Connectedness of the basin of attraction for rational maps

Barański, Krzysztof

doi:10.1090/S0002-9939-98-04184-7

Connectedness of the basin of attraction for rational maps
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by Krzysztof Barański PDF

Proc. Amer. Math. Soc. 126 (1998), 1857-1866 Request permission

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

K. Barański, Ph.D. thesis, in preparation.
Adrien Douady and John Hamal Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 2, 287–343. MR 816367
John W. Green, Harmonic functions in domains with multiple boundary points, Amer. J. Math. 61 (1939), 609–632. MR 90, DOI 10.2307/2371316
Mikhail Lyubich and John Milnor, The Fibonacci unimodal map, J. Amer. Math. Soc. 6 (1993), no. 2, 425–457. MR 1182670, DOI 10.1090/S0894-0347-1993-1182670-0
P. Makienko, Pinching and plumbing deformations of quadratic rational maps, preprint, Internat. Centre Theoret. Phys., Miramare-Trieste, 1993
J. Milnor, Dynamics in one complex variable: introductory lectures, preprint, SUNY at Stony Brook, IMS # 1990/5.
Feliks Przytycki, Iterations of rational functions: which hyperbolic components contain polynomials?, Fund. Math. 149 (1996), no. 2, 95–118. MR 1376666, DOI 10.4064/fm-149-2-95-118
Feliks Przytycki, Remarks on the simple connectedness of basins of sinks for iterations of rational maps, Dynamical systems and ergodic theory (Warsaw, 1986) Banach Center Publ., vol. 23, PWN, Warsaw, 1989, pp. 229–235. MR 1102717
M. Shishikura, The connectivity of the Julia set of rational maps and fixed points, preprint, Inst. Hautes Études Sci., Bures-sur-Yvette, 1990.