Pisot substitutions and beyond.

*(English)*Zbl 1210.93006
Bielefeld: Univ. Bielefeld, Fakultät für Mathematik. 445 p. (2006).

Summary: This thesis consists (mainly) of three parts: At first, we define Hausdorff measures for product spaces of local fields. We look at iterated function systems on such spaces, the Hausdorff dimension of their attractor is estimated. Afterwards, model sets (respectively cut and project sets) are introduced. For point sets generated by a substitution rule, we state criteria that ensure that the point set in question is a model set. These results are applied to Pisot substitutions. It is conjectured that all (one-dimensional) point sets generated by a Pisot substitution are model sets. A list of equivalent statements for this conjecture is given.
In the first part of Chapter 4 we show that the Haar measure on a product space of local fields is a Hausdorff measure (Theorem 4.56). Then, iterated function systems and their attractors are introduced. Generalising results of K. J. Falconer (on \(R^n\)), upper (Prop. 4.122) and in some cases lower (Lemma 4.126 & Props. 4.127 & 4.129) bounds for the Hausdorff dimension (as well as the box-counting dimension (Lemma 4.133)) for these attractors are derived.

The main theme of Chapter 5 are model sets (cut and project sets) and Delone point sets that are generated by a substitution, and their interplay. Following Baake-Moody, the construction of a cut and project scheme for a multi-component Delone set is derived in Section 5.3. This construction is extended and modified for substitution point sets (Section 5.7.3) if there is a so-called algebraic or overlap coincidence; indeed, one obtains an extended internal space here (Prop. 5.137). The main theorem of this chapter is Theorem 5.154 which states equivalent conditions under which a substitution point set is a model set: Either the substitution has an algebraic or overlap coincidence, or there exist certain aperiodic tilings of the (extended) internal space.

These results are applied to sequences generated by Pisot substitutions in Chapter 6 (Theorem 6.77). However, one can derive additional equivalent conditions, e.g., that a certain periodic tiling of the internal space exists (Prop. 6.72), or that the so-called “Geometric Coincidence Condition” (GCC) is satisfied (for the latter, we also state a graph-theoretic formulation, see Section 6.9). The complete list – and thus the central theorem of this thesis – of all equivalent conditions is given in Theorem 6.116. Here, we have also included statements from Chapter 7. In that chapter, we mainly summarise results about the diffraction measure of Delone sets, the spectrum of the dynamical system generated by a Delone set and the torus parametrisation for a model set already present in the literature. To contextualize the achieved results, we look at and apply our methods to “visible lattice points” (Chapter 5a), tilings of the plane (Chapter 6a), lattice substitution systems (Chapter 6b) and reducible Pisot substitutions and [beta]-substitutions (Chapter 6c).

The main theme of Chapter 5 are model sets (cut and project sets) and Delone point sets that are generated by a substitution, and their interplay. Following Baake-Moody, the construction of a cut and project scheme for a multi-component Delone set is derived in Section 5.3. This construction is extended and modified for substitution point sets (Section 5.7.3) if there is a so-called algebraic or overlap coincidence; indeed, one obtains an extended internal space here (Prop. 5.137). The main theorem of this chapter is Theorem 5.154 which states equivalent conditions under which a substitution point set is a model set: Either the substitution has an algebraic or overlap coincidence, or there exist certain aperiodic tilings of the (extended) internal space.

These results are applied to sequences generated by Pisot substitutions in Chapter 6 (Theorem 6.77). However, one can derive additional equivalent conditions, e.g., that a certain periodic tiling of the internal space exists (Prop. 6.72), or that the so-called “Geometric Coincidence Condition” (GCC) is satisfied (for the latter, we also state a graph-theoretic formulation, see Section 6.9). The complete list – and thus the central theorem of this thesis – of all equivalent conditions is given in Theorem 6.116. Here, we have also included statements from Chapter 7. In that chapter, we mainly summarise results about the diffraction measure of Delone sets, the spectrum of the dynamical system generated by a Delone set and the torus parametrisation for a model set already present in the literature. To contextualize the achieved results, we look at and apply our methods to “visible lattice points” (Chapter 5a), tilings of the plane (Chapter 6a), lattice substitution systems (Chapter 6b) and reducible Pisot substitutions and [beta]-substitutions (Chapter 6c).