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Rational map associated with the Sigmoid Beverton-Holt model on the projective line over $\mathbb{Q}_p$

Cheng Liu

TL;DR

The paper addresses the $p$-adic dynamics of the Sigmoid Beverton-Holt model on the projective line over $\mathbb{Q}_p$ by reducing the rational map to the polynomial map $f_N(x)=\frac{x^N+1}{a}$ via a topological conjugacy, enabling a comprehensive analysis without the gcd$(N,p)$ constraint. It then classifies the dynamics across the regimes $|a|_p>1$, $|a|_p<1$, and $|a|_p=1$, using minimal decomposition and $p$-adic repeller theory to obtain fixed-point structures, invariant sets, and, in many cases, conjugacies to shift spaces. The main contributions include explicit descriptions of asymptotic behavior, fixed points, and minimal invariant sets (with detailed $p=2$ and $p=3$ treatments) and a systematic reduction of rational dynamics to polynomial dynamics. This work advances the understanding of $p$-adic dynamical systems for biologically motivated models and demonstrates the effectiveness of minimal decomposition and shift-space conjugacies in resolving the long-term behavior of seemingly complex $p$-adic maps.

Abstract

We describe the dynamical structure of the $p$-adic rational dynamical systems associated with the Sigmoid Beverton-Holt model on the projective line over the field $\mathbb{Q}_p$ of $p$-adic numbers. Our methods are minimal decomposition of $p$-adic polynomials with coefficients in $\mathbb{Z}_p$ established by Fan and Liao and the chaotic description of $p$-adic repellers of Fan, Liao, Wang and Zhou.

Rational map associated with the Sigmoid Beverton-Holt model on the projective line over $\mathbb{Q}_p$

TL;DR

The paper addresses the -adic dynamics of the Sigmoid Beverton-Holt model on the projective line over by reducing the rational map to the polynomial map via a topological conjugacy, enabling a comprehensive analysis without the gcd constraint. It then classifies the dynamics across the regimes , , and , using minimal decomposition and -adic repeller theory to obtain fixed-point structures, invariant sets, and, in many cases, conjugacies to shift spaces. The main contributions include explicit descriptions of asymptotic behavior, fixed points, and minimal invariant sets (with detailed and treatments) and a systematic reduction of rational dynamics to polynomial dynamics. This work advances the understanding of -adic dynamical systems for biologically motivated models and demonstrates the effectiveness of minimal decomposition and shift-space conjugacies in resolving the long-term behavior of seemingly complex -adic maps.

Abstract

We describe the dynamical structure of the -adic rational dynamical systems associated with the Sigmoid Beverton-Holt model on the projective line over the field of -adic numbers. Our methods are minimal decomposition of -adic polynomials with coefficients in established by Fan and Liao and the chaotic description of -adic repellers of Fan, Liao, Wang and Zhou.
Paper Structure (10 sections, 36 theorems, 101 equations)

This paper contains 10 sections, 36 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. P-adic numbers and the projective line $\mathbb{P}^1(\mathbb{Q}_p)$
  4. p-adic dynamical systems
  5. p-adic polynomial dynamical systems
  6. Dynamical systems of $\left(\mathbb{P}^1(\mathbb{Q}_p),f_N\right)$
  7. Case $|a|_p>1$
  8. Case $|a|_p<1$
  9. Case $|a|_p=1$
  10. Three $p$-adic polynomial dynamical systems

Key Result

Proposition 1.1

Let $F(x)=ax^n\in\mathbb{Q}_p[x]$ with $n\ge 1$. Suppoes $G(x)=b^mx^m+\cdots+b_0\in\mathbb{Q}_p[x]$ with $b_mb_0\neq 0$. Let $\phi(x)=\frac{F(x)}{G(x)}$. If $m\le n$, then $\left(\mathbb{P}^1(\mathbb{Q}_p),\phi\right)$ is topologically conjugate to $\left(\mathbb{P}^1(\mathbb{Q}_p),H\right)$ with and $deg(H)=n$.

Theorems & Definitions (64)

  • Proposition 1.1
  • proof
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • ...and 54 more