Dynamics over Local Fields: Good Reduction

Abstract

The study of the dynamics of polynomial and rational maps over ℝ and ℂ has a long history and includes many deep theorems, some of which were briefly discussed in Chapter 1. A more recent development is the creation of an analogous theory over complete local fields such as the p-adic rational numbers ℚ _p and the completion ℂ _p of an algebraic closure of ℚ _p . The nonarchimedean nature of the absolute value on ℚ _p and ℂ _p makes some parts of the theory easier than when working over ℂ or ℝ. But as usual, there is a price to pay. For example, the theory of nonarchimedean dynamics must deal with the fact that ℚ _p is totally disconnected and far from being algebraically closed, while ℂ _p is not locally compact.