Moments of balanced measures on Julia sets

Abstract

By a theorem of S. Demko there exists a balanced measure on the Julia set of an arbitrary nonlinear rational transformation on the Riemann sphere. It is proved here that if the transformation admits an attractive or indifferent cycle, then there is a point with respect to which all the moments of a balanced measure exist; moreover, these moments can be calculated exactly. An explicit balanced measure is exhibited in an example where the Julia set is the whole sphere and for which the moments, with respect to any point, do not all exist.