Characterizations of the quasi-periodicity in the quasi-crystal structure

研究结果总结如下：1.Ammann-Beenker瓦片是平面的准周期瓦片，它是由Ammann匹配规则构造的。证明了Ammann-Beenker瓦片可以由一个具有4个状态的自动机组成，并从符号动力学的角度指出了有关合成序列的一些结果。2.在格的正交投影的限制映射是内射的假设下，我们确定了由投影法得到的两个瓦片何时属于同一同构类。3.证明了(2n+1)维mod3标准透镜空间的切丛对于任意m=n或m&gt；n是稳定可扩到(2m+1)维mod3标准透镜空间当且仅当n=0，3或0&lt；n&lt；3$。4.得到了透镜空间上R-向量丛的稳定不可扩张性的一个定理，改进了一些结果，研究了稳定可扩性与透镜空间上向量丛的跨度之间的关系，证明了复化对每个m&gt；n是可扩化的N当且仅当n=0，5或0&lt；n&lt；5，并证明了张量积的复化对于每一个m&gt；n是可扩张的当且仅当0&lt；n&lt；13或n=0，13，15.5.我们研究了由匹配规则构造的PenRose瓷砖的结构，给出了瓷砖的局部构型、非周期性的初等证明、局部同构性质、不可数性以及由匹配规则得到的全部可通过自上而下生成的事实.

The summary of research results is as follows.1.The Ammann-Beenker tilings are quasiperiodic tilings of the plane, which is constructed by using the Ammann's matching rules. We show that the Ammann-Beenker tilings can be composed by an automaton with 4 states, and note some results concerning composition sequences from the viewpoint of symbolic dynamics.2.Under the assumption that the restriction map of the orthogonal projection to a lattice is injective, we determine when two tilings obtained by the projection method belong to the same isomorphism class. As its application we have uncountably many isomorphism classes of quasiperiodic tilings by the projection method.3.We prove that the tangent bundle of the (2n+1)-dimensional mod 3 standard lens space is stably extendible to the (2m+1)-dimensional mod 3 standard lens space for every m=n or m>n if and only if n=0,3 or 0<n<3 $.4.We obtain a theorem on stable unextendibility of R-vector bundles over lens space improving some results, study relations between stable extendibility and span of vector bundles over lens space, and prove that the complexification is extendible for every m>n if and only if n=0,5 or 0<n<5, and prove that the complexification of the tensor product is extendible for every m>n if and only if 0<n<13 or n=0,13,15.5.We study the structure of the Penrose tiling constructed by the matching rule, and a substitution rule, which gives us the local configuration of the tiles, the elementary proofs of the aperiodicity, the locally isomorphic property, the uncountability and the fact that all obtained by the matching rule can be constructed via the up-down generation.