ON A CLASS OF RAUZY FRACTALS WITHOUT THE FINITENESS PROPERTY

Abstract

We present some topological and arithmetical aspects of a class of Rauzy fractals R-a,R-b related to the polynomials of the form P-a,P-b(x) = x(3) - ax(2) - 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 R-a,R-b. We construct explicit finite automata that recognize the boundaries of these fractals. This allows to establish the number of neighbors of R-a,R-b in the tiling it generates. Furthermore, we prove that if 2a + 3b + 4 <= 0 then R-a,R-b is not homeomorphic to a topological disk. We also show that the boundary of the set R-3,R--2 is generated by two infinite iterated function systems.