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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Julia sets are uniformly perfect


Authors: R. Mañé and L. F. da Rocha
Journal: Proc. Amer. Math. Soc. 116 (1992), 251-257
MSC: Primary 58F23; Secondary 30D05, 31A25, 58F11
MathSciNet review: 1106180
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Abstract: We prove that Julia sets are uniformly perfect in the sense of Pommerenke (Arch. Math. 32 (1979), 192-199). This implies that their linear density of logarithmic capacity is strictly positive, thus implying that Julia sets are regular in the sense of Dirichlet. Using this we obtain a formula for the entropy of invariant harmonic measures on Julia sets. As a corollary we give a very short proof of Lopes converse to Brolin's theorem.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1992-1106180-2
PII: S 0002-9939(1992)1106180-2
Article copyright: © Copyright 1992 American Mathematical Society