On minimal decomposition of $p$-adic polynomial dynamical systems

TL;DR

This paper studies the dynamics of polynomial systems over p-adic integers, showing that their behavior is fully characterized by minimal subsystems, and explicitly describes these for quadratic polynomials over _2.

Contribution

It provides a complete description of the minimal decomposition of p-adic polynomial dynamical systems, including explicit characterization for quadratic cases.

Findings

Dynamical behavior is fully determined by minimal subsystems.

Explicit minimal subsystems are identified for quadratic polynomials over _2.

The structure of p-adic polynomial systems is clarified through minimal decomposition.

Abstract

A polynomial of degree $\ge 2$ with coefficients in the ring of $p$-adic numbers $\mathbb{Z}_p$ is studied as a dynamical system on $\mathbb{Z}_p$. It is proved that the dynamical behavior of such a system is totally described by its minimal subsystems. For an arbitrary quadratic polynomial on $\mathbb{Z}_2$, we exhibit all its minimal subsystems.