Dynamics of the square mapping on the ring of $p$-adic integers

TL;DR

This paper investigates the dynamical properties of the square mapping on the ring of p-adic integers, revealing distinct behaviors for p=2 and p≥3, including fixed points, periodic points, and minimal components.

Contribution

It provides a detailed analysis of the fixed points, periodic points, and minimal components of the square map on p-adic integers for all primes p, with explicit descriptions for each case.

Findings

For p=2, only attracting fixed points are present.

For p≥3, existence of a fixed point 0 with an attracting basin.

For p≥3, finitely many periodic points and countably many minimal components.

Abstract

For each prime number $p$, the dynamical behavior of the square mapping on the ring $\mathbb{Z}_p$ of $p$-adic integers is studied. For $p=2$, there are only attracting fixed points with their attracting basins. For $p\geq 3$, there are a fixed point $0$ with its attracting basin, finitely many periodic points around which there are countably many minimal components and some balls of radius $1/p$ being attracting basins. All these minimal components are precisely exhibited for different primes $p$.