Rational map $ax+1/x$ on the projective line over $\mathbb{Q}\_{p}$

TL;DR

This paper analyzes the dynamical behavior of the rational map $ax+1/x$ on the projective line over the $p$-adic numbers, providing a detailed description for primes $p \,\geq\, 3$.

Contribution

It offers a new detailed characterization of the dynamics of the map $ax+1/x$ over $ ext{P}^1( ext{Q}_p)$ for $p \,\geq\, 3$, expanding understanding of $p$-adic dynamical systems.

Findings

Describes the dynamical structure for $p \,\geq\, 3$

Provides explicit descriptions of orbits and invariant sets

Analyzes stability and periodic points

Abstract

The dynamical structure of the rational map $ax+1/x$ on the projective line $\P$ over the field $\mathbb{Q}\_p$ of $p$-adic numbers is described for $p\geq 3$.