Computability of Brolin-Lyubich Measure

TL;DR

This paper proves that the Brolin-Lyubich measure for rational functions is always computable via an algorithm, even when the Julia set itself is not computable, and explores conditions for the computability of harmonic measures.

Contribution

It establishes the universal computability of the Brolin-Lyubich measure for rational maps and provides conditions for harmonic measure computability in polynomial basins.

Findings

Brolin-Lyubich measure is always computable from coefficients of R.

Harmonic measure computability depends on domain properties.

Computability may fail for general domains, but holds for polynomial basins.

Abstract

Brolin-Lyubich measure $\lambda_R$ of a rational endomorphism $R:\riem\to\riem$ with $\deg R\geq 2$ 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.