Symbolic Dynamic Formulation for the Collatz Conjecture: I. Local and Quasi-global Behavior

TL;DR

This paper introduces a symbolic dynamical framework for analyzing the Collatz conjecture, revealing cyclic behaviors and methods for generating divergent sequences, which may aid in understanding its global stability properties.

Contribution

It presents a novel symbolic dynamical formulation based on discrete mappings, enabling characterization of Collatz behaviors and grouping of similar dynamical sequences.

Findings

Identification of cyclic behaviors in Collatz itineraries

Techniques for generating arbitrarily long divergent sequences

Grouping of itineraries into families with similar dynamics

Abstract

In this work, a symbolic dynamical formulation based upon discrete iterative mappings derived from the Collatz conjecture is introduced. It is demonstrated that this formulation naturally induces a ternary alphabet useful for characterizing the expansive and dissipative behavior of generated itineraries. Furthermore, local and quasi-global analyses indicate cyclic behaviors that should prove useful for describing global stability properties of itineraries. Additionally, techniques for generating arbitrarily long divergent itineraries are presented. Finally, this symbolic formulation allows itineraries to be grouped into sequence families that retain equivalent dynamical behavior.