On a class of Rauzy fractals without the finiteness property

Abstract

We present some topological and arithmetical aspects of a class of Rauzy fractals a,b related to the polynomials of the form Pa,b (x) = x3 − ax2 − bx − 1, where a and b are integers satisfying −a + 1 ≤ b ≤ −2. This class has the property that 0 lies on the boundary of a,b. We construct explicit finite automata that recognize the boundaries of these fractals. This allows to establish the number of neighbors of a,b in the tiling it generates. Furthermore, we prove that if 2a + 3b + 4 ≤ 0 then a,b is not homeomorphic to a topological disk. We also show that the boundary of the set 3,−2 is generated by two infinite iterated function systems.