| author: | Shigeki Akiyama and Nertila Gjini |
|---|---|
| title: | Connectedness of number theoretical tilings |
| keywords: | Tile, Connectedness, Pisot number, number system |
| abstract: | Let T=T(A,D) be a self-affine tile in ℝ
defined by
an integral expanding matrix n A and a digit set D . In
connection with canonical number systems, we study connectedness
of T when D corresponds to the set of consecutive integers
{0,1,..., |det(A)|-1} . It is shown that in ℝ
and 3 ℝ , for any integral expanding matrix 4 A ,
T(A,D) is connected.
We also study the connectedness of Pisot dual tilings which play
an important role in the study of β -expansion, substitution
and symbolic dynamical system. It is shown that each tile
generated by a Pisot unit of degree 3 is arcwise connected. This
is naturally expected since the digit set consists of consecutive
integers as above. However surprisingly, we found families of
disconnected Pisot dual tiles of degree 4 . Also we give a simple
necessary and sufficient condition for the connectedness of the
Pisot dual tiles of degree 4 . As a byproduct, a complete
classification of the β -expansion of 1 for quartic Pisot
units is given. |
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| reference: | Shigeki Akiyama and Nertila Gjini (2005), Connectedness of number theoretical tilings, Discrete Mathematics and Theoretical Computer Science 7, pp. 269-312 |
| bibtex: | For a corresponding BibTeX entry, please consider our BibTeX-file. |
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| pdf-source: | dm070116.pdf (1009 K) |
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