Computability of Brolin-Lyubich Measure

Abstract

Brolin-Lyubich measure ${\lambda }_R$ of a rational endomorphism $R:\riem\to\riem$ with $°R\ge 2$ is the unique invariant measure of maximal entropy $h_{{\lambda }_R}=h_{\text{top}}(R)=\log d$. Its support is the Julia set $J(R)$. We demonstrate that ${\lambda }_R$ 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.

References in corpus (1)

• On computational complexity of Siegel Julia sets

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