### Newton Polygons and a Nonarchimedean Fatou Lemma

By Alon Levy

(Joint with Patrick Ingram and Rafe Jones)

November 9, 2011

#### Abstract

Let f be a rational function in one variable over a field. Over C, it is known that every attracting fixed point x of f attracts some critical point z, and furthermore we can choose z such that the forward images of z approach but are distinct from x. We use Newton polygons to prove an analog over non-archimedean fields, with applications including the finiteness of postcritically finite maps over a number field, and a bound on the number of attracting cycles over a non-archimedean field.