Connectedness of number theoretical tilings (dual tilings generated by Pisot units of degree 3 and 4) |
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| Authors: | Shigeki Akiyama Nertila Gjini |
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| Institution: | 1. Department of Mathematics, Faculty of Science, Niigata University, Japan;2. University of New York Tirana, Rr.“Komuna e Parisit” Tirana, Albania |
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| Abstract: | We study the connectedness of Pisot dual tilings. It is shown that each tile generated by a Pisot unit of degree 3 is arcwise connected. However surprisingly, we found families of disconnected Pisot dual tiles of degree 4 which have infinitely many connected components. Also we give a simple necessary and sufficient condition for the connectedness of the Pisot dual tiles of degree 4. As a byproduct, we give a complete classification of the β expansion of 1 for quartic Pisot units. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) |
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