Dynamical Systems with Bounded Condition and $C^{*}$-algebras
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914173027352576 |
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| author | Mori, Takehiko |
| author_facet | Mori, Takehiko |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_05713 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dynamical Systems with Bounded Condition and $C^{*}$-algebras Mori, Takehiko Operator Algebras Dynamical Systems Number Theory 47L30, 47L90 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. |
| title | Dynamical Systems with Bounded Condition and $C^{*}$-algebras |
| topic | Operator Algebras Dynamical Systems Number Theory 47L30, 47L90 |
| url | https://arxiv.org/abs/2508.05713 |