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

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 $p\ge 3$, 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 $1\mathrm{/}p$ being attracting basins. All these minimal components are precisely exhibited for different primes $p$.