Eremenko, A.; Lyubich, M. Dynamical properties of some classes of entire functions. (English) Zbl 0735.58031 Ann. Inst. Fourier 42, No. 4, 989-1020 (1992). The paper contains the detailed exposition in English of the results obtained in 1983-1984. It is concerned with the dynamics of an entire transcendental function whose inverse has only finitely many singularities. It is proven that there are no escaping orbits on the Fatou set. Together with absence of non-wandering domains this yields a complete picture of the dynamics on the Fatou set. It is shown under some extra assumptions that the set of escaping orbits has actually zero Lebesgue measure. If a function depends analytically on parameters then no periodic points escape to \(\infty\). This yields the \(\lambda\)-lemma and the Structural Stability Theorem. The main tool in this paper is a ”logarithmic change of variable at \(\infty\)”. Reviewer: A.Eremenko Cited in 11 ReviewsCited in 199 Documents MSC: 37F99 Dynamical systems over complex numbers 30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable Keywords:dynamics; periodic points; entire function; Julia set; structural stability PDFBibTeX XMLCite \textit{A. Eremenko} and \textit{M. Lyubich}, Ann. Inst. Fourier 42, No. 4, 989--1020 (1992; Zbl 0735.58031) Full Text: DOI Numdam EuDML References: [1] L. AHLFORS, Untersuchungen zur Theorie der konformen Abbildung und der ganzen Funktionen, Acta soc. sci. fenn., N.S.I., 9 (1930), 1-40. · JFM 56.0984.02 [2] L. AHLFORS, Conformal invariants, McGraw-Hill, New York, 1973. · Zbl 0272.30012 [3] L. AHLFORS, L. BERS, Riemann mapping theorem for variable metrics, Ann. Math., 72, 2 (1960), 385-404. · Zbl 0104.29902 [4] I.N. BAKER, Repulsive fixpoints of entire functions, Math. Z., 104 (1968), 252-256. · Zbl 0172.09502 [5] I.N. BAKER, The domains of normality of an entire function, Ann. Acad. Sci. Fenn., S.A.I., 1 (1975), 277-283. · Zbl 0329.30019 [6] I.N. BAKER, An entire function which has wandering domains, J. Austral. Math. Soc., 22, 2 (1976), 173-176. · Zbl 0335.30001 [7] I.N. BAKER, Wandering domains in the iteration of entire functions, Proc. London Math. Soc., 49 (1984), 563-576. · Zbl 0523.30017 [8] I.N. BAKER, P. RIPPON, Iteration of exponential functions, Ann. Acad. Sci. Fenn., S.A.I., 9 (1984), 49-77. · Zbl 0558.30029 [9] L. BERS, H.L. ROYDEN, Holomorphic families of injections, Acta Math., 3-4 (1986), 259-286. · Zbl 0619.30027 [10] P. BLANCHARD, Complex analytic dynamics on the Riemann sphere, Bull. Amer. Math. Soc., II, 1 (1984), 85-141. · Zbl 0558.58017 [11] R.L. DEVANEY, Structural instability of exp z, Proc. Amer. Math. Soc., 94, 3 (1985), 545-548. · Zbl 0575.58027 [12] R.L. DEVANEY, M. KRYCH, Dynamics of exp z, Ergod. Theory and Dyn. Syst., 4 (1984), 35-52. · Zbl 0567.58025 [13] R.L. DEVANEY, L. GOLDBERG, J.H. HUBBARD, A dynamical approximation to the exponential map by polynomials, Preprint, Berkeley, 1986. [14] A. DOUADY, J.H. HUBBARD, Itération des polynômes quadratiques complexes, C.R. Acad. Sci., 294, 3 (1982), 123-126. · Zbl 0483.30014 [15] A. DOUADY, J.H. HUBBARD, Étude dynamique des polynômes complexes (Première partie), Publ. Math. d’Orsay, 84-02. · Zbl 0552.30018 [16] A.E. EREMENKO, On the iteration of entire functions, Proc. Banach Center Publ. Warsaw, 1989. · Zbl 0692.30021 [17] A.E. EREMENKO, M. Yu. LYUBICH, Iterates of entire functions, Soviet Math. Dokl., 30, 3 (1984), 592-594. · Zbl 0588.30027 [18] A.E. EREMENKO, M. Yu. LYUBICH, Iterates of entire functions, Preprint, Inst. for Low Temperatures, Kharkov, 6, 1984 (Russian). · Zbl 0588.30027 [19] A.E. EREMENKO, M. Yu. LYUBICH, Structural stability in some families on entire functions, Functional Anal. Appl., 19, 4 (1985), 323-324. · Zbl 0597.30032 [20] A.E. EREMENKO, M. Yu. LYUBICH, Structural stability in some families of entire functions, Preprint, Inst. for Low Temperatures, Kharkov, 29, 1984 (Russian). [21] A.E. EREMENKO, M. Yu. LYUBICH, Examples of entire functions with pathological dynamics, J. London Math. Soc., 36 (1987), 458-468. · Zbl 0601.30033 [22] A.E. EREMENKO, M. Yu. LYUBICH, Dynamics of analytic transformations, Algebra and Analysis (Leningrad J. Math.), 1, 3 (1989), 1-70. · Zbl 0712.58036 [23] P. FATOU, Sur les équations fonctionnelles, Bull. Soc. Math. France, 47 (1919), 161-271. · JFM 47.0921.02 [24] P. FATOU, Sur les équations fonctionnelles, Bull. Soc. Math. France, 48 (1920), 33-94, 208-314. · JFM 47.0921.02 [25] P. FATOU, Sur l’itération des fonctions transcendantes entières, Acta math., 47 (1926), 337-370. · JFM 52.0309.01 [26] L.R. GOLDBERG, L. KEEN, A finiteness theorem for a dynamical class of entire functions, Ergod. Theory and Dyn. Syst., 6 (1986), 183-192. · Zbl 0657.58011 [27] M.R. HERMAN, Are there critical points on the boundaries of singular domains ? Commun. Math. Phys., 99 (1985), 593-612. · Zbl 0587.30040 [28] G. JULIA, Mémoire sur l’itération des fonctions rationnelles, J. de Math. Pures et Appliquées (8), 1 (1918), 47-245. · JFM 46.0520.06 [29] O. LEHTO, K. VIRTANEN, Quasiconformal mappings in the plane, Springer-Verlag, Berlin, 1973. · Zbl 0267.30016 [30] M. Yu. LYUBICH, On a typical behavior of trajectories of a rational mapping of the sphere, Soviet Math. Dokl., 27 (1983), 22-25. · Zbl 0595.30034 [31] M. Yu. LYUBICH, Some typical properties of the dynamics of rational maps, Russian Math. Surveys, 8, 5 (1983), 154-155. · Zbl 0598.58028 [32] M. Yu. LYUBICH, The dynamics of rational transforms: the topological picture, Russian Math. Surveys, 41, 4 (1986), 43-117. · Zbl 0619.30033 [33] M. Yu. LYUBICH, The measurable dynamics of the exponential map, Siberian Journ. Math., 28, 5 (1987), 111-127. · Zbl 0667.58037 [34] J. MILNOR, Dynamics in one complex variable, Introductory lectures, Preprint SUNY Stony Brook, 1990/5. [35] R. MAÑÉ, P. SAD, D. SULLIVAN, On the dynamics of rational maps, Ann. Scient. Ec. Norm. Sup. 4e série, 16 (1983), 193-217. · Zbl 0524.58025 [36] C. MCMULLEN, Area and Hausdorff dimension of Julia sets of entire functions, Trans. Amer. Math. Soc., 300, 1 (1987), 325-342. · Zbl 0618.30027 [37] M. MISIUREWICZ, On iterates of ez, Ergod. Theory and Dyn. Syst., 1 (1981), 103-106. · Zbl 0466.30019 [38] P. MONTEL, Leçons sur les familles normales de fonctions analytiques et leurs applications, Gauthier-Villars, Paris, 1927. · JFM 53.0303.02 [39] R. NEVANLINNA, Analytic functions, Springer, Berlin-Heidelberg-New York, 1970. · Zbl 0199.12501 [40] M. REES, The exponential map is not recurrent, Math. Z., 191, 4 (1986), 593-598. · Zbl 0595.30033 [41] M. SHISHIKURA, On the quasi-conformal surgery of rational functions, Ann. Sci. École Norm. Sup., 20 (1987), 1-29. · Zbl 0621.58030 [42] D. SULLIVAN, Quasi-conformal homeomorphisms and dynamics, I. Ann. Math., 122, 3 (1985), 402-418. [43] D. SULLIVAN, Quasi-conformal homeomorphisms and dynamics, III. Preprint, IHES, 1982. [44] H. TÖPFER, Über die Iteration der ganzen transzendenten Funktionen, insbesondere von sinz und cosz, Math. Ann., 117 (1939), 65-84. · JFM 65.0327.05 [45] G. VALIRON, Fonctions analytiques, Presses Universitaires de France, Paris, 1954. · Zbl 0055.06702 [46] H. WITTICH, Neuere Untersuchungen Über eindeutige analytische Funktionen, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1955. · Zbl 0067.05501 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.