Download PDF - On biaccessible points in Julia sets of polynomialsAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

On biaccessible points in Julia sets of polynomials

Authors 1

Affiliations

  1. Institute of Mathematics, Warsaw University, Banacha 2, 02-097 Warszawa, Poland

Abstract

Let f be a polynomial of one complex variable so that its Julia set is connected. We show that the harmonic (Brolin) measure of the set of biaccessible points in J is zero except for the case when J is an interval.

Bibliography

  1. [DMNU] M. Denker, R. D. Mauldin, Z. Nitecki and M. Urbański, Conformal measures for rational functions revisited, Fund. Math. 157 (1998), 161-173.
  2. [DU] M. Denker and M. Urbański, Ergodic theory of equillibrium states for rational maps, Nonlinearity 4 (1991), 103-134.
  3. [DH] A. Douady et J. Hubbard, Etude dynamique des polynômes complexes (première partie), Publ. Math. d'Orsay, 84-02.
  4. [FKS] S. Fomin, I. Kornfeld and Ya. Sinai, Ergodic Theory, Springer, Berlin, 1982.
  5. [FLM] A. Freire, A. Lopes and R. Mañé, An invariant measure for rational maps, Bol. Soc. Brasil. Mat. 14 (1983), 45-62.
  6. [Ja] M. Jakobson, On the classification of polynomial endomorphisms of the plane, Mat. Sb. 80 (1969), 365-387 (in Russian).
  7. [Ma] R. Mañé, On the Bernoulli property for rational maps, Ergodic Theory Dynam. Systems 5 (1985), 71-88.
  8. [Po] C. Pommerenke, Boundary Behaviour of Conformal Maps, Springer, Berlin, 1992.
  9. [P] F. Przytycki, Remarks on simple connectedness of basins of sinks for iterations of rational maps, in: Banach Center Publ. 23, PWN, 1989, 229-235.
  10. [PUbook] F. Przytycki and M. Urbański, Fractals in the Plane-Ergodic Theory Methods, to appear; preliminary version at www.math.unt.edu/urbanski.
  11. [Th] W. Thurston, On the dynamics of iterated rational maps, preprint.
  12. [Za] S. Zakeri, Biaccessibility in quadratic Julia sets, I: the locally connected case, preprint SUNY, Stony Brook, 1998.
  13. [Z] A. Zdunik, Parabolic orbifolds and the dimension of the maximal measure for rational maps, Invent. Math. 99 (1990), 627-649.
Fundamenta Mathematicae
2000/163/3
Export to PBN, Opens in new tab
Pages:
277-286
Main language of publication
English
Received
1999-05-25
Accepted
1999-09-20
Published
2000
Exact and natural sciences