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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
##### Keywords:
external rays; Julia st; fixed point
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