Analysis for pre-fractal structures in guasiperiodic filings

研究结果总结如下1.我们在2维和更高维的绿洲中构造了一个序列紧的spam图块。利用这个构造，我们分析了由替换规则得到的图块的某些对称性，我们在一些例子中明确给出了具有旋转对称性的图块的构造（Penrose图块，Ammann-Beenker图块，Danzer图块）。给出了一个n元环烃分子的数学模型，并研究了该模型的构型空间的拓扑结构。在假定环状分子所需条件的前提下，证明了当n> 4时，环状分子的构型空间是n-4维球面。这一结果对n= 5，6.3时n元环烃分子的构型空间给出了合理的解释。我们考虑了由特征值和相应的局部化中心组成的点过程，并证明了在无限体积极限下它收敛于Poisson点过程.研究了一维无理旋转的单位区间任意分解编码问题。编码序列具有递归更新的结构，当这个过程用平稳过程描述时，我们进行分类。证明了当且仅当所有分解点、初始值和旋转速度都在同一个真实的二次域上时，才会发生这种情况.我们研究了受激拉曼散射区边界上的点是否总是产生最终周期轨道。值得注意的是，我们证明了这种周期性是有效的黄金平均值的情况下。这项研究扩展到最近对具有自包含结构的离散旋转的进一步研究.得到了（2n+1）维模3标准透镜空间上R-向量丛的稳定不可扩性定理，改进了某些已知结果.

The summary of research results is as follows1. We construct a sequentially compact spam of patches in 2 and higher dimensional oases By using this constnction, we analyze certain symmetries of tilings obtained by substitution rules we give explicitly the construction of tilings with rotational symmetries in some examples(a Penrose tiling, an Ammann-Beenker tiling a Danzer tiling).2. We give a mathematical model of n-membered ringed hydrocarbon molecules, and study the topology of a configuration space of the model. Assuming requited conditions for ringed molecules, we prove that its configuration space is diffeomorphic to (n-4)-dimensional sphere for n> 4. This result gives an appropriate explanation of the configuration space of n-membered ringed hydrocarbon molecules when n= 5, 6.3. We consider the point process composed of the eigenvalues and corresponding localization centers, and showed that in the infinite volume limit it converges to the Poisson point process.4. We studied the coding of 1-dimensional irrational rotation with arbitray decomposition of the unit interval. The coded sequence has recursively renewable structure, and we then classify when this procedure is described by stationaly process. It tured out that this happens when and only when all decomposition points, the initial value and the speed of rotations are in the same real quadratic field.5. We studied the points on the boundary of SRS region whether they always produces eventually periodic orbits or not. Remarkably we prove that this periodicity is valid in the case of golden mean. This study extends to recent further researches on discretized rotation having self-indudng structures.6. We obtain a theorem on stable unextendibility of R-vector bundles over (2n+1)-dimensional standard lens space mod 3 improving some known results.