Albanian Journal of Mathematics (ISNN: 1930-1235), Vol 6, No 1 (2012)

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Pisot dual tilings of low degree and their disconnectedness

Nertila Gjini


We study the connectedness of the graph-directed self-affine tiles associated to β-expansions , called Pisot dual tilings. These tiles are examples of Rauzy fractals and play an important role in the study of β-expansion, substitution and symbolic dynamical system. Using the complete classification of the β-expansion of 1 for quartic Pisot units and the classification of the connected tilings given in [4] and [5], here we continue studying connectedness of Pisot dual tilings generated by a Pisot unit with integral minimal equation x4 - ax3 - bx2 - cx - 1 = 0 in the special case when a + c - 2⌊β⌋ = 1. It is shown that every tile is disconnected having infinitely many connected components.

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