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Discrete Mathematics and Theoretical Computer Science DMTCS vol. 7, 2005, 269312 Connectedness of number theoretic tilings
 

Summary: Discrete Mathematics and Theoretical Computer Science DMTCS vol. 7, 2005, 269312
Connectedness of number theoretic tilings
Shigeki Akiyama1
and Nertila Gjini2
1
Department of Mathematics, Faculty of Science, Niigata University, Ikarashi 2-8050, Niigata 950-2181, Japan,
e-mail address: akiyama@math.sc.niigata-u.ac.jp
2
Department of Mathematics, University of New York Tirana, Rr. Komuna e Parisit, Tirana, Albania,
e-mail address: ngjini@unyt.edu.al
received May 26, 2003, accepted Nov 14, 2005.
Let T = T(A, D) be a self-affine tile in Rn
defined by an integral expanding matrix 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 R3
and R4
, for any integral expanding matrix A, T(A, D) is connected.
We also study the connectedness of Pisot dual tilings which play an important role in the study of -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

  

Source: Akiyama, Shigeki - Department of Mathematics, Niigata University

 

Collections: Mathematics