Ergodic Theory and Dynamical Systems

Research Article

Equilibrium measures for rational maps

Artur Oscar Lopesa1

a1 Instituto de Matemática, Universidade Federal do Rio Grande do Sul, Av. Bento Gonçalves, 9.500-Campus do Vale-91.500, Porte Alegre-RS-Brasil

Abstract

For a polynomial map the measure of maximal entropy is the equilibrium measure for the logarithm potential in the Julia set [1], [4].

Here we will show that in the case where f is a rational map such that f(∞) = ∞ and the Julia set is bounded, then the two measures mentioned above are equal if and only if f is a polynomial.

(Received October 05 1984)

(Revised May 03 1985)

(Revised July 20 1985)