$p$-Adic Monomial Dynamical Systems

We consider discrete dynamical systems in the field of $p$-adic numbers, $\mathbb{Q}_p$, for prime numbers $p\geq 3$. We study systems that are given by iterations of the monomial function $x\mapsto x^n$, where $n\geq 2$ is an integer. The dynamics looks totally different depending on whether ${p\mid n}$ or not. In both cases, interesting dynamics occurs on the unit sphere, $S_1(0)$ in $\mathbb {Q}_p$. In this article, we state some results about cycles and fuzzy cycles.