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Moments of balanced measures on Julia sets


Authors: M. F. Barnsley and A. N. Harrington
Journal: Trans. Amer. Math. Soc. 284 (1984), 271-280
MSC: Primary 30D05; Secondary 30E05, 58F11
DOI: https://doi.org/10.1090/S0002-9947-1984-0742425-6
MathSciNet review: 742425
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Abstract: By a theorem of S. Demko there exists a balanced measure on the Julia set of an arbitrary nonlinear rational transformation on the Riemann sphere. It is proved here that if the transformation admits an attractive or indifferent cycle, then there is a point with respect to which all the moments of a balanced measure exist; moreover, these moments can be calculated exactly. An explicit balanced measure is exhibited in an example where the Julia set is the whole sphere and for which the moments, with respect to any point, do not all exist.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1984-0742425-6
Keywords: Julia sets, orthogonal polynomials
Article copyright: © Copyright 1984 American Mathematical Society

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