Wandering orbit portraits
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- by Jan Kiwi
- Trans. Amer. Math. Soc. 354 (2002), 1473-1485
- DOI: https://doi.org/10.1090/S0002-9947-01-02896-3
- Published electronically: November 20, 2001
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Abstract:
We study a counting problem in holomorphic dynamics related to external rays of complex polynomials. We give upper bounds on the number of external rays that land at a point $z$ in the Julia set of a polynomial, provided that $z$ has an infinite forward orbit.References
- Pau Atela, Bifurcations of dynamic rays in complex polynomials of degree two, Ergodic Theory Dynam. Systems 12 (1992), no. 3, 401–423. MR 1182654, DOI 10.1017/S0143385700006854
- A. Blokh and G. Levin, Growing trees, laminations and the dynamics on the Julia set, Preprint IHES, September 1999.
- Lennart Carleson and Theodore W. Gamelin, Complex dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR 1230383, DOI 10.1007/978-1-4612-4364-9
- Adrien Douady, Descriptions of compact sets in $\textbf {C}$, Topological methods in modern mathematics (Stony Brook, NY, 1991) Publish or Perish, Houston, TX, 1993, pp. 429–465. MR 1215973
- Radu Bǎdescu, On a problem of Goursat, Gaz. Mat. 44 (1939), 571–577. MR 0000087
- De Hai Zhang, $q$-deformed Gel′fand-Dikiĭ potentials of quantum deformation KdV equation, Proceedings of the 1992 Nonlinear Science Symposium (Chinese) (Hefei, 1992), 1993, pp. 97–101 (English, with English and Chinese summaries). MR 1228289
- Lisa R. Goldberg and John Milnor, Fixed points of polynomial maps. II. Fixed point portraits, Ann. Sci. École Norm. Sup. (4) 26 (1993), no. 1, 51–98. MR 1209913, DOI 10.24033/asens.1667
- K. Keller. Invariant factors, Julia equivalences and the (abstract) Mandelbrot set. Lecture Notes in Mathematics, 1732. Springer-Verlag, Berlin, 2000.
- J. Kiwi, Rational Rays and Critical Portraits of Complex Polynomials, Thesis, SUNY at Stony Brook, 1997. (Stony Brook IMS Preprint 1997/15)
- J. Kiwi, Rational laminations of complex polynomials, pp 111–154 in Laminations and Foliations in Geometry, Topology and Dynamics, ed. M. Lyubich et al., Contemporary Mathematics 269, 2001.
- G. Levin, On backward stability of holomorphic dynamical systems, Fund. Math. 158 (1998), no. 2, 97–107. MR 1656942, DOI 10.4064/fm-158-2-97-107
- J. Milnor, Dynamics in one complex variable: Introductory Lectures, Vieweg, 1999.
- J. Milnor, Periodic orbits, external rays and the Mandelbrot set: an expository account, pp 277-331 in Géométrie complexe et sytèmes dynamiques (Orsay, 1995), edited by M. Flexor et al., Astérique 261, 2000.
- W. P. Thurston, On the combinatorics of iterated rational maps, Manuscript, 1985.
Bibliographic Information
- Jan Kiwi
- Affiliation: Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile
- Email: jkiwi@mat.puc.cl
- Received by editor(s): April 11, 2000
- Received by editor(s) in revised form: March 29, 2001
- Published electronically: November 20, 2001
- Additional Notes: Supported by “Proyecto Fondecyt #1990436”, “Fundación Andes, Chile” and “Cátedra Presidencial en Geometría”.
- © Copyright 2001 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 354 (2002), 1473-1485
- MSC (2000): Primary 37F10, 37F20
- DOI: https://doi.org/10.1090/S0002-9947-01-02896-3
- MathSciNet review: 1873015