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

Takehiko Mori

TL;DR

The paper develops a framework linking dynamical systems on discrete, countable phase spaces to operator algebras by introducing a bounded condition and constructing C*-algebras from associated partial isometries. It shows that irreducibility of these algebras corresponds to minimality under separating conditions, and introduces uniqueness and the stronger totally uniqueness conditions to establish bijections between invariant sets and reducing subspaces, with the Collatz and qx+d families as primary examples. A key advancement is a symbolic representation that connects discrete systems to symbolic dynamics, enabling a faithful translation to topological dynamical concepts under totally uniqueness. Collectively, these results provide a structured bridge between discrete dynamics, C*-algebras, and symbolic/topological representations, with explicit maps between invariant sets and operator-theoretic reducing subspaces.

Abstract

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.

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

TL;DR

The paper develops a framework linking dynamical systems on discrete, countable phase spaces to operator algebras by introducing a bounded condition and constructing C*-algebras from associated partial isometries. It shows that irreducibility of these algebras corresponds to minimality under separating conditions, and introduces uniqueness and the stronger totally uniqueness conditions to establish bijections between invariant sets and reducing subspaces, with the Collatz and qx+d families as primary examples. A key advancement is a symbolic representation that connects discrete systems to symbolic dynamics, enabling a faithful translation to topological dynamical concepts under totally uniqueness. Collectively, these results provide a structured bridge between discrete dynamics, C*-algebras, and symbolic/topological representations, with explicit maps between invariant sets and operator-theoretic reducing subspaces.

Abstract

In this paper, we study abstract dynamical systems with discrete phase spaces. One example of such a system is induced by the -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 -algebras on certain Hilbert spaces. For a map on a general discrete phase space, we consider -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 -invariant sets to the family of reducing subspaces for the corresponding -algebra. By introducing the totally uniqueness condition for , we show that this injection is a bijection if satisfies this condition. This condition is crucial in providing a symbolic representation of the dynamical system induced by , and we discuss the relationship between this symbolic representation and that of a topological dynamical system.

Paper Structure

This paper contains 5 sections, 22 theorems, 44 equations.

Table of Contents

  1. Dynamical systems with discrete phase spaces
  2. Bounded condition and $C^{*}$-algebras
  3. Uniqueness condition
  4. Totally uniqueness condition
  5. Symbolic representation

Key Result

Theorem 2

Let $f$ be the Collatz map, $\mathcal{H}$ be a Hilbert space with $\dim\mathcal{H}=\aleph_{0}$ and $\{e_{n}\}_{n \in \mathbb{N}}$ be a complete orthonormal system for $\mathcal{H}$. When $T_{1},T_{2} \in \mathcal{B}(\mathcal{H})$ are defined by: where $n \in \mathbb{N}$, the following are equivalent:

Theorems & Definitions (47)

  • Definition 1: CK80IM, The Cuntz-Krieger algebras $\mathcal{O}_{A}$
  • Theorem 2: Mo25AOT
  • Theorem 3: Mo25AOT
  • Definition 1.1: Total orbits
  • Definition 1.2: Invariant sets
  • Lemma 1.3
  • proof
  • Definition 2.1: Mo25AOT, Bounded condition
  • Lemma 2.2: Mo25AOT
  • Theorem 2.3: Mo25AOT
  • ...and 37 more