## Wandering orbit portraits

HTML articles powered by AMS MathViewer

- by Jan Kiwi PDF
- Trans. Amer. Math. Soc.
**354**(2002), 1473-1485 Request permission

## 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.

## Additional 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