arXiv subjects
Publications and source records attributed to Shousuke Ohmori.
We discuss here geometric structures of condensed matters by means of a fundamental topological method. Any geometric pattern can be universally represented by a decomposition space of a topological space consisting of the infinite product space of 0 and 1, in which a partition with a specific topological structure determines a character of each geometric structure.
We discuss Devaney chaos on compact metric spaces using a decomposition space characterized by topological nature of symbolic dynamics. A chaotic map obtained here is defined as a topologically conjugate of the chaotic map on a decomposition space which is induced by a chaotic map of symbolic dynamics. In particular, the chaotic character of the tent map and the baker map on [0,1] are reconsidered based on decomposition dynamics involving symbolic dynamics with different two chaotic maps. As an example of compact metric space we exhibit a chaotic map existing on any given finite graph.
Two distinct structures of aggregates of atoms connected by anisotropic bonds with a network configuration are discussed from the viewpoint of a point set topology. A specific topological space connects the two types of topological structures based on the equivalence classes, which show the main difference of the structures between the aggregates of atoms with or without a clusterized structure.
We discuss a sufficient condition for a space to be filled with an arbitrary finite number of self-similar spaces using a topological concept.
A Lambda (Card Lambda > aleph)-product space of {0,1} has a partition {X1,...,Xn} for any n a decomposition space of each Xi of which is self-similar.(February 16, 2015)
A partition of a $Λ(CardΛ\succ \aleph_0)$-product space of $\{0,1\}$ defines an abstract polycrystal composed of abstract singlecrystals a decomposition space (space of equivalence classes) of each of which is self-similar.
Both Cantor middle-third set and Sierpiński carpet are self-similar, perfect, compact metric spaces. In spite of the similarity of the mathematical procedure of construction, there exists between them a fundamental difference in topological nature, and this difference affects the methods of construction of an interesting non-trivial quotient space of them. The totally disconnectedness (or, more generally, zero-dimensional) enables Cantor middle-third set to have a non-trivial quotient space which is self-similar. On the other hand, concerning Sierpiński carpet, because of the connectedness of its structure, no non-trivial quotient space which is self-similar can be constructed by such an elegant procedure as that for Cantor middle-third set. Various topologically significant nature specific to Cantor middle-third set owe mainly to the totally disconnectedness of the set.
The existence of a decomposition space with a dendritic structure of a topological space $(\{0,1\}^Λ,τ_{0}^Λ)$ is discussed. Here, $Λ$ is any set with the cardinal number $\succ \aleph , \{0,1\}^{Λ}=\{φ:Λ\rightarrow \{0,1\}\}, τ_0$ is the discrete topology for $\{0,1\}$ and the topology $τ_0^{Λ}$ for $\{0,1\}^Λ$ is the topology with the base $β=\{ ~;~G_{λ_1}\in τ_0,\dots,G_{λ_n}\in τ_0, \{λ_1,\dots,λ_n\}\subset Λ,n\in {\bf N}\}$ where the notation $ $ concerning the subset $E_{λ_i}, i=1,\dots,n$ of $\{0,1\}$ denotes the set $\{φ:Λ\rightarrow \{0,1\}~;~φ(λ_1)\in E_{λ_1},\dots,φ(λ_n)\in E_{λ_n}, φ(λ)\in \{0,1\}, λ\in Λ-\{λ_1,\dots,λ_n\}\}$.
A topological space homeomorphic to a self-similar space is demonstrated to be self-similar. There exists a self-similar space $S$ whose coarse graining is homeomorphic to $S$. The coarse graining of $S$ is, therefore, self-similar again. In the same way, the coarse graining of the self-similar coarse graining of $S$ is, furthermore, self-similar. These situations succeed endlessly. Such a self-similar $S$ is generated actually from an intense quadratic dynamics.