Communications in Mathematical Physics

, Volume 308, Issue 3, pp 743–771 | Cite as

Computability of Brolin-Lyubich Measure

  • Ilia Binder
  • Mark Braverman
  • Cristobal Rojas
  • Michael Yampolsky
Article

Abstract

Brolin-Lyubich measure λR of a rational endomorphism \({R:{\hat{\mathbb {C}}}\to {\hat{\mathbb {C}}}}\) with deg R ≥ 2 is the unique invariant measure of maximal entropy \({h_{\lambda_R}=h_{{\rm top}}(R)=\log d}\) . 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.

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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Ilia Binder
    • 1
  • Mark Braverman
    • 1
  • Cristobal Rojas
    • 1
  • Michael Yampolsky
    • 1
  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada

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