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On the Collatz Conjecture: Topological and Ergodic Approach

Eduardo Santana

Abstract

We study the Collatz function famously related to the Collatz Conjecture under the topological and ergodic perpectives, including an approach with thermodynamic formalism. By introducing a key topology and its Borel sigma-algebra we show that recurrence implies periodicity. Moreover, we establish that the set of periodic orbits is finite if, and only if, every continuous potential possesses some equilibrium state. The uniqueness of periodic orbits is equivalent to the uniqueness of equilibrium state for every bounded and continuous potential. Additionaly, by using the Alexandroff compactification of the established topology, we prove finiteness of cycles, which is a significant advance to the conjecture itself. Finally, we apply our technique to the Baker and Syracuse maps, obtaining a similar result for a general class of important maps.

On the Collatz Conjecture: Topological and Ergodic Approach

Abstract

We study the Collatz function famously related to the Collatz Conjecture under the topological and ergodic perpectives, including an approach with thermodynamic formalism. By introducing a key topology and its Borel sigma-algebra we show that recurrence implies periodicity. Moreover, we establish that the set of periodic orbits is finite if, and only if, every continuous potential possesses some equilibrium state. The uniqueness of periodic orbits is equivalent to the uniqueness of equilibrium state for every bounded and continuous potential. Additionaly, by using the Alexandroff compactification of the established topology, we prove finiteness of cycles, which is a significant advance to the conjecture itself. Finally, we apply our technique to the Baker and Syracuse maps, obtaining a similar result for a general class of important maps.
Paper Structure (15 sections, 10 theorems, 29 equations)

This paper contains 15 sections, 10 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Proof of Theorem \ref{['A']}
  4. Invoking foundations
  5. Trapping the orbits towards periodicity
  6. The coarseness of the key topology
  7. Peculiarity of the discrete topology
  8. Some key remarks
  9. Exploring periodicity towards integrals
  10. A key potential
  11. The ergodic decomposition
  12. Finiteness of periodic orbits
  13. Uniqueness of periodic orbits: the final step
  14. Finiteness of Cycles
  15. Baker and Syracuse Maps

Key Result

Theorem A

We have the following facts with respect to $f$

Theorems & Definitions (26)

  • Theorem A
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Remark 5
  • ...and 16 more