TY - BOOK

T1 - Tilings related to Number Systems and Substitutions

AU - Rossi, Lucia

N1 - no embargo

PY - 2023

Y1 - 2023

N2 - The present doctoral thesis contains results that can be grouped into two areas. The first one corresponds to digit systems and tilings. Lagarias and Wang considered integral self-affine tiles F = F(A, D) related to an expanding integer matrix A and a digit set D satisfying |D|= |det (A)|; these tiles are defined as attractors of an iterated function system, and in general have intricate shapes and a fractal boundary. They proved, under the assumption that F has positive measure, the existence of a tiling and a multiple tiling of R^n. Steiner and Thuswaldner introduced rational self-affine tiles associated with expanding algebraic numbers or, equivalently, expanding rational matrices with irreducible characteristic polynomials. They considered only a particular type of digit set, where D is obtained as a complete residue set modulo the base A: this is a strong assumption because it guarantees the existence of a tile of positive measure. The challenge of this theory is that rational self-affine tiles are not subsets of R^n, and instead are defined in a representation space which is a subring of a certain Adèle ring. A central part of our work consisted in proving analogous results to the ones by Lagarias and Wang, in the setting of Steiner and Thuswaldner: we considered an expanding rational matrix A (without the irreducibility restriction) and a set of digits D (not necessarily a residue set) which induced a rational self-affine tile F(A, D), and proved topological properties and the existence of a tiling and a lattice multiple tiling. The representation space in this more general case is defined in terms of a projective limit. On top of expanding the existing theory with new results, we made a thorough survey of the mathematical tools necessary to understand it, computed many specific examples in detail, and present multiple illustrations. The second area covered in this thesis is word combinatorics, in particular, we considered a family of infinite words over a two-letter alphabet that we called N-continued fraction sequences, obtained in terms of symbolic substitutions. They are related to N-continued fraction expansions of real numbers and constitute a generalization of Sturmian sequences. We prove combinatorial results of these sequences: in particular, we give balance constants and compute their complexity function, as well as other dynamical results. As a bridge between the two areas covered, at the end of the thesis we relate N-continued fraction sequences to Rauzy fractals, which are also self-affine sets. For non-unimodular substitutions, the representation space for the corresponding Rauzy fractals has a factor defined in terms of a projective limit which is analogous to the one used for rational self-affine tiles.

AB - The present doctoral thesis contains results that can be grouped into two areas. The first one corresponds to digit systems and tilings. Lagarias and Wang considered integral self-affine tiles F = F(A, D) related to an expanding integer matrix A and a digit set D satisfying |D|= |det (A)|; these tiles are defined as attractors of an iterated function system, and in general have intricate shapes and a fractal boundary. They proved, under the assumption that F has positive measure, the existence of a tiling and a multiple tiling of R^n. Steiner and Thuswaldner introduced rational self-affine tiles associated with expanding algebraic numbers or, equivalently, expanding rational matrices with irreducible characteristic polynomials. They considered only a particular type of digit set, where D is obtained as a complete residue set modulo the base A: this is a strong assumption because it guarantees the existence of a tile of positive measure. The challenge of this theory is that rational self-affine tiles are not subsets of R^n, and instead are defined in a representation space which is a subring of a certain Adèle ring. A central part of our work consisted in proving analogous results to the ones by Lagarias and Wang, in the setting of Steiner and Thuswaldner: we considered an expanding rational matrix A (without the irreducibility restriction) and a set of digits D (not necessarily a residue set) which induced a rational self-affine tile F(A, D), and proved topological properties and the existence of a tiling and a lattice multiple tiling. The representation space in this more general case is defined in terms of a projective limit. On top of expanding the existing theory with new results, we made a thorough survey of the mathematical tools necessary to understand it, computed many specific examples in detail, and present multiple illustrations. The second area covered in this thesis is word combinatorics, in particular, we considered a family of infinite words over a two-letter alphabet that we called N-continued fraction sequences, obtained in terms of symbolic substitutions. They are related to N-continued fraction expansions of real numbers and constitute a generalization of Sturmian sequences. We prove combinatorial results of these sequences: in particular, we give balance constants and compute their complexity function, as well as other dynamical results. As a bridge between the two areas covered, at the end of the thesis we relate N-continued fraction sequences to Rauzy fractals, which are also self-affine sets. For non-unimodular substitutions, the representation space for the corresponding Rauzy fractals has a factor defined in terms of a projective limit which is analogous to the one used for rational self-affine tiles.

KW - Number Systems

KW - Tilings

KW - Substitutions

KW - Radix Expansions

KW - Word Combinatorics

KW - Self-affine Sets

KW - Zahlensysteme

KW - Tilings

KW - Substitutionen

KW - Stellenwertentwicklungen

KW - Kettenbrüche

KW - Wortkombinatorik

KW - Selbstähnliche Mengen

U2 - 10.34901/mul.pub.2023.192

DO - 10.34901/mul.pub.2023.192

M3 - Doctoral Thesis

ER -