Computability of Brolin-Lyubich Measure

Brolin-Lyubich measure λ _R of a rational endomorphism \({R:{\hat{\mathbb {C}}}\to {\hat{\mathbb {C}}}}\) with deg R ≥ 2 is the unique invariant measure of maximal entropy \({h_{\lambda_R}=h_{{\rm top}}(R)=\log d}\) . 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.