INDEXING THE DIFFRACTION SPECTRUM OF A NON-PISOT SELF-SIMILAR STRUCTURE

We study the Fourier transform, or diffraction spectrum, of a self-similar structure in one dimension, generated by a substitution rule, which does not possess the Pisot property. Several consequences of this number-theoretical property are investigated. The structure exhibits unbounded density fluctuations with respect to its average lattice, which diverge as a power of the system size. The diffraction spectrum consists neither of Bragg peaks, nor of diffuse scattering; it is a singular continuous and multifractal measure. The Fourier transform exhibits peaks of singular scattering and is shown to possess scaling properties-in a weak sense-around those values of the wave vector. These local properties are analyzed in detail and put in perspective with open questions concerning the chaotic behavior of a generalized Arnold-Sinai cat map. We argue that the peaks of the Fourier intensity cannot be indexed by any simple scheme, such as a finite number of integers. In that respect, the present structure is therefore different from periodic or quasiperiodic structures and from most singular continuous cases studied previously.