与移位基数系统相关的分形图块
Fractal tiles associated with shift radix systems
摘要：
移位基数系统根据在$d$维实向量空间中变化的参数$\mathbf{r}$形成动力系统的集合。他们推广了著名的计数系统，如β-展开，关于有理基的展开，以及规范数系统。Beta计数和规范数系统与分形形状密切相关，例如经典的Rauzy分形和双龙。这些分形被证明是重要的研究性质的扩张在几个设置。在本文中，我们将一组分形瓦片与移位基数系统相关联。我们表明，对于某些类的参数$\mathbf{r}$这些瓷砖符合仿射副本的著名瓷砖与β-扩展和规范的数字系统。另一方面，这些瓦片为具有（非单位）皮索数的β-展开以及具有（非monic）展开多项式的规范数系统提供了瓦片的自然族。我们还证明了与移位基数系统瓷砖的基本属性。事实上，我们证明，在一些代数条件下的参数$\mathbf{r}$的移位基数系统，这些瓷砖提供多个平铺，甚至平铺的$d$维实向量空间。这些拼接原来有一个更复杂的结构比从上述已知的数字系统产生的拼接。这样的瓦片可以由具有无限多个不同形状的瓦片组成。此外，瓦片不需要是自仿射的（或图定向自仿射的）。
英文摘要：
Shift radix systems form a collection of dynamical systems depending on a parameter $\mathbf{r}$ which varies in the $d$-dimensional real vector space. They generalize well-known numeration systems such as beta-expansions, expansions with respect to rational bases, and canonical number systems. Beta-numeration and canonical number systems are known to be intimately related to fractal shapes, such as the classical Rauzy fractal and the twin dragon. These fractals turned out to be important for studying properties of expansions in several settings. In the present paper we associate a collection of fractal tiles with shift radix systems. We show that for certain classes of parameters $\mathbf{r}$ these tiles coincide with affine copies of the well-known tiles associated with beta-expansions and canonical number systems. On the other hand, these tiles provide natural families of tiles for beta-expansions with (non-unit) Pisot numbers as well as canonical number systems with (non-monic) expanding polynomials. We also prove basic properties for tiles associated with shift radix systems. Indeed, we prove that under some algebraic conditions on the parameter $\mathbf{r}$ of the shift radix system, these tiles provide multiple tilings and even tilings of the $d$-dimensional real vector space. These tilings turn out to have a more complicated structure than the tilings arising from the known number systems mentioned above. Such a tiling may consist of tiles having infinitely many different shapes. Moreover, the tiles need not be self-affine (or graph directed self-affine).
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