Rational Interpreters for Discrete Dynamics: Existence, Exactness, and Decomposition over $p$-adic Fields

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.

💡 Research Summary

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The paper tackles an inverse problem in non‑Archimedean dynamics: given a finite discrete dynamical system (a functional graph on a set of N states), construct a continuous p‑adic dynamical system whose reduction at a prescribed resolution reproduces exactly the prescribed transitions. The author encodes each state as a closed p‑adic ball (a “cylinder”) in the valuation ring 𝒪_K of an unramified finite extension K/ℚ_p. These cylinders are pairwise disjoint and form a hierarchical partition: a ball of radius p⁻ⁿ corresponds to a depth‑n truncation of a p‑adic integer, and balls of finer depth are nested inside coarser ones.

An interpreter is defined as a map φ:𝒪_K→𝒪_K such that for every state ball B_x the image φ(B_x) is contained in the target ball B_{F(x)}. When the inclusion is an equality (exact interpretation), the reduction map π_n:𝒪_K→𝒪_K/pⁿ𝒪_K satisfies π_n∘φ = F∘π_n, i.e. φ lifts the discrete map F to the p‑adic level. The paper distinguishes three local dynamical types for each ball: contractive (image radius smaller), indifferent (equal radius), and expansive (larger radius), mirroring the classical p‑adic classification of fixed points.

The existence theorem proceeds in two stages. First, on each ball a piecewise affine map ψ_x(z)=β_x+u_x(z−α_x) is chosen so that ψ_x maps B_x into B_{F(x)} with the desired linear factor u_x. Second, a global rational function φ is obtained by applying the rigid‑analytic Runge approximation theorem, which guarantees that rational functions are dense on affinoid domains. The Runge step yields a rational function that coincides with the piecewise affine model on the prescribed union of balls, is analytic (hence pole‑free) there, and respects the inclusion condition. The construction is non‑effective: no explicit bounds on degree or height are provided, a point the author acknowledges as an open problem.

To achieve exact ball mapping, the paper introduces a linear dominance condition. Roughly, the linear coefficient u_x must dominate the radius of the source ball in a way that guarantees |u_x|·p⁻ⁿ = p^{-(n+δ)} (or the reverse inequality) for some δ>0. Under this hypothesis, φ maps each source ball precisely onto its target ball, not merely into it. The Robustness Theorem shows that linear dominance defines an open condition in the space of analytic maps, so small perturbations of φ preserve exactness. Consequently, the interpreter’s local type (contractive, indifferent, expansive) is stable under perturbations.

The paper then extends the framework to composite alphabets such as ℤ/mℤ with m=∏p_i^{k_i}. A Dynamic Chinese Remainder Theorem is proved: the CRT isomorphism Θ:ℤ/mℤ≅∏ℤ/p_i^{k_i}ℤ lifts to dynamics, i.e. a map F on ℤ/mℤ decomposes into component maps F_i on each prime‑power factor, and an interpreter φ on ℤ/mℤ corresponds to the product of interpreters φ_i on the factors. This yields a clean “horizontal” decomposition (splitting the alphabet) and a “vertical” decomposition (refining the depth of the cylinders). The author discusses how these two decompositions are independent and can be combined to analyze more intricate systems.

A profinite (inverse‑limit) perspective follows. If a family of interpreters {φ_n} on the residue rings ℤ/pⁿℤ is compatible under the natural reduction maps, then the inverse limit defines a 1‑Lipschitz map ψ:ℤ_p→ℤ_p. This provides a global continuous dynamical system that simultaneously lifts all finite‑level dynamics. However, constructing a compatible family of analytic (or rational) interpreters remains an open problem; the paper isolates this as a separate lifting question.

The notion of good reduction is employed as a selection principle. When many interpreters exist, those with strict good reduction (the reduction map preserves degree and does not collapse distinct balls) are singled out as “natural” lifts. Good reduction also guarantees that the interpreter’s dynamics on the residue field is faithfully reflected at the p‑adic level.

The author illustrates the theory with several worked examples: (1) a concrete cycle‑and‑transient system, (2) dynamics on Witt vectors over GF(4), (3) interpreting the Frobenius endomorphism as an exact interpreter and the Verschiebung as a shift‑register interpreter, (4) stability analysis of the p‑adic quadratic family z↦z²+c, (5) connections to discrete cellular‑automaton‑type maps (DCRT, PDS). These examples demonstrate how the abstract framework translates into concrete arithmetic and dynamical phenomena.

In the concluding sections, the paper reflects on the structural meaning of the results, relates them to existing work on forward p‑adic dynamics, reduction theory, and Berkovich spaces, and outlines future directions: effective degree/height bounds, compatible analytic lifting across levels, extensions to multivariate maps, and deeper exploration of the interplay between the horizontal (CRT) and vertical (depth) decompositions.

Overall, the work provides a systematic methodology for “lifting” finite discrete dynamics to continuous p‑adic dynamics, establishes existence and robustness of rational interpreters, introduces a dynamic CRT for composite alphabets, and opens a pathway toward a unified profinite dynamical theory.