Generalized Rauzy tilings and bounded remainder sets

Rauzy introduced a fractal set associted with the toric shift by the vector $(\beta^{-1},\beta^{-2})$, where $\beta$ is the real root of the equation $\beta^3=\beta^2+\beta+1$. He show that this fractal can be partitioned into three fractal sets that are bounded remaider sets with respect to the considered toric shift. Later, the introduced set was named as the Rauzy fractal. Further, many generalizations of Rauzy fractal are discovered. There are many applications of the generalized Rauzy fractals to problems in number theory, dynamical systems and combinatorics.
Zhuravlev propose an infinite sequence of tilings of the original Rauzy fractal and show that these tilings also consist of bounded remainder sets. In this paper we consider the problem of constructing similar tilings for the generalized Rauzy fractals associated with algebraic Pisot units.
We introduce an infinite sequence of tilings of the $d-1$-dimensional Rauzy fractals associated with the algebraic Pisot units of the degree $d$ into fractal sets of $d$ types. Each subsequent tiling is a subdivision of the previous one. Some results describing the self-similarity properties of the introduced tilings are proved.
Also, it is proved that the introduced tilings are so called generalized exchanding tilings with respect to some toric shift. In particular, the action of this shift on the tiling is reduced to exchanging of $d$ central tiles. As a corollary, we obtain that the Rauzy tiling of an arbitrary order consist of bounded remainder sets with respect to the considered toric shift.
In addition, some self-similarity property of the orbit of considered toric shift is established.