Collatz猜想与非阿基米德谱理论：第一部分——算术动力系统与非阿基米德值分布理论
The Collatz Conjecture & Non-Archimedean Spectral Theory: Part I -- Arithmetic Dynamical Systems and Non-Archimedean Value Distribution Theory
AI总结：
本文提出利用(p,q)-adic分析研究算术动力系统，为缩短的qx+1映射构造Numen函数并证明对应原理，为Collatz猜想提供新范式。
摘要：
设$q$为奇素数，令$T_{q}:\mathbb{Z}\rightarrow\mathbb{Z}$为缩短的$qx+1$映射，定义为：若$n$为偶数则$T_{q}\left(n\right)=n/2$，若$n$为奇数则$T_{q}\left(n\right)=\left(qn+1\right)/2$。研究这些映射的动力学因极其困难而声名狼藉，其中对$T_{3}$动力学的刻画正是著名的Collatz猜想的等价表述。本系列论文通过一种被忽视的超度量分析领域，为研究此类算术动力系统提出了新范式，我们将其称为$\left(p,q\right)$-adic分析，即研究从$p$-adic到$q$-adic的函数，其中$p$和$q$为不同素数。在本文（第一篇）中，我们将$T_{q}$映射作为更一般理论的玩具模型，对于每个奇素数$q$，构造了函数$χ_{q}:\mathbb{Z}_{2}\rightarrow\mathbb{Z}_{q}$（即$T_{q}$的Numen），并证明了对应原理（CP）：$x\in\mathbb{Z}\backslash\left\{ 0\right\}$是$T_{q}$的周期点，当且仅当存在$\mathfrak{z}\in\mathbb{Z}_{2}\backslash\left\{ 0,1,2,\ldots\right\}$使得$χ_{q}\left(\mathfrak{z}\right)=x$。此外，若$\mathfrak{z}\in\mathbb{Z}_{2}\backslash\mathbb{Q}$使得$χ_{q}\left(\mathfrak{z}\right)\in\mathbb{Z}$，则$χ_{q}\left(\mathfrak{z}\right)$在$T_{q}$下的迭代趋向于$+\infty$或$-\infty$。
英文摘要：
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 $χ_{q}:\mathbb{Z}_{2}\rightarrow\mathbb{Z}_{q}$ (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 $χ_{q}\left(\mathfrak{z}\right)=x$. Additionally, if $\mathfrak{z}\in\mathbb{Z}_{2}\backslash\mathbb{Q}$ makes $χ_{q}\left(\mathfrak{z}\right)\in\mathbb{Z}$, then the iterates of $χ_{q}\left(\mathfrak{z}\right)$ under $T_{q}$ tend to $+\infty$ or $-\infty$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
