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Are there critical points on the boundaries of singular domains? (English) Zbl 0587.30040
In continuation with the works of M. P. Fatou, C. L. Siegel and E. Ghys, the author proves that if a rational function f of the Riemann sphere of degree not less than two leaves invariant a singular domain C on which the rotation number of f satisfies a diophantine condition, provided that on \(\bar C\) f is injective, then each boundary component of C contains critical points of f. Several applications of the main theorem are pointed out. Furthermore a survey of the theory of iteration of entire functions of \({\mathbb{C}}\) is made.
Reviewer: S.K.Chatterjea

30E99 Miscellaneous topics of analysis in the complex plane
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
58C05 Real-valued functions on manifolds
critical points
Full Text: DOI
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