Equidistribution and generalized Mahler measures

Abstract

If K is a number field and is a rational map of degree d > 1, then at each place v of K, one can associate to φ a generalized Mahler measure for polynomials F K[t]. These Mahler measures give rise to a formula for the canonical height h φ(β) of an element; this formula generalizes Mahler's formula for the usual Weil height h(β). In this paper, we use Diophantine approximation to show that the generalized Mahler measure of a polynomial F at a place v can be computed by averaging log | F | v over the periodic points of φ.