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

Let q be an odd prime, and let T-q: Z -> Z be the Shortened qx+1 map, defined by T-q(n)=n/2 if n is even and T-q(n)=(qn+1)/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 (p,q)-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):Z(2) -> Z(q) (the Numen of T-q) and prove the Correspondence Principle (CP): x is an element of Z\{0} is a periodic point of T-q if and only there is a z is an element of Z(2)\{0,1,2, ...} so that chi(q)(z ) =x. Additionally, if z is an element of Z(2)\Q makes chi(q)(z)is an element of Z, then the iterates of chi(q)(z) under T-q tend to +infinity or -infinity.