Interior components of a tile associated to a quadratic canonical number system
Interior components of a tile associated to a quadratic canonical number system
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DOI：
10.1016/j.topol.2007.10.007
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发表时间：
2008-03
影响因子：
0.6
通讯作者：
B. Loridant;J. Thuswaldner
中科院分区：
文献类型：
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作者：
B. Loridant;J. Thuswaldner
设α= - 2+ - 1是多项式p(x)=x2+4x+5的一个根。众所周知，对（p(x),{0,1,2,3,4}）形成一个正则数系统，即每个x∈Z[α]允许形状x=a0+a1α+⋯+a α α α与ai∈{0,1,2,3,4}的有限表示。这个数系统中整数部分为0的点的集合T称为这个正则数系统的基本定义域。它在文献中得到了广泛的研究。到目前为止，我们知道它是一个平面连续体，其内部是非空的，引起平面的平铺。然而，它的内部是不相连的。在本文中，我们描述了图向自相似结构的吸引子中它内部的一些分量的闭包。关联图也可以用来确定这些组件边界的豪斯多夫维数。令人惊奇的是，这个维数严格小于T边界的豪斯多夫维数。
Let α=−2+−1 be a root of the polynomial p(x)=x2+4x+5. It is well known that the pair (p(x),{0,1,2,3,4}) forms a canonical number system, i.e., that each x∈Z[α] admits a finite representation of the shape x=a0+a1α+⋯+aℓαℓwith ai∈{0,1,2,3,4}. The set T of points with integer part 0 in this number system 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 T.