The Collatz Conjecture & Non-Archimedean Spectral Theory: Part I -- Arithmetic Dynamical Systems and Non-Archimedean Value Distribution Theory

Abstract

Let $q$ be an odd prime, and let $T_{q}:\mathbb{Z}\rightarrow\mathbb{Z}$ be the Shortened $qx+1$ map, defined by $T_{q}\left(n\right)=n/2$ if $n$ is even and $T_{q}\left(n\right)=\left(qn+1\right)/2$ if $n$ is odd. The study of the dynamics of these maps is infamous for its difficulty, with the characterization of the dynamics of $T_{3}$ being an alternative formulation of the famous Collatz Conjecture. This series of papers presents a new paradigm for studying such arithmetic dynamical systems by way of a neglected area of ultrametric analysis which we have termed $\left(p,q\right)$-adic analysis, the study of functions from the $p$-adics to the $q$-adics, where $p$ and $q$ are distinct primes. In this, the first paper, working with the $T_{q}$ maps as a toy model for the more general theory, for each odd prime $q$, we construct a function $\chi_{q}:\mathbb{Z}_{2}\rightarrow\mathbb{Z}_{q}$ (the Numen of $T_{q}$) and prove the Correspondence Principle (CP): $x\in\mathbb{Z}\backslash\left\{ 0\right\}$ is a periodic point of $T_{q}$ if and only there is a $\mathfrak{z}\in\mathbb{Z}_{2}\backslash\left\{ 0,1,2,\ldots\right\}$ so that $\chi_{q}\left(\mathfrak{z}\right)=x$. Additionally, if $\mathfrak{z}\in\mathbb{Z}_{2}\backslash\mathbb{Q}$ makes $\chi_{q}\left(\mathfrak{z}\right)\in\mathbb{Z}$, then the iterates of $\chi_{q}\left(\mathfrak{z}\right)$ under $T_{q}$ tend to $+\infty$ or $-\infty$.