Analysis for pre-fractal structures in guasiperiodic filings

研究成果总结如下：1.研究成果综述。我们在2维及更高维绿洲中构造了一个由面片组成的序列紧Spam。利用这个构造，我们分析了由置换规则得到的镶嵌的某些对称性，并在一些例子(Penrose瓦片、Ammann-Beenker瓦片和Danzer瓦片)中明确地给出了具有旋转对称性的瓦片的构造。给出了n元环烃分子的数学模型，并研究了该模型构型空间的拓扑结构。在假定环状分子满足条件的情况下，证明了n-4维环烃分子的构型空间与(n-4)维球面同构。这一结果较好地解释了n=5，6.3时n元环状碳氢分子的构型空间。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到泊松点过程。研究了单位区间任意分解的一维无理旋转编码问题。编码序列具有递归更新结构，当该过程被平稳过程描述时，我们对其进行分类。结果表明，当且仅当所有分解点、初值和旋转速度都在同一实二次域中时，才会发生这种情况。我们研究了SRS区域边界上的点是否总是产生最终的周期轨道。值得注意的是，我们证明了这种周期性在中值情形下是有效的。这一研究延伸到了最近对具有自约束结构的离散化旋转的进一步研究。我们得到了(2n+1)维标准透镜空间mod3上R-向量丛的稳定不可扩性的一个定理，改进了一些已知结果。

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.