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Connectedness of the basin of attraction
for rational maps


Author: Krzysztof Baranski
Journal: Proc. Amer. Math. Soc. 126 (1998), 1857-1866
MSC (1991): Primary 58F23
DOI: https://doi.org/10.1090/S0002-9939-98-04184-7
MathSciNet review: 1443144
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Abstract: We prove some results concerning degree of a rational map on the immediate basin $B(s)$ of an attracting fixed point $s$. In particular, if $B(s)$ contains all but two critical points or values counted with multiplicity, then the entire basin of attraction is connected. For any number $k \geq 3$ we give examples of rational maps with disconnected basin of an attracting fixed point such that there are exactly $k$ critical points outside the immediate basin of attraction.


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

  • [Ba] K. Bara\'{n}ski, Ph.D. thesis, in preparation.
  • [DH] A. Douady and J. Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. École Norm. Sup. 18 (1985), 287-343. MR 87f:58083
  • [HP] F. von Haeseler and H.-O. Peitgen, Newton's method and complex dynamical systems, Acta Appl. Math. 13 (1988), 3-58. MR 90a:58102
  • [LM] M. Lyubich and J. Milnor, The Fibonacci unimodal maps, J. Amer. Math. Soc. 6 (1993), 425-457. MR 93h:58080
  • [Ma] P. Makienko, Pinching and plumbing deformations of quadratic rational maps, preprint, Internat. Centre Theoret. Phys., Miramare-Trieste, 1993
  • [Mi] J. Milnor, Dynamics in one complex variable: introductory lectures, preprint, SUNY at Stony Brook, IMS # 1990/5.
  • [P1] F. Przytycki, Iterations of rational functions: which hyperbolic components contain polynomials?, Fund. Math. 149 (1996), 95-118. MR 97e:58199
  • [P2] -, Remarks on simple-connectedness of basins of sinks for iterations of rational maps, in: Banach Center Publ. 23, PWN, 1989, 229-235. MR 92e:58180
  • [Sh] M. Shishikura, The connectivity of the Julia set of rational maps and fixed points, preprint, Inst. Hautes Études Sci., Bures-sur-Yvette, 1990.

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Additional Information

Krzysztof Baranski
Affiliation: Institute of Mathematics, Warsaw University, ul. Banacha 2, 02-097 Warsaw, Poland
Email: baranski@mimuw.edu.pl

DOI: https://doi.org/10.1090/S0002-9939-98-04184-7
Received by editor(s): May 1, 1996
Received by editor(s) in revised form: November 14, 1996
Additional Notes: Research supported by Polish KBN Grant No 2 P301 01307.
Communicated by: Mary Rees
Article copyright: © Copyright 1998 American Mathematical Society

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