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

TL;DR

This work studies the inverse lifting problem in non-Archimedean dynamics: given a finite functional graph, it constructs continuous $p$-adic interpreters whose residue-level transitions reproduce the prescribed dynamics on a finite cylinder partition. The method combines piecewise-affine local models with rigid-analytic Runge approximation to yield pole-free rational interpreters on unions of disjoint balls, with robustness guaranteed by a linear-dominance condition that yields exact ball mappings. It introduces Witt-vector encoding to handle composite alphabets and proves a Dynamic Chinese Remainder Theorem that factorizes dynamics across prime-power components, while a pro-finite perspective links compatible towers to 1-Lipschitz maps on $ olinebreak[0]\\mathbb{Z}_p$ and potential profinite limits. Reduction theory, via strict good reduction, serves as a selection principle to pick interpreters that preserve discrete structure across residues, depths, and towers. Overall, the paper provides existence, stability, and decomposition results for finite-resolution $p$-adic interpreters and outlines a program toward moduli spaces and radius-zero interpretations, with several worked arithmetic examples illustrating Frobenius, Verschiebung, and dynamic-systems decompositions.

Abstract

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.

Figures (5)

• Figure 1: Counterexample $f(x)=x+1$ mod $2^n$ for $n=1,2,3$: period grows (2-cycle, 4-cycle, 8-cycle). Each edge is a single arrow; the 2-cycle is drawn as two curved arcs with the same curvature so they do not overlap. The multiplier $\lambda=f'(x)=1$ (parabolic), so Hensel's non-degeneracy fails and cycle length is not preserved. Vertical: reduction (down).
• Figure 2: Four compatible levels of the functional graphs of $f(x)=x^2+1$ modulo $2^n$ for $n=1,2,3,4$. Each panel shows the digraph at that modulus (arrows: $x\mapsto f(x)$). Colours: one colour per vertex label; the same colour in different panels indicates the same residue class (lifting: a vertex at level $n$ projects to the vertex of the same colour at level $n-1$). Reading the tower upward, this encodes lifting toward the limit map on $\mathbb{Z}_2$; commutativity $\pi_{n+1,n}\circ f_{n+1}=f_n\circ\pi_{n+1,n}$ holds. Under the non-degeneracy condition of Theorem \ref{['thm:hensel_exact_cycle']}, periodic points lift to $\mathbb{Z}_2$.
• Figure 3: Functional graph $G_F$ with a cycle $\{0,1\}$, a transient state $2$, and a fixed point $3$ (self-loop at $3$, by convention in Section \ref{['sec:encoding']}).
• Figure 4: DCRT "going" (reduction): the global graph $G_F$ on $\mathbb{Z}/6\mathbb{Z}$ (top) reduces to the local graphs $G_{F_2}$ (mod $2$) and $G_{F_3}$ (mod $3$) (bottom). Vertices are bicolored by residues modulo $2$ (left half) and modulo $3$ (right half), making the reduction $x\mapsto(x\bmod 2,\;x\bmod 3)$ visible.
• Figure 5: DCRT "return" (lifting): the local graphs $G_{f_3}$ (mod $3$) and $G_{f_4}$ (mod $4$) (bottom) are lifted by $\Theta^{-1}$ to the global functional digraph$G_f$ on $\mathbb{Z}/12\mathbb{Z}$ (top): each vertex has exactly one outgoing edge $x \to f(x)$, where $f$ is the unique map that reduces to $f_3$ mod $3$ and $f_4$ mod $4$. Vertices are bicolored by residues modulo $3$ and modulo $4$. Edges in $\mathbb{Z}/3\mathbb{Z}$ and $\mathbb{Z}/4\mathbb{Z}$ are labeled $d_i$ and $e_j$ respectively; each edge in the global graph is labeled $s_{i,j}$ for the unique common lift (gluing) of $d_i$ and $e_j$.

Theorems & Definitions (109)

• Lemma 2.1: Cylinder Partition and Nesting
• Lemma 3.1: Nesting Property
• proof
• Remark 3.2: Nesting and Profinite Towers
• Lemma 3.3: Affine Images
• proof
• Lemma 3.4: Cauchy Bound
• proof
• Lemma 3.5: Coefficient Stability
• proof