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Fixed points of polynomial maps. II: Fixed point portraits. (English) Zbl 0771.30028

[For a review of part I see the preceding review.]
Let \(f\) be a monic polynomial whose filled-in Julia set \(K(f)\) is connected and denote by \(\varphi\) the conformal map from \(\widehat {\mathbb{C}}\setminus D\) to \(\widehat {\mathbb{C}}\setminus K\) which is asymptotic to the identity at \(\infty\). It turns out that (1) \(\varphi(z^ d)=f(\varphi(z))\). For \(t\in\mathbb{R}\setminus\mathbb{Z}\) the external ray \(R_ t=\{\varphi(r\exp(2\pi it))\), \(1<r<-\infty\}\). If \(t\) is rational \(R_ t\) lands at a point \(\xi=\lim_{r\to 1}\varphi(r\exp(2\pi t))\) in the Julia set \(J(f)\). Every repelling or parabolic fixed point of \(f\) is the landing point of a finite (non-empty) set of external rays. If no rational external rays land at a fixed point (for example because it is attracting), the point is called rationally invisible. By (1) the rays with \(t=j/(d-1)\), \(0\leq j<d\), are fixed under \(f\). If at least one of these last rays lands at a fixed point, then it is said to have rotation number \(\rho=0\) and its type \(T\) is the set of such fixed rays which land at the point. Finally, if the set \(T\) of rational rays which land at a fixed point is non-empty and does not contain fixed rays, then \(T\) forms a degree \(d\) rotation set in the sense of Part I and has some rational combinatorial rotation number \({p\over q}\neq 0\) in \(\mathbb{Q}/\mathbb{Z}\). The fixed point portrait of \(f\) is the collection \(P=\{T_ 1,\dots,T_ k\}\) of the types of its rationally visible fixed points. The authors obtain four necessary conditions for \(P\):
P1. Each \(T_ j\), \(1\leq j\leq k\) (\(\leq d\)) is a degree \(d\) rational rotation set with some well-defined rotation number \(\rho_ j\).
P2. If \(i\neq j\), \(T_ i\subset\;\) a single connected component of \((\mathbb{R}/\mathbb{Z})-T_ j\).
P3. The union of \(T_ j\) with rotation number zero is the set of fixed external rays.
P4. If \(T_ i\), \(T_ j\) have non-zero rotation numbers and \(i\neq j\), then \(T_ i\), \(T_ j\) are separated in \(\mathbb{R}/\mathbb{Z}\) by at least one \(T_ \ell\) with rotation number zero.
The question arises whether these conditions are also sufficient and the main result of the paper is that this is indeed so if \(k=d\): given \(d\) non-vacuous rational types satisfying P1-4 there is a critically preperiodic polynomial of degree \(d\) for which this is the fixed point portrait. (The general case has been solved subsequently by A. Poirier, Stony Brook I.M.S., Preprint 1991/20).
Appendices give extensions to cases where the Julia set is not connected, to examining the change in the fixed point portrait under change in the parameters of \(f\) and to applying the results in the special case of degree two.
Reviewer: I.N.Baker (London)

MSC:

30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
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References:

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