Computability of Brolin-Lyubich Measure

Abstract

Brolin-Lyubich measure $λ_R$ of a rational endomorphism $R:\riem\to\riem$ with $°R\geq 2$ is the unique invariant measure of maximal entropy $h_{λ_R}=h_{\text{top}}(R)=\log d$. 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, 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.