The Hydra Map and Numen Formalisms for Collatz-Type Problems

TL;DR

This paper extends the Hydra map and Numen formalism to algebraic number fields, providing a technical manual for future research on Collatz-type problems using non-Archimedean spectral theory.

Contribution

It generalizes the Hydra map and Numen formalism to the ring of integers in global fields, expanding tools for analyzing Collatz-type problems in algebraic number theory.

Findings

Formalism extended to algebraic number fields

Provides background on p-adic analysis and number theory

Serves as a technical manual for future research

Abstract

This paper details a generalization of the formalism presented in the author's 2024 paper, "The Collatz Conjecture and Non-Archimedean Spectral Theory - Part I - Arithmetic Dynamical Systems and Non-Archimedean Value Distribution Theory", to the case of Hydra maps on the ring of integers $\mathcal{O}_{K}$ of a global field $K$. In addition to recounting these definitions, background material is presented for the necessary standard material in algebraic number theory and integration and Fourier analysis with respect to the $p$-adic Haar measure. This paper is meant to serve as a technical manual for use of Hydra maps and numens in future research.