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

Abstract

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