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Takehiko Mori

Publications and source records attributed to Takehiko Mori.

5 recordsLinked to original sources
arXiv2025

Dynamical Systems with Bounded Condition and $C^{*}$-algebras

In this paper, we study abstract dynamical systems with discrete phase spaces. One example of such a system is induced by the $3 x{+}1$-map on the set of all natural numbers, also known as the Collatz map. Our main focus is on dynamical systems induced by maps on countable discrete sets that satisfy a bounded condition. When these maps satisfy the bounded and a separating conditions, a minimality of the induced dynamical systems is equivalent to the irreducibility of certain $C^{*}$-algebras on certain Hilbert spaces. For a map $f$ on a general discrete phase space, we consider $f$-invariant sets and investigate their properties. When the phase space is countable and the map satisfies the bounded condition, we construct an order-preserving injection from the family of $f$-invariant sets to the family of reducing subspaces for the corresponding $C^{*}$-algebra. By introducing the totally uniqueness condition for $f$, we show that this injection is a bijection if $f$ satisfies this condition. This condition is crucial in providing a symbolic representation of the dynamical system induced by $f$, and we discuss the relationship between this symbolic representation and that of a topological dynamical system.

math.OA↗
arXiv2025

Cup and cap products for cohomology and homology groups of ample groupoids

This paper explores the cup and cap products within the cohomology and homology groups of ample groupoids, focusing on their applications and fundamental properties. Ample groupoids, which are étale groupoids with a totally disconnected unit space, play a crucial role in the study of topological dynamical systems and operator algebras. We introduce the cup product, which defines a bilinear map on cohomology classes, providing a graded ring structure, and the cap product, which defines a bilinear map relating homology and cohomology. The paper aims to make these concepts accessible to a broader mathematical audience, offering clear definitions and detailed explanations. We also demonstrate an application of the cap product in the analysis of automorphisms of groupoid $C^*$-algebras. Specifically, we show how it helps determine the asymptotic innerness of automorphisms. Our results include the first explicit computations of cup products in the cohomology of tiling spaces, which may pave the way for new research in this area.

math.OA↗
arXiv2025

Application of Operator Theory for the Collatz Conjecture

The Collatz map (or the $3n{+}1$-map) $f$ is defined on positive integers by setting $f(n)$ equal to $3n+1$ when $n$ is odd and $n/2$ when $n$ is even. The Collatz conjecture states that starting from any positive integer $n$, some iterate of $f$ takes value $1$. In this study, we discuss formulations of the Collatz conjecture by $C^{*}$-algebras in the following three ways: (1) single operator, (2) two operators, and (3) Cuntz algebra. For the $C^{*}$-algebra generated by each of these, we consider the condition that it has no non-trivial reducing subspaces. For (1), we prove that the condition implies the Collatz conjecture. In the cases (2) and (3), we prove that the condition is equivalent to the Collatz conjecture. For similar maps, we introduce equivalence relations by them and generalize connections between the Collatz conjecture and irreducibility of associated $C^{*}$-algebras.

math.OA↗
arXiv2014

Direct imaging of monovacancy-hydrogen complexes in single graphitic layer

Understanding how foreign chemical species bond to atomic vacancies in graphene layers can advance our ability to tailor the electronic and magnetic properties of defective graphenic materials. Here we use ultra-high vacuum scanning tunneling microscopy (UHV-STM) and density functional theory to identify the precise structure of hydrogenated single atomic vacancies in a topmost graphene layer of graphite and establish a connection between the details of hydrogen passivation and the electronic properties of a single atomic vacancy. Monovacancy-hydrogen complexes are prepared by sputtering of the graphite surface layer with low energy ions and then exposing it briefly to an atomic hydrogen environment. High-resolution experimental UHV-STM imaging allows us to determine unambiguously the positions of single missing atoms in the defective graphene lattice and, in combination with the ab initio calculations, provides detailed information about the distribution of low-energy electronic states on the periphery of the monovacancy-hydrogen complexes. We found that a single atomic vacancy where each sigma-dangling bond is passivated with one hydrogen atom shows a well-defined signal from the non-bonding pi-state which penetrates into the bulk with a (\sqrt 3 \times \sqrt 3)R30^ \circ periodicity. However, a single atomic vacancy with full hydrogen termination of sigma-dangling bonds and additional hydrogen passivation of the extended pi-state at one of the vacancy's monohydrogenated carbon atoms is characterized by complete quenching of low-energy localized states. In addition, we discuss the migration of hydrogen atoms at the periphery of the monovacancy-hydrogen complexes which dramatically change the vacancy's low-energy electronic properties, as observed in our low-bias high-resolution STM imaging.

cond-mat.mtrl-sci↗