Almost Newton, sometimes Lattès

Abstract

Self-maps everywhere defined on the projective space PN over a global field are the basic objects of study in the arithmetic of dynamical systems. In this paper we study the natural self-maps defined the following way: F is a homogeneous polynomial of degree d in (N+1) variables Xi defining a smooth hypersurface. Suppose the characteristic of the field does not divide d and define the map of partial derivatives ϕF=(FX0,…,FXN). One can also compose such a map with an element of PGLN+1. In the case N=1, the smoothness condition means that F has only simple zeroes and we prove that a self-map of P1 has constant multipliers if and only if it has the form ϕ(X,Y)=(FY,−FX). We recover in this manner classical dynamical systems like the Newton method for finding roots of polynomials or the Lattès map corresponding to the multiplication by 2 on an elliptic curve and the multiplication by n on the multiplicative group.