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

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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.