Connectedness of number theoretical tilings (dual tilings generated by Pisot units of degree 3 and 4)
Shigeki Akiyama, Nertila Gjini · PAMM · 2007
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)