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

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.