An Invariant Measure For Finitely Generated Rational Semigroups

. For a finitely generated rational semigroup G we establish the existence of a probability measure ¯ = ¯G on the Julia set J(G) which has a certain invariance property with respect to the semigroup. 1. Introduction In [7], M. Lyubich established the existence and uniqueness of a regular Borel probability measure ¯ f supported on the Julia set J(f) of a rational function f of degree at least 2, satisfying ¯ f (E) = ¯ f (f \Gamma1 (E)) for any Borel subset E of C . This measure has applications to computer algorithms for generating pictures of the Julia set of a rational function. In particular, the process of calculating and plotting inverse images of a point under the sequence of iterates will give a good approximation of the Julia set except in certain situations. See [10], pp. 35--38 and [9], Appendix E for a discussion. The measure ¯ f shows which parts of the Julia set are unlikely to be shown by such a computer scheme. Results similar to those of Lyubich were established inde..