Analysis of substitution and its application

1)在假设置换满足(1)么模、Pisot、不可约条件的前提下，我们成功地证明了存在称为原子表面的集族，原子表面上的动力系统，称为区域交换变换，与环面上的旋转同构。2)在替换不满足不可约条件的情况下，我们得到了区域交换变换不同构于旋转，但它与旋转的诱导变换同构。3)我们对不满足Pisot条件的替换进行了深入的研究，并成功地得到了结果(见参考文献11)。“与非Pisot矩阵相关的瓦片”(Anial.Instiiut傅立叶，格勒诺布尔(2006)与M.Furukado和E.Arhr。小罗宾逊)

1)On the assumption that the substitution satifies (1) unimodular, Pisot, irreducible condition, we succeeded to prove that, there exists the family of sets called atomic surfaces and the dynamical system on the atomic surfaces, which is called domain exchange transformation, is isomorphic to a rotation on torus.2)In the case that substitution does not satisfy the irreducible condition, we have the result that the domain exchange transformation is not isomorphic to the rotation, but it is isomorphic to the induced transformation of the rotation.3)We are going to study deeply the substitution which is not satisfied the Pisot condition, and we succeeded to obtain the result (see 11. REFERENCES, "Tilings Associated with non-Pisot Matrices"(Annal. Instiiut Fourier, Grenoble (2006) with M. Furukado and E. Arhour. ROBINSON, Jr.)