Dynamics of polynomial semigroups: measures, potentials, and external fields

Abstract

In this paper, we give a description of a natural invariant measure associated with a finitely generated polynomial semigroup (which we shall call the Dinh-Sibony measure) in terms of potential theory. This requires the theory of logarithmic potentials in the presence of an external field, which, in our case, is explicitly determined by the choice of a set of generators. Along the way, we establish the continuity of the logarithmic potential for the Dinh-Sibony measure, which might be of independent interest. We then use the F-functional of Mhaskar and Saff to discuss bounds on the capacity and diameter of the Julia sets of such semigroups.