- Authors: D. van Melkebeek and A. Bultheel.

- Reference:
* Journal de Mathématiques Pures et Appliquées *,
73(2): 191-221, 1994.

#### Abstract

To study finite Borel measures on the Riemann sphere invariant under a
rational function *R* of degree greater than one, we decompose them in
an *R*-invariant component measure supported on the Julia set and a
finite number of mutually singular *R*-invariant
component measures vanishing on the Julia set. The latter ones can be
described easily. For a characterization of the former one, we use a general
approach based on a weight function for *R* on the Riemann sphere. We
investigate the relation between weight functions for *R* and
*R-*invariant Borel
probability measures on the Riemann sphere in both directions and discuss
how such a measure can be constructed, given a weight function for *R*.