Dynamical properties of some classes of entire functions

A. Eremenko; M. Yu Lyubich

Annales de l'institut Fourier (1992)

  • Volume: 42, Issue: 4, page 989-1020
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Eremenko, A., and Lyubich, M. Yu. "Dynamical properties of some classes of entire functions." Annales de l'institut Fourier 42.4 (1992): 989-1020. <http://eudml.org/doc/74982>.

@article{Eremenko1992,
abstract = {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.},
author = {Eremenko, A., Lyubich, M. Yu},
journal = {Annales de l'institut Fourier},
keywords = {dynamics; periodic points; entire function; Julia set; structural stability},
language = {eng},
number = {4},
pages = {989-1020},
publisher = {Association des Annales de l'Institut Fourier},
title = {Dynamical properties of some classes of entire functions},
url = {http://eudml.org/doc/74982},
volume = {42},
year = {1992},
}

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
PB - Association des Annales de l'Institut Fourier
VL - 42
IS - 4
SP - 989
EP - 1020
AB - 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.
LA - eng
KW - dynamics; periodic points; entire function; Julia set; structural stability
UR - http://eudml.org/doc/74982
ER -

References

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  1. [A1] L. AHLFORS, Untersuchungen zur Theorie der konformen Abbildung und der ganzen Funktionen, Acta soc. sci. fenn., N.S.I., 9 (1930), 1-40. Zbl56.0984.02JFM56.0984.02
  2. [A2] L. AHLFORS, Conformal invariants, McGraw-Hill, New York, 1973. Zbl0272.30012
  3. [AB] L. AHLFORS, L. BERS, Riemann mapping theorem for variable metrics, Ann. Math., 72, 2 (1960), 385-404. Zbl0104.29902MR22 #5813
  4. [B1] I.N. BAKER, Repulsive fixpoints of entire functions, Math. Z., 104 (1968), 252-256. Zbl0172.09502MR37 #1599
  5. [B2] I.N. BAKER, The domains of normality of an entire function, Ann. Acad. Sci. Fenn., S.A.I., 1 (1975), 277-283. Zbl0329.30019MR53 #5867
  6. [B3] I.N. BAKER, An entire function which has wandering domains, J. Austral. Math. Soc., 22, 2 (1976), 173-176. Zbl0335.30001MR54 #7777
  7. [B4] I.N. BAKER, Wandering domains in the iteration of entire functions, Proc. London Math. Soc., 49 (1984), 563-576. Zbl0523.30017MR86d:58066
  8. [BR] I.N. BAKER, P. RIPPON, Iteration of exponential functions, Ann. Acad. Sci. Fenn., S.A.I., 9 (1984), 49-77. Zbl0558.30029MR86d:58065
  9. [BRo] L. BERS, H.L. ROYDEN, Holomorphic families of injections, Acta Math., 3-4 (1986), 259-286. Zbl0619.30027MR88i:30034
  10. [Bla] P. BLANCHARD, Complex analytic dynamics on the Riemann sphere, Bull. Amer. Math. Soc., II, 1 (1984), 85-141. Zbl0558.58017MR85h:58001
  11. [D] R.L. DEVANEY, Structural instability of exp z, Proc. Amer. Math. Soc., 94, 3 (1985), 545-548. Zbl0575.58027MR86m:58093
  12. [DK] R.L. DEVANEY, M. KRYCH, Dynamics of exp z, Ergod. Theory and Dyn. Syst., 4 (1984), 35-52. Zbl0567.58025MR86b:58069
  13. [DGH] R.L. DEVANEY, L. GOLDBERG, J.H. HUBBARD, A dynamical approximation to the exponential map by polynomials, Preprint, Berkeley, 1986. 
  14. [DH1] A. DOUADY, J.H. HUBBARD, Itération des polynômes quadratiques complexes, C.R. Acad. Sci., 294, 3 (1982), 123-126. Zbl0483.30014MR83m:58046
  15. [DH2] A. DOUADY, J.H. HUBBARD, Étude dynamique des polynômes complexes (Première partie), Publ. Math. d'Orsay, 84-02. Zbl0552.30018MR87f:58072a
  16. [E] A.E. EREMENKO, On the iteration of entire functions, Proc. Banach Center Publ. Warsaw, 1989. Zbl0692.30021MR92c:30027
  17. [EL1] A.E. EREMENKO, M. Yu. LYUBICH, Iterates of entire functions, Soviet Math. Dokl., 30, 3 (1984), 592-594. Zbl0588.30027
  18. [EL2] A.E. EREMENKO, M. Yu. LYUBICH, Iterates of entire functions, Preprint, Inst. for Low Temperatures, Kharkov, 6, 1984 (Russian). Zbl0588.30027
  19. [EL3] A.E. EREMENKO, M. Yu. LYUBICH, Structural stability in some families on entire functions, Functional Anal. Appl., 19, 4 (1985), 323-324. Zbl0597.30032
  20. [EL4] A.E. EREMENKO, M. Yu. LYUBICH, Structural stability in some families of entire functions, Preprint, Inst. for Low Temperatures, Kharkov, 29, 1984 (Russian). 
  21. [EL5] A.E. EREMENKO, M. Yu. LYUBICH, Examples of entire functions with pathological dynamics, J. London Math. Soc., 36 (1987), 458-468. Zbl0601.30033MR89e:30047
  22. [EL6] A.E. EREMENKO, M. Yu. LYUBICH, Dynamics of analytic transformations, Algebra and Analysis (Leningrad J. Math.), 1, 3 (1989), 1-70. Zbl0712.58036MR91b:58109
  23. [F1] P. FATOU, Sur les équations fonctionnelles, Bull. Soc. Math. France, 47 (1919), 161-271. Zbl47.0921.02JFM47.0921.02
  24. [F2] P. FATOU, Sur les équations fonctionnelles, Bull. Soc. Math. France, 48 (1920), 33-94, 208-314. JFM47.0921.02
  25. [F3] P. FATOU, Sur l'itération des fonctions transcendantes entières, Acta math., 47 (1926), 337-370. Zbl52.0309.01JFM52.0309.01
  26. [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. Zbl0657.58011MR88b:58126
  27. [H] M.R. HERMAN, Are there critical points on the boundaries of singular domains ? Commun. Math. Phys., 99 (1985), 593-612. Zbl0587.30040MR86j:58067
  28. [J] G. JULIA, Mémoire sur l'itération des fonctions rationnelles, J. de Math. Pures et Appliquées (8), 1 (1918), 47-245. Zbl46.0520.06JFM46.0520.06
  29. [LV] O. LEHTO, K. VIRTANEN, Quasiconformal mappings in the plane, Springer-Verlag, Berlin, 1973. Zbl0267.30016MR49 #9202
  30. [L1] M. Yu. LYUBICH, On a typical behavior of trajectories of a rational mapping of the sphere, Soviet Math. Dokl., 27 (1983), 22-25. Zbl0595.30034
  31. [L2] M. Yu. LYUBICH, Some typical properties of the dynamics of rational maps, Russian Math. Surveys, 8, 5 (1983), 154-155. Zbl0598.58028
  32. [L3] M. Yu. LYUBICH, The dynamics of rational transforms: the topological picture, Russian Math. Surveys, 41, 4 (1986), 43-117. Zbl0619.30033
  33. [L4] M. Yu. LYUBICH, The measurable dynamics of the exponential map, Siberian Journ. Math., 28, 5 (1987), 111-127. Zbl0667.58037
  34. [Mi] J. MILNOR, Dynamics in one complex variable, Introductory lectures, Preprint SUNY Stony Brook, 1990/5. 
  35. [MSS] R. MAÑÉ, P. SAD, D. SULLIVAN, On the dynamics of rational maps, Ann. Scient. Ec. Norm. Sup. 4e série, 16 (1983), 193-217. Zbl0524.58025MR85j:58089
  36. [McM] C. MCMULLEN, Area and Hausdorff dimension of Julia sets of entire functions, Trans. Amer. Math. Soc., 300, 1 (1987), 325-342. Zbl0618.30027MR88a:30057
  37. [M] M. MISIUREWICZ, On iterates of ez, Ergod. Theory and Dyn. Syst., 1 (1981), 103-106. Zbl0466.30019MR82i:58058
  38. [Mo] P. MONTEL, Leçons sur les familles normales de fonctions analytiques et leurs applications, Gauthier-Villars, Paris, 1927. Zbl53.0303.02JFM53.0303.02
  39. [N] R. NEVANLINNA, Analytic functions, Springer, Berlin-Heidelberg-New York, 1970. Zbl0199.12501
  40. [R] M. REES, The exponential map is not recurrent, Math. Z., 191, 4 (1986), 593-598. Zbl0595.30033MR87g:58063
  41. [S] M. SHISHIKURA, On the quasi-conformal surgery of rational functions, Ann. Sci. École Norm. Sup., 20 (1987), 1-29. Zbl0621.58030MR88i:58099
  42. [S1] D. SULLIVAN, Quasi-conformal homeomorphisms and dynamics, I. Ann. Math., 122, 3 (1985), 402-418. 
  43. [S2] D. SULLIVAN, Quasi-conformal homeomorphisms and dynamics, III. Preprint, IHES, 1982. 
  44. [T] H. TÖPFER, Über die Iteration der ganzen transzendenten Funktionen, insbesondere von sinz und cosz, Math. Ann., 117 (1939), 65-84. Zbl0021.41602JFM65.0327.05
  45. [V] G. VALIRON, Fonctions analytiques, Presses Universitaires de France, Paris, 1954. Zbl0055.06702MR15,861a
  46. [W] H. WITTICH, Neuere Untersuchungen Über eindeutige analytische Funktionen, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1955. Zbl0067.05501

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