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Application of Operator Theory for the Collatz Conjecture

Takehiko Mori

TL;DR

This work reframes the Collatz $3n{+}1$ problem as an operator-algebraic question, linking discrete dynamics to $C^*$-algebras generated by natural dynamical liftings. It shows that irreducibility of the generated algebras provides a spectrum of results: a no-nontrivial-reducing-subspaces condition suffices in the single-operator formulation, while irreducibility (and cyclic vectors) are equivalent to the conjecture in the two-operator and Cuntz-algebra formulations. The paper further generalizes the framework to bounded-condition dynamical systems and Cuntz-Krieger settings, via first-return maps and orbit equivalence, suggesting a broad operator-theoretic pathway to analyze orbit connectivity. By connecting orbit structure, cyclic vectors, and irreducibility, the authors offer a novel theoretical lens for the Collatz problem with potential implications for future advances in dynamical systems and operator algebras. Overall, the work deepens the interplay between dynamical systems and operator algebras and provides rigorous reformulations that could inform new approaches to the conjecture.

Abstract

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.

Application of Operator Theory for the Collatz Conjecture

TL;DR

This work reframes the Collatz problem as an operator-algebraic question, linking discrete dynamics to -algebras generated by natural dynamical liftings. It shows that irreducibility of the generated algebras provides a spectrum of results: a no-nontrivial-reducing-subspaces condition suffices in the single-operator formulation, while irreducibility (and cyclic vectors) are equivalent to the conjecture in the two-operator and Cuntz-algebra formulations. The paper further generalizes the framework to bounded-condition dynamical systems and Cuntz-Krieger settings, via first-return maps and orbit equivalence, suggesting a broad operator-theoretic pathway to analyze orbit connectivity. By connecting orbit structure, cyclic vectors, and irreducibility, the authors offer a novel theoretical lens for the Collatz problem with potential implications for future advances in dynamical systems and operator algebras. Overall, the work deepens the interplay between dynamical systems and operator algebras and provides rigorous reformulations that could inform new approaches to the conjecture.

Abstract

The Collatz map (or the -map) is defined on positive integers by setting equal to when is odd and when is even. The Collatz conjecture states that starting from any positive integer , some iterate of takes value . In this study, we discuss formulations of the Collatz conjecture by -algebras in the following three ways: (1) single operator, (2) two operators, and (3) Cuntz algebra. For the -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 -algebras.

Paper Structure

This paper contains 19 sections, 35 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. The Collatz conjecture and dynamical systems
  3. Dynamical systems
  4. The Collatz conjecture
  5. Equivalence relations
  6. Hilbert spaces and operator theory
  7. Hilbert spaces
  8. Bounded linear operators
  9. Reducing subspaces
  10. $C^*$-algebras
  11. Formulation of the Collatz conjecture by operators
  12. Formulation by a single operator
  13. Formulation by two operators
  14. Formulation by the Cuntz algebra
  15. Generalization to dynamical systems with bounded condition
  16. ...and 4 more sections

Key Result

Lemma 2.3.2

If $\sim$ is defined as above, then $\sim$ is an equivalence relation.

Theorems & Definitions (89)

  • Definition 2.1.1: Orbits
  • Definition 2.1.2: First-return maps
  • Definition 2.2.1: The Collatz map
  • Conjecture 2.2.2: The Collatz conjecture
  • Definition 2.3.1
  • Lemma 2.3.2: Equivalence relations
  • Definition 3.1.1: Hilbert spaces
  • Definition 3.1.2: Orthogonal complements
  • Theorem 3.1.3: Co96GTM, Orthogonal decomposition
  • Definition 3.1.4: Completely orthonormal systems
  • ...and 79 more