$\boldsymbol{p}$-adic rational dynamical systems

Abstract

In this paper, we survey recent developments in the dynamical systems of rational maps in the field $\\Q_p$ of $p$-adic numbers, the field $\\cp$of $p$-adic complex numbers and the Berkovich space over $\\cp$.For a rational map $\\phi\\in~\\Q_p(z)$, we mainly investigate the minimality and the chaoticity of the dynamical system $(\\P^1(\\Q_p),~\\phi)$ on the projective line $(\\P^1(\\Q_p),\\phi)$ over $\\Q_p$.For a complex rational map $\\phi\\in~\\cp$, the dynamical systems $(\\P^1(\\cp),~\\phi)$ and $(\\Pb,~\\phi)$ are stuided, where $\\P^1(\\cp)$ is the projective line over $\\cp$ and$\\Pb$ is the Berkovich projective line over $\\cp$. In this case, the main subjects are the Fatou set andthe Julia set. Some open problems are presented.