The Hydra Map and Numen Formalisms for Collatz-Type Problems
Maxwell C. Siegel
TL;DR
This work generalizes Collatz-type dynamics to Hydra maps on the ring of integers $\mathcal{O}_{K}$ of a global field, unifying archimedean and non-archimedean analysis within a single framework and paving the way toward an adèlic formulation. It introduces Hydra maps and their numen $X_H$, establishes a central functional equation $X_H(pn+j)=r_j X_H(n)+c_j$, and develops conditions under which $X_H$ extends to $\mathbb{Z}_p$ with measurability or continuity. A Correspondence Principle links periodic points of Hydra maps to rational $p$-adic integers and connects $X_H$-values to divergent trajectories under archimedean metrics. The paper also delivers a comprehensive $p$-adic Fourier-analytic apparatus, showing the distribution of $X_H$ is governed by a self-similar measure $\mu_{H,\ell}$ with an ell-adic characteristic function $\hat{\mu}_{H,\ell}$, enabling cross-place analysis essential for arithmetic dynamics on global fields.
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.