The geometry of non-unit Pisot substitutions

Milton Minervino[1]; Jörg Thuswaldner[1]

  • [1] University of Leoben Department of Mathematics and Information Technology Chair of Mathematics and Statistics Franz-Josef-Strasse 18, A-8700 Leoben (Austria)

Annales de l’institut Fourier (2014)

  • Volume: 64, Issue: 4, page 1373-1417
  • ISSN: 0373-0956

Abstract

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It is known that with a non-unit Pisot substitution σ one can associate certain fractal tiles, so-called Rauzy fractals. In our setting, these fractals are subsets of a certain open subring of the adèle ring of the associated Pisot number field. We present several approaches on how to define Rauzy fractals and discuss the relations between them. In particular, we consider Rauzy fractals as the natural geometric objects of certain numeration systems, in terms of the dual of the one-dimensional realization of σ , and in the context of model sets for particular cut and project schemes. We also define stepped surfaces suited for non-unit Pisot substitutions. We provide basic topological and geometric properties of the Rauzy fractals, prove some tiling results for them, and provide relations to subshifts defined in terms of the periodic points of σ , to adic transformations, and a domain exchange.

How to cite

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Minervino, Milton, and Thuswaldner, Jörg. "The geometry of non-unit Pisot substitutions." Annales de l’institut Fourier 64.4 (2014): 1373-1417. <http://eudml.org/doc/275546>.

@article{Minervino2014,
abstract = {It is known that with a non-unit Pisot substitution $\sigma $ one can associate certain fractal tiles, so-called Rauzy fractals. In our setting, these fractals are subsets of a certain open subring of the adèle ring of the associated Pisot number field. We present several approaches on how to define Rauzy fractals and discuss the relations between them. In particular, we consider Rauzy fractals as the natural geometric objects of certain numeration systems, in terms of the dual of the one-dimensional realization of $\sigma $, and in the context of model sets for particular cut and project schemes. We also define stepped surfaces suited for non-unit Pisot substitutions. We provide basic topological and geometric properties of the Rauzy fractals, prove some tiling results for them, and provide relations to subshifts defined in terms of the periodic points of $\sigma $, to adic transformations, and a domain exchange.},
affiliation = {University of Leoben Department of Mathematics and Information Technology Chair of Mathematics and Statistics Franz-Josef-Strasse 18, A-8700 Leoben (Austria); University of Leoben Department of Mathematics and Information Technology Chair of Mathematics and Statistics Franz-Josef-Strasse 18, A-8700 Leoben (Austria)},
author = {Minervino, Milton, Thuswaldner, Jörg},
journal = {Annales de l’institut Fourier},
keywords = {Rauzy fractal; tiling; $p$-adic completion; beta-numeration; -adic completion},
language = {eng},
number = {4},
pages = {1373-1417},
publisher = {Association des Annales de l’institut Fourier},
title = {The geometry of non-unit Pisot substitutions},
url = {http://eudml.org/doc/275546},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Minervino, Milton
AU - Thuswaldner, Jörg
TI - The geometry of non-unit Pisot substitutions
JO - Annales de l’institut Fourier
PY - 2014
PB - Association des Annales de l’institut Fourier
VL - 64
IS - 4
SP - 1373
EP - 1417
AB - It is known that with a non-unit Pisot substitution $\sigma $ one can associate certain fractal tiles, so-called Rauzy fractals. In our setting, these fractals are subsets of a certain open subring of the adèle ring of the associated Pisot number field. We present several approaches on how to define Rauzy fractals and discuss the relations between them. In particular, we consider Rauzy fractals as the natural geometric objects of certain numeration systems, in terms of the dual of the one-dimensional realization of $\sigma $, and in the context of model sets for particular cut and project schemes. We also define stepped surfaces suited for non-unit Pisot substitutions. We provide basic topological and geometric properties of the Rauzy fractals, prove some tiling results for them, and provide relations to subshifts defined in terms of the periodic points of $\sigma $, to adic transformations, and a domain exchange.
LA - eng
KW - Rauzy fractal; tiling; $p$-adic completion; beta-numeration; -adic completion
UR - http://eudml.org/doc/275546
ER -

References

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