90 research outputs found
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 as coupling constant tends to . 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 .
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 as tends to
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.
Connectedness of fractals associated with Arnoux-Rauzy substitutions
Rauzy fractals are compact sets with fractal boundary that can be associated
with any unimodular Pisot irreducible substitution. These fractals can be
defined as the Hausdorff limit of a sequence of compact sets, where each set is
a renormalized projection of a finite union of faces of unit cubes. We exploit
this combinatorial definition to prove the connectedness of the Rauzy fractal
associated with any finite product of three-letter Arnoux-Rauzy substitutions.Comment: 15 pages, v2 includes minor corrections to match the published
versio
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