Lingmin LiaoDynamics of rational maps on the projective line of the field of $p$adic numbersAbstract
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 AiHua Fan, ShiLei Fan and YueFei Wang.
