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

Fan, Shilei; Liao, Lingmin

doi:10.1090/proc12777

Dynamics of the square mapping on the ring of $p$-adic integers
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Proc. Amer. Math. Soc. 144 (2016), 1183-1196 Request permission

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 $p$.

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