Interior components of a tile associated to a quadratic canonical number system

Let α=−2+−1 be a root of the polynomial p(x)=x2+4x+5p(x)=x2+4x+5. It is well known that the pair (p(x),{0,1,2,3,4})(p(x),{0,1,2,3,4}) forms a canonical number system  , i.e., that each x∈Z[α]x∈Z[α] admits a finite representation of the shape x=a0+a1α+⋯+aℓαℓx=a0+a1α+⋯+aℓαℓ with ai∈{0,1,2,3,4}ai∈{0,1,2,3,4}. The set TT of points with integer part 0 in this number systemT:={∑i=1∞aiα−i,ai∈{0,1,2,3,4}} is called the fundamental domain   of this canonical number system. It has been studied extensively in the literature. Up to now it is known that it is a plane continuum with nonempty interior which induces a tiling of the plane. However, its interior is disconnected. In the present paper we describe some of (the closures of) the components of its interior as attractors of graph directed self-similar constructions. The associated graph can also be used in order to determine the Hausdorff dimension of the boundary of these components. Amazingly, this dimension is strictly smaller than the Hausdorff dimension of the boundary of TT.