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 \(-\infty\).