INVARIANT FINITE BOREL MEASURES FOR RATIONAL FUNCTIONS ON THE RIEMANNSPHERE

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 fini te number of mutually singular R-invariant component measures vanishin g on the Julia set. The latter one can be described easily. For a char acterization of the former one, we use a general approach based on a w eight function for R on the Riemann sphere. We investigate the relatio n between weight functions for R and R-invariant Borel probability mea sures on the Riemann sphere in both directions and discuss how such a measure can be constructed, given a weight function for R.