The autoconjugacy of a generalized Collatz map

TL;DR

This paper explores the autoconjugacy map associated with generalized Collatz functions, extending known 2-adic properties and proposing new conjectures about their behavior, including periodicity and divergence.

Contribution

It introduces the autoconjugacy map for generalized mx+r Collatz functions, proving basic properties and formulating new conjectures in this broader setting.

Findings

Properties of the autoconjugacy map are established

Conjectures on periodicity and divergence are proposed

Generalization of 2-adic properties to mx+r maps

Abstract

Many of the 2-adic properties of the 3x+1 map generalize to the analogous mx+r map, where m and r are odd integers. We introduce the corresponding autoconjugacy map, prove some simple properties of it and make some further conjectures in the general setting, including weak versions of the periodicity and divergent trajectories conjectures.