On local connectivity for the Julia set of rational maps: Newton’s famous example
نویسندگان
چکیده
منابع مشابه
Connectivity of Julia Sets for Singularly Perturbed Rational Maps
In this paper we consider the family of rational maps of the form F λ (z) = z n + λ/z n where n ≥ 2. It is known that there are two cases where the Julia sets of these maps are not connected. If the critical values of F λ lie in the basin of ∞, then the Julia set is a Cantor set. And if the critical values lie in the preimage of the basin surrounding the pole at 0, then the Julia set is a Canto...
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In this paper, we consider the family of rational maps Fλ(z) = z n + λ zd , where n ≥ 2, d ≥ 1, and λ ∈ C. We consider the case where λ lies in the main cardioid of one of the n − 1 principal Mandelbrot sets in these families. We show that the Julia sets of these maps are always homeomorphic. However, two such maps Fλ and Fμ are conjugate on these Julia sets only if the parameters at the center...
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We prove that the Julia set J(f) of at most finitely renormalizable unicritical polynomial f : z 7→ z + c with all periodic points repelling is locally connected. (For d = 2 it was proved by Yoccoz around 1990.) It follows from a priori bounds in a modified Principle Nest of puzzle pieces. The proof of a priori bounds makes use of new analytic tools developed in [KL] that give control of moduli...
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ژورنال
عنوان ژورنال: Annals of Mathematics
سال: 2008
ISSN: 0003-486X
DOI: 10.4007/annals.2008.168.127