Rauzy fractals and tilings

Milton Minervino

Research output: ThesisDoctoral Thesis

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Abstract

This thesis is about Rauzy fractals, geometric objects arising in the study of symbolic dynamical systems generated by Pisot substitutions. One of the main open problems in this field is translated geometrically to a tiling problem by Rauzy fractals. The main subject of this thesis is to describe the principal issues and to present new advances when going beyond the main hypotheses, namely unimodularity and irreducibility. We introduce several approaches on how to define Rauzy fractals and stepped surfaces for non-unit Pisot substitutions and discuss the relations between them. Special emphasis will be given to Dumont-Thomas numeration, duals of geometrical realizations of substitutions and model sets. We provide basic topological and geometric properties of Rauzy fractals, prove some tiling results for them, and provide relations to substitutive subshifts, to adic transformations, and a domain exchange. Several collections of tiles can be defined in the framework of (non-unit) Pisot beta-numeration. This includes an aperiodic and a periodic one made of Rauzy fractals, a periodic one induced by the natural extension of the beta-transformation and a Euclidean one made of integral beta-tiles. We show that all these collections are tilings if one of them is a tiling or, equivalently, the weak finiteness property or a spectral condition on the boundary graph hold. We also obtain new results on rational numbers with purely periodic beta-expansions. Finally we set up a geometrical theory for the study of the dynamics of reducible Pisot substitutions based on Rauzy fractals generated by duals of higher dimensional extensions of substitutions. Under certain assumptions we obtain geometric representations of stepped surfaces and related polygonal tilings, self-replicating and periodic tilings made of Rauzy fractals. We analyse the codings of a domain exchange defined on these fractal domains and we interpret them in a new combinatorial way.
Translated title of the contributionRauzy Fraktale und Pflasterungen
Original languageEnglish
QualificationDr.mont.
Supervisors/Advisors
  • Thuswaldner, Jörg, Assessor A (internal)
  • Tichy, Robert, Assessor B (external), External person
Publication statusPublished - 2014

Bibliographical note

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Keywords

  • Rauzy fractal
  • tiling
  • beta-numeration
  • Pisot conjecture

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