### Abstract

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.

Original language | English (US) |
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Pages (from-to) | 743-771 |

Number of pages | 29 |

Journal | Communications In Mathematical Physics |

Volume | 308 |

Issue number | 3 |

DOIs | |

State | Published - Dec 1 2011 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

*Communications In Mathematical Physics*,

*308*(3), 743-771. https://doi.org/10.1007/s00220-011-1363-1