Skip to Main content Skip to Navigation
Journal articles

Connectedness of number theoretical tilings

Abstract : Let T=T(A,D) be a self-affine tile in \reals^n 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 \reals^3 and \reals^4, 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 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.
Document type :
Journal articles
Complete list of metadata

Cited literature [40 references]  Display  Hide  Download

https://hal.inria.fr/hal-00959042
Contributor : Service Ist Inria Sophia Antipolis-Méditerranée / I3s <>
Submitted on : Thursday, March 13, 2014 - 5:09:20 PM
Last modification on : Tuesday, August 13, 2019 - 2:34:02 PM
Long-term archiving on: : Friday, June 13, 2014 - 12:18:25 PM

File

dm070116.pdf
Files produced by the author(s)

Identifiers

  • HAL Id : hal-00959042, version 1

Collections

Citation

Shigeki Akiyama, Nertila Gjini. Connectedness of number theoretical tilings. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2005, 7, pp.269-312. ⟨hal-00959042⟩

Share

Metrics

Record views

182

Files downloads

887