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

Abstract

ABSTRACTThe 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.