On a class of Rauzy fractals without the finiteness property

We present some topological and arithmetical aspects of a class of Rauzy fractals ${\mathcal{R}}_{a,b}$ related to the polynomials of the form $P_{a,b}(x)=x^3-a{x}^2-bx-1$, where $a$ and $b$ are integers satisfying $-a+1 \leq b \leq -2$\mathrm{.}Thisclasshasthepropertythat$0$liesontheboundaryof$$\mathcal{R}$_{a,b}$. We construct explicit finite automata that recognize the boundaries of these fractals, which allows to establish the number of neighbors of ${\mathcal{R}}_{a,b}$. In particular, we prove that if $2a+3b+4\le 0$ then ${\mathcal{R}}_{a,b}$ is not homeomorphic to a topological disk. We also show that the boundary of the set ${\mathcal{R}}_{3,-2}$ is generated by two infinite iterated function systems.