutions on compact Hausdorff alphabets. Focussing on implications of primitivity, we provide a self-contained introduction to the topological dynamics of their subshifts. We then reframe questions from ergodic theory in terms of spectral properties of the corresponding substitution operator. The standard Perron-Frobenius theory in finite dimensions is no longer applicable. To overcome this, we exploit the theory of positive operators on Banach lattices. As an application, we identify computable criteria that guarantee quasi-compactness of the substitution operator and hence unique ergodicity of the associated subshift.