Dynamics of rational functions over different fields

Abstract

In this article, we shall prove the following two basic results of arithmetic dynamics. Let ƒ: P¹(<i>K</i>) → P¹(<i>K</i>) be a rational function of degree 𝑑 &gt; 1. Discovered during the research, there are different results shown with the different definitions of <i>K</i>. If <i>K</i> is an algebraically closed field of characteristic 0, then <i>#Fix</i>(ƒ^<i>n</i>) = 𝑑^<i>n</i> + <i>0</i>(1). If <i>K</i> is a number field, then there are only finitely many preperiodic points of ƒ.