Analogues of the 3 x + 1 Problem in Polynomial Rings of Characteristic 2

ABSTRACT

The Collatz conjecture (also known as the 3x + 1 problem) concerns the behavior of the discrete dynamical system on the positive integers defined by iteration of the so-called 3x + 1 function. We investigate analogous dynamical systems in rings of functions of algebraic curves over . We prove in this setting a generalized analogue of a theorem of Terras concerning the asymptotic distribution of stopping times. We also present experimental data on the behavior of these dynamical systems.

Keywords:

• Collatz conjecture
• discrete dynamical system
• random walk
• 3 x + 1

2000 AMS Subject Classification:

• Primary 11B83
• Secondary 60G40, 60G50, 11Y99

Acknowledgments

Professors Carl Pomerance and Jeffrey Lagarias provided helpful comments on an earlier draft of this article, for which the author is very grateful. The author also owes thanks to Professor Hans Johnston for his help with the computations. This article represents one part of the author’s dissertation, supervised and guided by Professor Siman Wong.

Notes

1 A complete proof of all four cases is given in a supplemental document available on the author’s website.

2 Full details of this proof are given in a supplemental document available on the author’s website.

3 Full details of this proof are given in a supplemental document available on the author’s website.

Additional information

Funding

This work was supported in part by NSA grant H98230-14-1-0307.