## International Conference on $p$-ADIC MATHEMATICAL PHYSICS AND ITS APPLICATIONS $p$-ADICS.2015, 07-12.09.2015, Belgrade, Serbia

Lingmin Liao

### Dynamics of rational maps on the projective line of the field of $p$-adic numbers

Abstract

Rational maps in the field $\mathbb{Q}_p$ of $p$-adic numbers are studied as dynamical systems on the projective line $\mathbb{P}^1(\mathbb{Q}_p)$ of $\mathbb{Q}_p$. First, the results on the rational maps of degree one will be recalled. Then we will mainly investigate the rational maps having good reduction. For general prime $p$, a criterion of the minimality of such a rational map will be given. For the case $p=2$, minimal rational maps with good reduction will be completely characterized in terms of their coefficients. It is also proved that a rational map of degree $2$ or $3$ can never be minimal on the whole space $\mathbb{P}^1(\mathbb{Q}_p)$. At last, the dynamics of the rational maps $ax+1/x$ with parameter $a\in \mathbb{Q}_p$ will be fully studied. The talk is based on some joint works with Ai-Hua Fan, Shi-Lei Fan and Yue-Fei Wang.