  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.
