Analogues of the $3x+1$ 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 ${\mathbb{F}}_2$ . 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.