Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-1992-1106180-2
MathSciNet review: 1106180
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] H. Brolin, Invariant sets under iteration of rational functions, Ark. Mat. 6 (1965), 103-144. MR 0194595 (33:2805)
  • [2] A. Freire, A. Lopes, and R. Manñé, An invariant measure for rational maps, Bol. Soc. Brasil Mat. 14 (1983), 45-62. MR 736568 (85m:58110b)
  • [3] A. Lopes, Equilibrium measures for rational maps, Ergodic Theory Dynamical Systems 6 (1986), 414-426. MR 863202 (88e:58055)
  • [4] M. Ju. Ljubich, Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory Dynamical Systems 3 (1983), 351-386. MR 741393 (85k:58049)
  • [5] R. Manñé, On the uniqueness of the maximizing measure for rational maps, Bol. Soc. Brasil Mat. 14 (1983), 27-43. MR 736567 (85m:58110a)
  • [6] Ch. Pommerenke, Uniformly perfect sets and the Poincaré metric, Arch. Math. 32 (1979), 192-199. MR 534933 (80j:30073)
  • [7] -, On uniformly perfect sets and Fuchsian groups, Analysis 4 (1986), 299-321. MR 780609 (86e:30044)
  • [8] M. Tsuji, Potential theory in modern function theory, Maruzen, Tokyo, 1959. MR 0114894 (22:5712)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 58F23, 30D05, 31A25, 58F11

Retrieve articles in all journals with MSC: 58F23, 30D05, 31A25, 58F11


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1992-1106180-2
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society