Symbolic Dynamical System and Number Theoretical Tilings

数论逼近算法的自然推广及其马尔可夫划分为数论与符号动力系统的联系提供了线索。在此研究过程中，证明了毕索数系统的弱有限性等价于上述构造中必不可少的不重叠平铺性质。在列表的第五篇论文中，我们证明了所有的三次Pisot单位都具有这种弱有限性。西多罗夫推测所有的皮索数都具有弱有限性质。另一方面，与正则数系统相关的平铺理论也得到了发展。在第一篇与H.Rao的论文中，研究了正则数系集合中有限性的一个变型，并给出了具有此性质的一族。我们得到了移位基系统的一个重要发展，它推广了已知的数系统，并提供了一类好的数系统的模空间。在这个问题上，与A.Pethoe、J.Thuswaldnder、H.Brunotte、K.Scheicher和W.Steiner的几个联合项目正在进行中。包括N.Gjini和j.o o o和j.s thuswaldner在内的几篇关于瓷砖拓扑的研究发表了。正则数系统的分类与幂积分基问题有关，这一问题是由t.a taku共同研究开始的，由Sh.Ito和p.a noux提出的取代动力学与该研究有很强的联系。它的代数结构允许同调研究。在H.Yoshihara的帮助下，我们正在尝试扩展这种方法。与Y.Tanigawa一起，我们研究了给定的与Pisot数密切相关的Salem数的幂的分数部分。引入与非周期结构相关的ζ函数是很有诱惑力的，我们正在与K.Matsumoto合作。日本数学学会出版的《Sugaku》上刊登了一份综合调查报告。

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 weak finiteness of Pisot number system is equivalent to the non-overlapping tiling property, which is essential in above construction. In the fifth paper of the list, we proved all cubic Pisot units have this weak finiteness. It is conjectured by N.Sidorov that all Pisot numbers have weakly finiteness property.On the other hand, the theory of tilings associated to canonical number systems was also developed. In the first paper with H.Rao, a variant of finiteness is studied in canonical number system setting and we gave some family with this property. One important development has been made that we reached to the concept of shift radix system, which generalize known number systems and provide us a kind of moduli space of good number systems. On this subject, several joint projects with A.Pethoe, J.Thuswaldnder, H.Brunotte, K.Scheicher and W.Steiner are on going. Several topological studies of tiles are published including the one with N.Gjini and the one with J.Luo and J.Thuswaldner. The classification of canonical number system is linked to the power integral basis problem on which a joint study had begun with T.TakeuchiSubstitution dynamics introduced by Sh.Ito and P.Arnoux has a strong connection to this study. Its algebraic structure permits homological study. With the help of H.Yoshihara, we are trying to extend of this method. With Y.Tanigawa, we studied the fractional part of powers of a given Salem number which intimately related to Pisot numbers. It is tempting to introduce zeta functions associated to aperiodic structures on which we are collaborating with K.Matsumoto. A comprehensive survey was written in ‘Sugaku' published by the Japanese Mathematical Society.