Boundary parametrization of planar self-affine tiles with collinear digit set

Abstract

We consider a class of planar self-affine tiles T = M −1 a∈D (T + a) generated by an expanding integral matrix M and a collinear digit set D as follows:We give a parametrization S 1 → ∂T of the boundary of T with the following standard properties. It is Hölder continuous and associated with a sequence of simple closed polygonal approximations whose vertices lie on ∂T and have algebraic preimages. We derive a new proof that T is homeomorphic to a disk if and only if 2|A| |B + 2|.

Topology of a class of p2-crystallographicreplication tiles

“…[3]). This also holds for the tiles T ℓ with A ≤ 0, as it is mentioned in [1] that changing A to −A, for a fixed B, results in an isometric transformation for the associated tiles T ℓ (see Equation (6.1)). Therefore, by Lemma 2.4, we just need to show that T and c(T ) = −T do not overlap.…”

Boundary Parametrization and the Topology of Tiles

“…, u 6 ) associated to β, that can be chosen such that u 1 + • • • + u 6 = 1. This is sufficient to perform the parametrization procedure (see [AL10] for more details). We just need to check the compatibility conditions of Definition 2.3 with p = 12, together with the additional compatibility condition (6.4) ψ(6; o m ) = ψ(1; 1).…”

Representations for complex numbers with integer digits

“…The setting fits into the framework of rotational beta-expansions recently introduced in [3,4] as a way to generalise the real (one-dimensional) beta-expansion to higher dimensions where the focus is set on ergodicity, soficness and invariant probability measures. Therefore, we will not discuss these topics here but refer to [3,4]. We rather concentrate on the special feature of the zeta-transformation, namely, that, due to the particular shape of our domain D, the produced digit sequences consist of integers only.…”