Density of periodic sources in the boundary of a basin of attraction for iteration of holomorphic maps: geometric coding trees technique

Feliks Przytycki; Anna Zdunik

Fundamenta Mathematicae (1994)

  • Volume: 145, Issue: 1, page 65-77
  • ISSN: 0016-2736

Abstract

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We prove that if A is a basin of immediate attraction to a periodic attracting or parabolic point for a rational map f on the Riemann sphere, then the periodic points in the boundary of A are dense in this boundary. To prove this in the non-simply connected or parabolic situations we prove a more abstract, geometric coding trees version.

How to cite

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Przytycki, Feliks, and Zdunik, Anna. "Density of periodic sources in the boundary of a basin of attraction for iteration of holomorphic maps: geometric coding trees technique." Fundamenta Mathematicae 145.1 (1994): 65-77. <http://eudml.org/doc/212034>.

@article{Przytycki1994,
abstract = {We prove that if A is a basin of immediate attraction to a periodic attracting or parabolic point for a rational map f on the Riemann sphere, then the periodic points in the boundary of A are dense in this boundary. To prove this in the non-simply connected or parabolic situations we prove a more abstract, geometric coding trees version.},
author = {Przytycki, Feliks, Zdunik, Anna},
journal = {Fundamenta Mathematicae},
keywords = {periodic sources; iteration of holomorphic maps; rational self-map; Riemann sphere; immediate basin of attraction},
language = {eng},
number = {1},
pages = {65-77},
title = {Density of periodic sources in the boundary of a basin of attraction for iteration of holomorphic maps: geometric coding trees technique},
url = {http://eudml.org/doc/212034},
volume = {145},
year = {1994},
}

TY - JOUR
AU - Przytycki, Feliks
AU - Zdunik, Anna
TI - Density of periodic sources in the boundary of a basin of attraction for iteration of holomorphic maps: geometric coding trees technique
JO - Fundamenta Mathematicae
PY - 1994
VL - 145
IS - 1
SP - 65
EP - 77
AB - We prove that if A is a basin of immediate attraction to a periodic attracting or parabolic point for a rational map f on the Riemann sphere, then the periodic points in the boundary of A are dense in this boundary. To prove this in the non-simply connected or parabolic situations we prove a more abstract, geometric coding trees version.
LA - eng
KW - periodic sources; iteration of holomorphic maps; rational self-map; Riemann sphere; immediate basin of attraction
UR - http://eudml.org/doc/212034
ER -

References

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  1. [D] A. Douady, informal talk at the Durham Symposium, 1988. 
  2. [Du] P. L. Duren, Theory of H p Spaces, Academic Press, New York, 1970. 
  3. [EL] A. È. Eremenko and G. M. Levin, On periodic points of polynomials, Ukrain. Mat. Zh. 41 (1989), 1467-1471 (in Russian). Zbl0686.30019
  4. [F] P. Fatou, Sur les équations fonctionnelles, Bull. Soc. Math. France 48 (1920), 33-94. 
  5. [K] A. Katok, Lyapunov exponents, entropy and periodic points for diffeomorphisms, Publ. Math. IHES 51 (1980), 137-173. Zbl0445.58015
  6. [LP] G. Levin and F. Przytycki, External rays to periodic points, preprint 24 (1992/93), the Hebrew University of Jerusalem. Zbl0854.30020
  7. [Pe] C. L. Petersen, On the Pommerenke-Levin-Yoccoz inequality, Ergodic Theory Dynamical Systems 13 (1993), 785-806. Zbl0802.30022
  8. [Po] Ch. Pommerenke, On conformal mapping and iteration of rational functions, Complex Variables 5 (1986), 117-126. 
  9. [P1] F. Przytycki, Hausdorff dimension of harmonic measure on the boundary of an attractive basin for a holomorphic map, Invent. Math. 80 (1985), 161-179. Zbl0569.58024
  10. [P2] F. Przytycki, Riemann map and holomorphic dynamics, ibid. 85 (1986), 439-455. Zbl0616.58029
  11. [P3] F. Przytycki, On invariant measures for iterations of holomorphic maps, in: Problems in Holomorphic Dynamics, preprint IMS 1992/7, SUNY at Stony Brook. 
  12. [P4] F. Przytycki, Accessability of typical points for invariant measures of positive Lyapunov exponents for iterations of holomorphic maps, Fund. Math., to appear. 
  13. [PS] F. Przytycki and J. Skrzypczak, Convergence and pre-images of limit points for coding trees for iterations of holomorphic maps, Math. Ann. 290 (1991), 425-440. Zbl0704.30035
  14. [PUZ] F. Przytycki, M. Urbański and A. Zdunik, Harmonic, Gibbs and Hausdorff measures for holomorphic maps, Part 1: Ann. of Math. 130 (1989), 1-40; Part 2: Studia Math. 97 (1991), 189-225. Zbl0703.58036
  15. [Ro] V. A. Rokhlin, Lectures on the entropy theory of transformations with invariant measures, Uspekhi Mat. Nauk 22 (5) (1967), 3-56 (in Russian); English transl.: Russian Math. Surveys 22 (5) (1967), 1-52. 
  16. [R] D. Ruelle, An inequality for the entropy of differentiable maps, Bol. Soc. Brasil. Math. 9 (1978), 83-87. Zbl0432.58013

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