Analysis for pre-fractal structures in guasiperiodic filings

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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.