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**On local connectivity for the Julia set of rational maps: Newton’s famous example.**
*(English)*
Zbl 1180.30033

This paper is a very detailed study of Julia sets of a one-parameter family of rational maps of the Riemann sphere that, in a very precise sense, can be considered the next natural family to study after the quadratic family. Indeed, the first result of this paper shows that a rational map of degree \(d\geq 2\) whose critical points are all simple, and all fixed except one, necessarily is either a quadratic map or a cubic Newton map, that is a map of the form \(N(z)=z-P(z)/P'(z)\), where \(P\) is a polynomial of degree 3 with distinct roots (and hence, up to an affine conjugation, we can assume that \(P(z)=z^3+pz+1\) with \(p\in{\mathbb C}\)). The paper is thus devoted to the study of the dynamics of cubic Newton maps (or, more precisely, of what the author calls “genuine” cubic Newton maps; he excludes a few particular cases where the nonfixed critical point \(x_0\in{\mathbb C}\) belongs to the immediate basin of one of the roots of the polynomial \(P\), and the very special case when \(N(x_0)=\infty\)).

The study is based on Yoccoz’s puzzles, built using “articulated rays”, an interesting technical generalization of external rays. Using these, the author shows that the connected Fatou components of a genuine cubic Newton map without Siegel points are Jordan domains, and that in this case the Julia set is locally connected if the orbit of the nonfixed critical point does not accumulate the boundary of an invariant immediate basin of attraction (actually, the author proves local connectivity of the Julia set in a few other cases too; and he conjectures that the Julia set of a genuine cubic Newton map should always be locally connected). Furthermore, the author also provides interesting examples of cubic Newton maps with locally connected Julia set and Cremer points; and of cubic Newton maps with locally connected Julia set containing a wandering nontrivial continuum (and having repelling periodic points only). Finally, he shows that in this family the Brjuno condition characterizes Siegel points, in the sense that if a genuine cubic Newton map \(N\) has a periodic point \(x\) of multiplier \(\lambda=e^{2\pi i\theta}\), then \(x\) is a Siegel point if and only if \(\theta\) is a Brjuno number; moreover, if \(\theta\) is neither a Brjuno number nor rational, then \(N\) has small cycles accumulating at \(x\).

The study is based on Yoccoz’s puzzles, built using “articulated rays”, an interesting technical generalization of external rays. Using these, the author shows that the connected Fatou components of a genuine cubic Newton map without Siegel points are Jordan domains, and that in this case the Julia set is locally connected if the orbit of the nonfixed critical point does not accumulate the boundary of an invariant immediate basin of attraction (actually, the author proves local connectivity of the Julia set in a few other cases too; and he conjectures that the Julia set of a genuine cubic Newton map should always be locally connected). Furthermore, the author also provides interesting examples of cubic Newton maps with locally connected Julia set and Cremer points; and of cubic Newton maps with locally connected Julia set containing a wandering nontrivial continuum (and having repelling periodic points only). Finally, he shows that in this family the Brjuno condition characterizes Siegel points, in the sense that if a genuine cubic Newton map \(N\) has a periodic point \(x\) of multiplier \(\lambda=e^{2\pi i\theta}\), then \(x\) is a Siegel point if and only if \(\theta\) is a Brjuno number; moreover, if \(\theta\) is neither a Brjuno number nor rational, then \(N\) has small cycles accumulating at \(x\).

Reviewer: Marco Abate (Pisa)