Topological structure of fractal tilings attached to number system

数论近似算法及其马尔可夫分解的自然推广为数论与符号动力系统的联系提供了线索。研究表明，Pisot数系的有限性及其弱形式在构造这种Beta展开式的自然扩张中起着重要作用。在列表的第一篇文章中，我们对所有具有这种有限性的三次Pisot单位进行了分类。在文献[6]中，证明了弱有限性保证了Beta展开式上的非重叠自仿射平铺。猜想所有的Pisot单位都具有弱有限性。另一方面，也发展了标准数系上的平铺理论。在与J.THuswaldner的第二篇文章中，我们对二次标准数系上的平铺的拓扑结构进行了深入的研究。如果一个瓦片的三点数为4或6，则该瓦片应该与圆盘同胚。最后，我们根据不可约多项式的系数对这些盘状情形进行了分类。在第五篇文章中，我们用A.Pethoe给出了标准数系的一个新的充分条件。当常数项足够大时，这一结果给出了一个尖锐的界限。Y.Tanigawa和S.Egami的其他论文致力于Euler-Zagier多重Zeta函数及其非负值的解析延拓。K.MatSumoto给出了它们的推广。我们希望这些研究将为定义与平铺相关的自然Zeta函数提供积极的想法。

Natural extension of number theoretical approximating algorithm and its Markov partition give clues to connect number theory and symbolic dynamical system. During this research period it is shown that finiteness and its weaker form of Pisot number system play important roles in constructing such natural extension of beta expansion. In the first paper of the list, we classified all cubic Pisot units having this finiteness property. In the 6-th paper, it is shown that weakly finiteness assures a non-overlapping self-affine tiling attached to beta expansions. It is conjectured that all Pisot units have weakly finiteness property. On the other hand, the theory of tilings attached to canonical number systems was also developed. In the 2-nd paper with J.Thuswaldner, we have studied closely topological structure of tilings attached to quadratic canonical number systems. If the number of triple points of a tile is 4 or 6 then the tile should be homeomorphic to a disk. We classified at last these disklike cases by coefficients of irreducible polynomials. The 5-th paper with A.Pethoe, we gave a new sufficient condition of canonical number system. When the constant term is sufficiently large, this result gives a sharp bound. Other papers with Y.Tanigawa and S.Egami are devoted to the analytic continuation of Euler-Zagier's multiple zeta function and their non-negative values. K.Matsumoto gave a generalization of them. We expect these researches would give positive ideas to define a natural zeta function attached to tilings.