A geometric approach to divergent points of higher dimensional Collatz mappings

TL;DR

This paper introduces a geometric framework for analyzing higher-dimensional Collatz mappings, identifying divergent trajectories and their densities, and demonstrating varied behaviors in algebraic integer rings compared to integers.

Contribution

It provides a geometric approach to study generalized Collatz mappings in higher dimensions and offers new insights into their divergent behaviors and densities.

Findings

Identified cones of points with divergent trajectories

Established lower bounds for the density of divergent points

Showed different behaviors in algebraic integer rings compared to integers

Abstract

We define generalized Collatz mappings on free abelian groups of finite rank and study their iteration trajectories. Using geometric arguments we describe cones of points having a divergent trajectory and we deduce lower bounds for the density of the set of divergent points. As an application we give examples which show that the iteration of generalized Collatz mappings on rings of algebraic integers can behave quite differently from the conjectured behavior in $Z$.