Communications in Mathematical Physics

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

Computability of Brolin-Lyubich Measure

  • Ilia BinderEmail author
  • Mark Braverman
  • Cristobal Rojas
  • Michael Yampolsky


Brolin-Lyubich measure λ R 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 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.


Invariant Measure Periodic Point Ideal Point Harmonic Measure Computable Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Springer-Verlag 2011

Authors and Affiliations

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

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