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

TL;DR

This paper explores analogues of the Collatz conjecture within polynomial rings over finite fields of characteristic 2, proving a generalized theorem and providing experimental data on their dynamics.

Contribution

It introduces a new framework for Collatz-like problems in algebraic function rings over  and proves a key asymptotic distribution theorem.

Findings

Proved a generalized analogue of Terras's theorem for these systems

Presented experimental data supporting the dynamical behavior

Extended Collatz conjecture concepts to polynomial rings over 

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.