2020/07/19 by Maxwell C. Siegel, Siegel, Maxwell Charles · 1 citation
Mathematics · #30G06 #37P99 #Benford’s Law and Fraud Detection #Dynamical Systems (math.DS) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2007.15936
openalex publication_date 2020/07/19 · openalex created_date 2023/03/31 · openalex updated_date 2026/07/28
Let q be an odd prime, and let Tq:ℤ→ℤ be the Shortened qx+1 map, defined by Tq(n)=n/2 if n is even and Tq(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 T3 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 Tq maps as a toy model for the more general theory, for each odd prime q, we construct a function χq:ℤ2→ℤq (the Numen of Tq) and prove the Correspondence Principle (CP): x∈ℤ\backslash\ 0\ is a periodic point of Tq if and only there is a \mathfrakz∈ℤ2\backslash\ 0,1,2,…\ so that χq(\mathfrakz)=x. Additionally, if \mathfrakz∈ℤ2\backslashℚ makes χq(\mathfrakz)∈ℤ, then the iterates of χq(\mathfrakz) under Tq tend to +∞ or -∞.