Dynamical properties of some classes of entire functions
Annales de l'Institut Fourier, Tome 42 (1992) no. 4, pp. 989-1020.

Cet article étudie la dynamique d’une fonction entière transcendante dont l’inverse n’a qu’un nombre fini de singularités. On prouve qu’il n’y a pas d’orbites qui tendent à l’infini sur l’ensemble de Fatou. Sous certaines conditions supplémentaires l’ensemble des orbites tendant vers l’infini est de mesure de Lebesgue nulle. Si une fonction dépend analytiquement de paramètres, alors un point périodique comme fonction des paramètres n’a seulement que des singularités algébriques. Ceci implique le théorèm’e de stabilité structurelle.

The paper is concerned with the dynamics of an entire transcendental function whose inverse has only finitely many singularities. It is rpoven that there are no escaping orbits on the Fatou set. Under some extra assumptions the set of escaping orbits has zero Lebesgue measure. If a function depends analytically on parameters then a periodic point as a function of parameters has only algebraic singularities. This yields the Structural Stability Theorem.

@article{AIF_1992__42_4_989_0,
     author = {Eremenko, A. and Lyubich, M. Yu},
     title = {Dynamical properties of some classes of entire functions},
     journal = {Annales de l'Institut Fourier},
     pages = {989--1020},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {42},
     number = {4},
     year = {1992},
     doi = {10.5802/aif.1318},
     mrnumber = {93k:30034},
     zbl = {0735.58031},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.1318/}
}
TY  - JOUR
AU  - Eremenko, A.
AU  - Lyubich, M. Yu
TI  - Dynamical properties of some classes of entire functions
JO  - Annales de l'Institut Fourier
PY  - 1992
SP  - 989
EP  - 1020
VL  - 42
IS  - 4
PB  - Institut Fourier
PP  - Grenoble
UR  - http://archive.numdam.org/articles/10.5802/aif.1318/
DO  - 10.5802/aif.1318
LA  - en
ID  - AIF_1992__42_4_989_0
ER  - 
%0 Journal Article
%A Eremenko, A.
%A Lyubich, M. Yu
%T Dynamical properties of some classes of entire functions
%J Annales de l'Institut Fourier
%D 1992
%P 989-1020
%V 42
%N 4
%I Institut Fourier
%C Grenoble
%U http://archive.numdam.org/articles/10.5802/aif.1318/
%R 10.5802/aif.1318
%G en
%F AIF_1992__42_4_989_0
Eremenko, A.; Lyubich, M. Yu. Dynamical properties of some classes of entire functions. Annales de l'Institut Fourier, Tome 42 (1992) no. 4, pp. 989-1020. doi : 10.5802/aif.1318. http://archive.numdam.org/articles/10.5802/aif.1318/

[A1] L. Ahlfors, Untersuchungen zur Theorie der konformen Abbildung und der ganzen Funktionen, Acta soc. sci. fenn., N.S.I., 9 (1930), 1-40. | JFM

[A2] L. Ahlfors, Conformal invariants, McGraw-Hill, New York, 1973. | Zbl

[AB] L. Ahlfors, L. Bers, Riemann mapping theorem for variable metrics, Ann. Math., 72, 2 (1960), 385-404. | MR | Zbl

[B1] I.N. Baker, Repulsive fixpoints of entire functions, Math. Z., 104 (1968), 252-256. | MR | Zbl

[B2] I.N. Baker, The domains of normality of an entire function, Ann. Acad. Sci. Fenn., S.A.I., 1 (1975), 277-283. | MR | Zbl

[B3] I.N. Baker, An entire function which has wandering domains, J. Austral. Math. Soc., 22, 2 (1976), 173-176. | MR | Zbl

[B4] I.N. Baker, Wandering domains in the iteration of entire functions, Proc. London Math. Soc., 49 (1984), 563-576. | MR | Zbl

[BR] I.N. Baker, P. Rippon, Iteration of exponential functions, Ann. Acad. Sci. Fenn., S.A.I., 9 (1984), 49-77. | MR | Zbl

[BRo] L. Bers, H.L. Royden, Holomorphic families of injections, Acta Math., 3-4 (1986), 259-286. | MR | Zbl

[Bla] P. Blanchard, Complex analytic dynamics on the Riemann sphere, Bull. Amer. Math. Soc., II, 1 (1984), 85-141. | MR | Zbl

[D] R.L. Devaney, Structural instability of exp z, Proc. Amer. Math. Soc., 94, 3 (1985), 545-548. | MR | Zbl

[DK] R.L. Devaney, M. Krych, Dynamics of exp z, Ergod. Theory and Dyn. Syst., 4 (1984), 35-52. | MR | Zbl

[DGH] R.L. Devaney, L. Goldberg, J.H. Hubbard, A dynamical approximation to the exponential map by polynomials, Preprint, Berkeley, 1986.

[DH1] A. Douady, J.H. Hubbard, Itération des polynômes quadratiques complexes, C.R. Acad. Sci., 294, 3 (1982), 123-126. | MR | Zbl

[DH2] A. Douady, J.H. Hubbard, Étude dynamique des polynômes complexes (Première partie), Publ. Math. d'Orsay, 84-02. | MR | Zbl

[E] A.E. Eremenko, On the iteration of entire functions, Proc. Banach Center Publ. Warsaw, 1989. | MR | Zbl

[EL1] A.E. Eremenko, M. Yu. Lyubich, Iterates of entire functions, Soviet Math. Dokl., 30, 3 (1984), 592-594. | Zbl

[EL2] A.E. Eremenko, M. Yu. Lyubich, Iterates of entire functions, Preprint, Inst. for Low Temperatures, Kharkov, 6, 1984 (Russian). | Zbl

[EL3] A.E. Eremenko, M. Yu. Lyubich, Structural stability in some families on entire functions, Functional Anal. Appl., 19, 4 (1985), 323-324. | Zbl

[EL4] A.E. Eremenko, M. Yu. Lyubich, Structural stability in some families of entire functions, Preprint, Inst. for Low Temperatures, Kharkov, 29, 1984 (Russian).

[EL5] A.E. Eremenko, M. Yu. Lyubich, Examples of entire functions with pathological dynamics, J. London Math. Soc., 36 (1987), 458-468. | MR | Zbl

[EL6] A.E. Eremenko, M. Yu. Lyubich, Dynamics of analytic transformations, Algebra and Analysis (Leningrad J. Math.), 1, 3 (1989), 1-70. | MR | Zbl

[F1] P. Fatou, Sur les équations fonctionnelles, Bull. Soc. Math. France, 47 (1919), 161-271. | JFM | Numdam

[F2] P. Fatou, Sur les équations fonctionnelles, Bull. Soc. Math. France, 48 (1920), 33-94, 208-314. | JFM | Numdam

[F3] P. Fatou, Sur l'itération des fonctions transcendantes entières, Acta math., 47 (1926), 337-370. | JFM

[GK] L.R. Goldberg, L. Keen, A finiteness theorem for a dynamical class of entire functions, Ergod. Theory and Dyn. Syst., 6 (1986), 183-192. | MR | Zbl

[H] M.R. Herman, Are there critical points on the boundaries of singular domains ? Commun. Math. Phys., 99 (1985), 593-612. | MR | Zbl

[J] G. Julia, Mémoire sur l'itération des fonctions rationnelles, J. de Math. Pures et Appliquées (8), 1 (1918), 47-245. | JFM | Numdam

[LV] O. Lehto, K. Virtanen, Quasiconformal mappings in the plane, Springer-Verlag, Berlin, 1973. | MR | Zbl

[L1] M. Yu. Lyubich, On a typical behavior of trajectories of a rational mapping of the sphere, Soviet Math. Dokl., 27 (1983), 22-25. | Zbl

[L2] M. Yu. Lyubich, Some typical properties of the dynamics of rational maps, Russian Math. Surveys, 8, 5 (1983), 154-155. | Zbl

[L3] M. Yu. Lyubich, The dynamics of rational transforms: the topological picture, Russian Math. Surveys, 41, 4 (1986), 43-117. | Zbl

[L4] M. Yu. Lyubich, The measurable dynamics of the exponential map, Siberian Journ. Math., 28, 5 (1987), 111-127. | Zbl

[Mi] J. Milnor, Dynamics in one complex variable, Introductory lectures, Preprint SUNY Stony Brook, 1990/5.

[MSS] R. Mañé, P. Sad, D. Sullivan, On the dynamics of rational maps, Ann. Scient. Ec. Norm. Sup. 4e série, 16 (1983), 193-217. | Numdam | MR | Zbl

[McM] C. Mcmullen, Area and Hausdorff dimension of Julia sets of entire functions, Trans. Amer. Math. Soc., 300, 1 (1987), 325-342. | MR | Zbl

[M] M. Misiurewicz, On iterates of ez, Ergod. Theory and Dyn. Syst., 1 (1981), 103-106. | MR | Zbl

[Mo] P. Montel, Leçons sur les familles normales de fonctions analytiques et leurs applications, Gauthier-Villars, Paris, 1927. | JFM

[N] R. Nevanlinna, Analytic functions, Springer, Berlin-Heidelberg-New York, 1970.

[R] M. Rees, The exponential map is not recurrent, Math. Z., 191, 4 (1986), 593-598. | MR | Zbl

[S] M. Shishikura, On the quasi-conformal surgery of rational functions, Ann. Sci. École Norm. Sup., 20 (1987), 1-29. | Numdam | MR | Zbl

[S1] D. Sullivan, Quasi-conformal homeomorphisms and dynamics, I. Ann. Math., 122, 3 (1985), 402-418.

[S2] D. Sullivan, Quasi-conformal homeomorphisms and dynamics, III. Preprint, IHES, 1982.

[T] H. Töpfer, Über die Iteration der ganzen transzendenten Funktionen, insbesondere von sinz und cosz, Math. Ann., 117 (1939), 65-84. | JFM | Zbl

[V] G. Valiron, Fonctions analytiques, Presses Universitaires de France, Paris, 1954. | MR | Zbl

[W] H. Wittich, Neuere Untersuchungen Über eindeutige analytische Funktionen, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1955. | Zbl

Cité par Sources :