论文标题
Collatz的猜想和非一切光光谱理论：第一部分 - 算术动力学系统和非Archimedean价值分布理论
The Collatz Conjecture & Non-Archimedean Spectral Theory: Part I -- Arithmetic Dynamical Systems and Non-Archimedean Value Distribution Theory
论文作者
论文摘要
令$ q $是一个奇怪的素数，让$ t_ {q}：\ mathbb {z} \ rightarrow \ rightArrow \ mathbb {z} $为缩短$ qx+1 $ map，由$ t_ {q} {q} \ left（n \ right）定义$ t_ {q} \ left（n \ right）= \ left（qn+1 \ right）/2 $如果$ n $是奇数。这些地图动力学的研究因其难度而臭名昭著，其表征$ t_ {3} $是著名的Collatz猜想的替代表述。这一系列论文提出了一种新的范式，用于通过被忽视的超级分析领域来研究这种算术动力学系统，我们称其为$ \ weft（p，q \ right）$ - ADIC分析，从$ p $ - 加法到$ q $ aDicts的功能的研究，其中$ p $ - $ p $和$ q $ $ $是不同的。在此中，第一篇论文，使用$ t_ {q} $作为更通用理论的玩具模型，对于每个奇数$ q $，我们构造了一个函数$χ_{q}：\ Mathbb {Z} _ {2} _ {2} \ rightArrow \ rightArrow \ rightArrow \ rightBb {z} _ {q} _ {q} $} （cp）：$ x \ in \ mathbb {z} \ backSlash \ left \ {0 \ right \} $是$ t_ {q} $的定期点，只有$ \ \ m athfrak {z} \ in \ mathbb {z} 0,1,2，\ ldots \ right \} $，因此$χ_{q} \ left（\ Mathfrak {z} \ right）= x $。此外，如果$ \ Mathfrak {z} \ in \ Mathbb {z} _ {2} \ BackSlash \ Mathbb {q} $ make $χ_{q} \ left（\ Mathfrak {z} {z} \ right） $χ_{q} \ left（\ Mathfrak {z} \ right）$下的$ t_ {q} $倾向于$+\ infty $或$ - \ \ \ \ infty $。
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 $-\infty$.
