Spectra of Discrete Schr\"odinger Operators with Primitive Invertible Substitution Potentials

We study the spectral properties of discrete Schr\"odinger operators with potentials given by primitive invertible substitution sequences (or by Sturmian sequences whose rotation angle has an eventually periodic continued fraction expansion, a strictly larger class than primitive invertible substitution sequences). It is known that operators from this family have spectra which are Cantor sets of zero Lebesgue measure. We show that the Hausdorff dimension of this set tends to $1$ as coupling constant $\lambda$ tends to $0$. Moreover, we also show that at small coupling constant, all gaps allowed by the gap labeling theorem are open and furthermore open linearly with respect to $\lambda$. Additionally, we show that, in the small coupling regime, the density of states measure for an operator in this family is exact dimensional. The dimension of the density of states measure is strictly smaller than the Hausdorff dimension of the spectrum and tends to $1$ as $\lambda$ tends to $0$

Tridiagonal substitution Hamiltonians

We consider a family of discrete Jacobi operators on the one-dimensional integer lattice with Laplacian and potential terms modulated by a primitive invertible two-letter substitution. We investigate the spectrum and the spectral type, the fractal structure and fractal dimensions of the spectrum, exact dimensionality of the integrated density of states, and the gap structure. We present a review of previous results, some applications, and open problems. Our investigation is based largely on the dynamics of trace maps. This work is an extension of similar results on Schroedinger operators, although some of the results that we obtain differ qualitatively and quantitatively from those for the Schoedinger operators. The nontrivialities of this extension lie in the dynamics of the associated trace map as one attempts to extend the trace map formalism from the Schroedinger cocycle to the Jacobi one. In fact, the Jacobi operators considered here are, in a sense, a test item, as many other models can be attacked via the same techniques, and we present an extensive discussion on this.Comment: 41 pages, 5 figures, 81 reference

Selfdual Substitutions in Dimension One

There are several notions of the 'dual' of a word/tile substitution. We show that the most common ones are equivalent for substitutions in dimension one, where we restrict ourselves to the case of two letters/tiles. Furthermore, we obtain necessary and sufficient arithmetic conditions for substitutions being selfdual in this case. Since many connections between the different notions of word/tile substitution are discussed, this paper may also serve as a survey paper on this topic.Comment: 28 pages, 5 figures. Several typos removed, some proofs shortened, thanks to the referees. The accepted version of this paper is shorter (22 pages, 4 figures), this arxiv version includes more examples, two appendices, plus a self-contained proof of Theorem 2.