On local connectivity for the Julia set of rational maps: Newton’s famous example
Annals of mathematics, Tome 168 (2008) no. 1, pp. 127-174
We show that Newton’s cubic methods (famous rational maps) have a locally connected Julia set except in some very specific cases. In particular, when those maps are infinitely renormalizable their Julia set is locally connected and contains small copies of nonlocally connected quadratic Julia sets. This also holds when Newton’s method is renormalizable and has Cremer points, unlike the polynomial case. After a dynamical description we show the necessity of the Brjuno condition within this family.
@article{10_4007_annals_2008_168_127,
author = {Pascale Roesch},
title = {On local connectivity for the {Julia} set of rational maps: {Newton{\textquoteright}s} famous example},
journal = {Annals of mathematics},
pages = {127--174},
year = {2008},
volume = {168},
number = {1},
doi = {10.4007/annals.2008.168.127},
mrnumber = {2415400},
zbl = {1180.30033},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2008.168.127/}
}
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%0 Journal Article %A Pascale Roesch %T On local connectivity for the Julia set of rational maps: Newton’s famous example %J Annals of mathematics %D 2008 %P 127-174 %V 168 %N 1 %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2008.168.127/ %R 10.4007/annals.2008.168.127 %G en %F 10_4007_annals_2008_168_127
Pascale Roesch. On local connectivity for the Julia set of rational maps: Newton’s famous example. Annals of mathematics, Tome 168 (2008) no. 1, pp. 127-174. doi: 10.4007/annals.2008.168.127
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