离散动力学的有理解释器：$p$-adic域上的存在性、精确性与分解
Rational Interpreters for Discrete Dynamics: Existence, Exactness, and Decomposition over $p$-adic Fields
AI总结：
本文研究了$p$-adic域上离散动力学的有理解释器的存在性、精确性与分解，提出动态中国剩余定理并探讨了逆极限扩展。
摘要：
我们针对非阿基米德动力学中的一个逆问题：给定一个有限的离散动力系统（等价于在$N$个状态上的函数图），构造一个连续的$p$-adic动力系统，其余数层面的行为能复现所规定的转移。利用$\mathcal{O}_K$的圆柱分割（视为无分歧$K/\mathbb{Q}_p$的魏特圆柱），我们将状态编码为成对不相交的闭球，并形式化解释器为一个映射，将每个状态球映射到其目标球。我们的主要存在性结果构造了有理解释器，这些解释器在规定的状态圆柱上是解析的（因此无极点），结合了刚性解析鲁涅分析与有限插值约束。在每个圆柱上满足线性主导条件时，球像是显式且局部仿射的，导致离散行为被稳健地分类为收缩、中性及扩展的区域。良好的约简提供了自然解释器的选择原理；一般有理解释器的有效次数和高度界仍待解决。对于复合字母表，我们证明了一个关于保持同余系统的动态中国剩余定理：同余同构$Θ:\mathbb{Z}/m\mathbb{Z}\xrightarrow{\sim}\prod_i\mathbb{Z}/p_i^{k_i}\mathbb{Z}$（对于$m=\prod p_i^{k_i}$）将$\mathbb{Z}/m\mathbb{Z}$上的动力学（等价于函数图）分解为各素数幂组件上的动力学，与约简相容。最后，我们讨论了逆极限（余有限）扩展：兼容的塔定义了$\mathbb{Z}_p$上的$1$-Lipschitz映射，而跨层级选择兼容的解析/有理解释器则成为另一个问题。
英文摘要：
We address an inverse problem in non-Archimedean dynamics: given a finite discrete dynamical system (equivalently, a functional graph on $N$ states), construct a continuous $p$-adic dynamical system whose residue-level behavior reproduces the prescribed transitions. Using the cylinder partition of $\mathcal{O}_K$ (viewed as \emph{Witt cylinders} for unramified $K/\mathbb{Q}_p$), we encode states by pairwise disjoint closed balls and formalize an \textbf{interpreter} as a map sending each state ball into its target ball. Our main existence result constructs rational interpreters that are analytic (hence pole-free) on the prescribed state cylinders, combining rigid-analytic Runge approximation with finite interpolation constraints. Under a linear-dominance condition on each cylinder, ball images are explicit and locally affine, leading to a robust classification of discrete behavior into contractive, indifferent, and expansive regimes. Good reduction provides a selection principle for natural interpreters; effective degree and height bounds for general rational interpreters remain open. For composite alphabets we prove a \textbf{Dynamic Chinese Remainder Theorem} for congruence-preserving systems: the CRT isomorphism $Θ:\mathbb{Z}/m\mathbb{Z}\xrightarrow{\sim}\prod_i\mathbb{Z}/p_i^{k_i}\mathbb{Z}$ (for $m=\prod p_i^{k_i}$) yields a factorization of the \emph{dynamics} (equivalently, the functional graph) on $\mathbb{Z}/m\mathbb{Z}$ into dynamics on the prime-power components, compatible with reduction. Finally, we discuss an inverse-limit (profinite) extension: compatible towers define a $1$-Lipschitz map on $\mathbb{Z}_p$, while selecting compatible analytic/rational interpreters across levels becomes a separate problem.
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