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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

MSC:
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
Keywords:
critical points
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[1] Ahlfors, L.V.: Lectures on quasiconformal mappings. Princeton: Van Nostrand 1966 · Zbl 0138.06002
[2] Carathéodory, C.: Theory of functions of a complex variable. Vol. II. New York: Chelsea 1954 · Zbl 0056.06703
[3] Fatou, M.P.: Sur les équations fonctionnelles. Bull. Soc. Math. Fr.47, 161-271 (1919);48, 33-94, 208-304 (1920) · JFM 47.0921.02
[4] Ghys, E.: Transformation holomorphe au voisinage d’une courbe de Jordan. C.R. Acad. Sc. Paris, t.289, 385-388 (1984) · Zbl 0573.58021
[5] Herman, M.R.: Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations. Publ. Math. I.H.E.S.49, 5-233 (1979) · Zbl 0448.58019
[6] Herman, M.R.: Examples de fractions rationnelles ayant une orbite dense sur la sphère de Riemann. Bull. Soc. Math. France112, 93-142 (1984) · Zbl 0559.58020
[7] Pommerenke, C.: Univalent functions. Göttingen: Vandenhoeck & Ruprecht 1975
[8] Rudin, W.: Real and complex analysis. New York: McGraw-Hill 1974 · Zbl 0278.26001
[9] Siegel, C.L.: Iteration of analytic functions. Ann. Math.43, 607-612 (1942) · Zbl 0061.14904 · doi:10.2307/1968952
[10] Yoccoz, J.C.: Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation véfifie une condition diophantienne. Ann. Sci. Ec. Norm. Sup., 4ème séries, t.17, 333-359 (1984) · Zbl 0595.57027
[11] Yoccoz, J.C.:C 1-conjugaison des difféomorphismes du cercle. Lecture Notes in Mathematics, Vol. 1007, pp. 814-827. Berlin, Heidelberg, New York: Springer 1983
[12] Ahlfors, L.V., Sario, L.: Riemann surfaces. Princeton, NJ: Princeton University Press 1960 · Zbl 0196.33801
[13] Baker, I.N.: Repulsive fixpoints of entire functions. Math. Z.104, 252-256 (1968) · Zbl 0172.09502 · doi:10.1007/BF01110294
[14] Baker, I.N.: The domains of normality of an entire function. Ann. Acad. Fennicae, Ser. A I1, 277-283 (1975) · Zbl 0329.30019
[15] Baker, I.N.: Limit functions and sets of non-normality in iteration theory. Ann. Acad. Sci. Fennicae, Ser. A I467, 1-11 (1970) · Zbl 0197.05302
[16] Baker, I.N.: An entire function which has wandering domains. J. Australian Math. Soc. (Ser. A)22, 173-176 (1976) · Zbl 0335.30001 · doi:10.1017/S1446788700015287
[17] Baker, I.N.: Wandering domains in the iteration of entire functions, Report No. 20, Institut Mittag-Leffler (1983)
[18] Baker, I.N., Rippon, P.J.: Iteration of exponential functions, Ann. Acad. Sci. Fennicae, Ser. AI9, 49-77 (1984) · Zbl 0558.30029
[19] Brjuno, A.D.: Analytical form of a differential equation. Trans. Moscow Math. Soc.25, 131-288 (1971)
[20] Brolin, H.: Invariant sets under iteration of rational functions. Ark. Mat.6, 103-144 (1966) · Zbl 0127.03401 · doi:10.1007/BF02591353
[21] Cremer, H.: Über die Häufigkeit der Nichtzentren. Math. Ann.115, 573-580 (1938) · JFM 64.0289.02 · doi:10.1007/BF01448957
[22] Douday, A., Hubbard, J.H.: On the dynamics of polynomial like mappings. Ann. Sci. Ec. Nor. Sup. (to appear)
[23] Duren, P.L.: Theory ofH p spaces. New York: Academic Press 1970 · Zbl 0215.20203
[24] Fatou, P.: Sur l’itération des fonctions transcendantes entières. Acta Math.47, 337-370 (1926) · JFM 52.0309.01 · doi:10.1007/BF02559517
[25] Ghys, E., Goldberg, L., Sullivan, D.: On the measurable dynamics ofz ?e z . Erg. Th. Dyna. Sys. (to appear) · Zbl 0616.58006
[26] Hocking, J.G., Young, G.S.: Topology, pp. 143-145. Reading, MA: Addison-Wesley 1961
[27] Misiurewicz, M.: On iterates ofe z . Erg. Th. Dyna. Syst.1, 103-106 (1981) · Zbl 0466.30019
[28] Sullivan, D.: Quasiconformal homeomorphisms and dynamics I, preprint I.H.E.S. (1982). Ann. Math. (to appear)
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