INTERIOR COMPONENTS OF A TILE ASSOCIATED TO A QUADRATIC CANONICAL NUMBER SYSTEM — PART II

Abstract

Let [Formula: see text] be a root of the polynomial p(x) = x2+ 4x + 5. It is well-known that the pair (α, {0, 1, 2, 3, 4}) forms a canonical number system, i.e., that each γ ∈ ℤ[α] admits a finite representation of the shape γ = a0+ a1α + ⋯ + aℓαℓwith ai∈ {0, 1, 2, 3, 4}. The set [Formula: see text] of points with integer part 0 in this number system [Formula: see text] is called the fundamental domain of this canonical number system. It is a plane continuum with nonempty interior which induces a tiling of ℂ. Moreover, it has a disconnected interior [Formula: see text]. In the first paper of this series we described the closures C0, C1, C2and C3of the four largest components of [Formula: see text] as attractors of graph-directed self-similar sets. Each of these four sets is a translation of C0. We conjectured that the closures of the other components are images of C0by similarity transformations. In this article we prove this conjecture. Moreover, we provide a graph from which the suitable similarities can be read off.