### Resumen

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.

Idioma original | English |
---|---|

Páginas (desde-hasta) | 743-771 |

Número de páginas | 29 |

Publicación | Communications in Mathematical Physics |

Volumen | 308 |

N.º | 3 |

DOI | |

Estado | Published - dic 2011 |

### Huella dactilar

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Citar esto

*Communications in Mathematical Physics*,

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

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*Communications in Mathematical Physics*, vol. 308, n.º 3, pp. 743-771. https://doi.org/10.1007/s00220-011-1363-1

**Computability of Brolin-Lyubich Measure.** / Binder, Ilia; Braverman, Mark; Rojas, Cristobal; Yampolsky, Michael.

Resultado de la investigación: Article

TY - JOUR

T1 - Computability of Brolin-Lyubich Measure

AU - Binder, Ilia

AU - Braverman, Mark

AU - Rojas, Cristobal

AU - Yampolsky, Michael

PY - 2011/12

Y1 - 2011/12

N2 - Brolin-Lyubich measure λR of a rational endomorphism R: Ĉ → Ĉ with deg R ≥ 2 is the unique invariant measure of maximal entropy hλR}=htop(R)=log d. Its support is the Julia set J(R). We demonstrate that λR 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.

AB - Brolin-Lyubich measure λR of a rational endomorphism R: Ĉ → Ĉ with deg R ≥ 2 is the unique invariant measure of maximal entropy hλR}=htop(R)=log d. Its support is the Julia set J(R). We demonstrate that λR 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.

UR - http://www.scopus.com/inward/record.url?scp=80955134005&partnerID=8YFLogxK

U2 - 10.1007/s00220-011-1363-1

DO - 10.1007/s00220-011-1363-1

M3 - Article

AN - SCOPUS:80955134005

VL - 308

SP - 743

EP - 771

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 3

ER -