arXiv subjects

J. Rogelio Pérez-Buendía

Publications and source records attributed to J. Rogelio Pérez-Buendía.

6 recordsLinked to original sources

arXiv2026

Hierarchical Expansion of Finite Discrete Dynamical Systems: A Non-Archimedean Variational Theory over Coordinate Orderings

A finite dynamical system on $\mathbb{F}_p^N$ has a functional graph and combinatorial basins, but nothing on which to run the local analysis a multiplier makes possible. We supply one. Orderings encode configurations as residue classes in $\mathbb{Z}_p$, and an exact rational interpreter over $\mathbb{C}_p$ carries each ball onto its image ball. Rational approximants send, resolution by resolution, every ball onto one of prescribed radius and center. A half-integer pole sphere, with no Archimedean counterpart, settles the expanding case. The prescribed radii are the missing multipliers. They sort balls into contracting, expanding or isometric, giving each configuration a word over three letters, one per scale. Fixed configurations all satisfy $f(a)=a$, yet their words refine them into scale-resolved classes. Contracting balls containing their images trap unique attracting fixed points of the approximant, expanding balls inside their images unique repelling ones. The counts become scores $(μ_E,μ_A,μ_I)$, Haar integrals over $\mathbb{Z}_p$, and minimizing $μ_E$ over orderings is the finite variational principle. The first score vanishes exactly on Anashin's $p$-adic automata, grading departures by resolution, while the ceiling $(N-1)p^N$ at every ordering isolates the $28$ outer-permutive rules among elementary cellular automata on rings of $N\ge5$ cells. The transition table and ordering determine radii, letters and scores. On a thirteen-gene Boolean network we certify the minimizer over all $13!$ orderings, whose hierarchy separates the ten fixed states by type.

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arXiv2026

La geometría de los monoides

In this article we present the basis of Monoid Theory from a categorical point of view, emphasizing the analogies and differences with the theory of modules over commutative rings. We present the generalization of affine schemes to affine monoidal schemes. We study the relationship between the latter and toric varieties, and we set up the basis for the study of Fontaine-Kato-Illusie's Logarithmic Geometry, widely used in Arithmetic Geometry. En este artículo presentamos las bases de la teoría de monoides desde el punto de vista categórico, haciendo énfasis en las analogías y diferencias con la teoría de módulos sobre anillos conmutativos. Se presenta la generalización de esquema afín a esquema afín monoidal; estudiamos la relación que éstos tienen con las variedades tóricas y sentamos las bases para el estudio de la Geometría Logarítmica de Fontaine-Kato-Illusie, ampliamente usada en Geometría Aritmética.

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arXiv2026

A Dynamical Néron--Ogg--Shafarevich Criterion via Orbital Arboreal Representations

Let $K$ be a non-archimedean local field and $φ: \mathbb{P}^1 \to \mathbb{P}^1$ a rational endomorphism of degree $d \geq 2$ over $K$. In the tame case ($p \nmid d$), we show that strict good reduction is equivalent to the existence of a nonempty Zariski open subset $U_k \subset \mathbb{P}^1_k \setminus \mathrm{PC}(\widetildeφ)$ over which the canonical residual morphism is finite étale of degree $d$. The criterion separates two complementary local invariants of a normalized integral lift: $\mathrm{Res}(F,G)$ controls residual degree drop, while the fiber discriminants $\mathrm{Disc}(F_{n,x})$ control étaleness of the residual fibers once full residual degree is ensured. Consequently, for every finite $x \in \mathcal{O}_K$ with $\bar{x} \in U_k$, the extensions $K(X_n(x))/K$ are unramified for all $n \geq 1$. We introduce the orbital preimage tree $T_{O^+(x)} = \varinjlim_n X_\infty(φ^n(x))$, the colimit in $G_K$-sets along the forward orbit, and the orbital arboreal Galois image $\mathcal{G}_{O^+(x)} = \mathrm{Im}(G_K \to \mathrm{Aut}(T_{O^+(x)}))$. On the forward-invariant safe locus $U_k^{\mathrm{safe}} = \bigcap_{m \geq 0} \widetildeφ^{-m}(U_k)$, strict good reduction is captured by the bijectivity of the orbital reduction map $X_n(x_m) \to \widetilde{X}_n(\bar{x}_m)$. This canonical orbit-invariant framework connects with arboreal Galois representations (Boston-Jones, Jones, and others) and yields pointwise and orbit-level reformulations. Explicit examples over $\mathbb{Q}_p$ illustrate the criterion.

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arXiv2026

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.

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arXiv2025

A $p$-adic Reaction--Diffusion Model of Branching Coral Growth and Calcification Dynamics

Coral colonies exhibit complex, self-similar branching architectures shaped by biochemical interactions and environmental constraints. To model their growth and calcification dynamics, we propose a novel p-adic reaction-diffusion framework defined over p-adic ultrametric spaces. The model incorporates biologically grounded reactions involving calcium and bicarbonate ions, whose interplay drives the precipitation of calcium carbonate (CaCO3). Nonlocal diffusion is governed by the Vladimirov operator over the p-adic integers, naturally capturing the hierarchical geometry of branching coral structures. Discretization over p-adic balls yields a high-dimensional nonlinear ODE system, which we solve numerically to examine how environmental and kinetic parameters, particularly CO2 concentration, influence morphogenetic outcomes. The resulting simulations reproduce structurally diverse and biologically plausible branching patterns. This approach bridges non-Archimedean analysis with morphogenesis modeling and provides a mathematically rigorous framework for investigating hierarchical structure formation in developmental biology.

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arXiv2024

La curva de Fargues--Fontaine: Una motivación al estudio de la teoría de representaciones de Galois $p$-ádicas

This article, written in Spanish, provides a comprehensive review of the Fargues-Fontaine curve, a cornerstone in $p$-adic Hodge theory, and its pivotal role in classifying $p$-adic Galois representations. We synthesize key developments surrounding this curve, emphasizing its connection between advanced concepts in arithmetic geometry and the practical theory of representations. We offer a detailed analysis of the Fontaine period rings ($B_{cris}, B_{st}, B_{dR}$), exploring their crucial algebraic and arithmetic properties and their contribution to the curve's construction and definition. Furthermore, we delve into the theory of admissible $p$-adic Galois representations, discussing how the curve, once defined, integrates with Harder-Narasimhan theory.

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