The talk is devoted to the study of connectivity of fractal tilings related to well-known radix representations. We study graph-directed self-aﬃne tiles associated to β-expansions with respect to Pisot units. These tiles are examples of Rauzy fractals. The objective is the case of Pisot units of degree three and four. In the case of cubic Pisot units we prove that the associated Rauzy fractal is connected. In the quartic case we ﬁnd examples of disconnected Rauzy fractals. The existence of such examples is surprising, as the digits of quartic β-expansions are consecutive integers, which leads one to expect connected tiles. Furthermore we give a complete description of all connected as well as disconnected Rauzy fractals corresponding to quartic β-expansions. In the case of discontinuity, we prove that each tile contains inﬁnitely many connected components.
In order to treat the case of β-expansions, a characterization of Pisot numbers in terms of the coeﬃcients of their minimal polynomials is needed. So ﬁrst we present such characterization. The connectivity proof yields another interesting result: it provides the complete description of the representation of 1 for β-expansions with respect to quartic Pisot units.