Computability of Brolin-Lyubich Measure

Brolin-Lyubich measure λ_R of a rational endomorphism R: Ĉ → Ĉ with deg R ≥ 2 is the unique invariant measure of maximal entropy h_λR}=h_top(R)=log d. Its support is the Julia set J(R). We demonstrate that λ_R is always computable by an algorithm which has access to coefficients of R, even when J(R) is not computable. In the case when R is a polynomial, the Brolin-Lyubich measure coincides with the harmonic measure of the basin of infinity. We find a sufficient condition for computability of the harmonic measure of a domain, which holds for the basin of infinity of a polynomial mapping, and show that computability may fail for a general domain.