Iteration of Quasiregular Mappings
详细信息    查看全文
  • 作者:Walter Bergweiler (1)
  • 关键词:Quasiregular ; uniformly quasiregular ; quasimeromorphic ; quasiconformal ; iteration ; dynamics ; Julia set ; Fatou set ; 30D05 ; 30C62 ; 30C65 ; 37F10
  • 刊名:Computational Methods and Function Theory
  • 出版年:2011
  • 出版时间:January 2011
  • 年:2011
  • 卷:10
  • 期:2
  • 页码:455-481
  • 参考文献:1. L. V. Ahlfors, Zur Theorie der 脺berlagerungsfl盲chen, Acta Math. 65 (1935), 157鈥?94; / Collected Papers, Vol. I, pp. 214鈥?51. CrossRef
    2. L. V. Ahlfors, Conformal Invariants: Topics in Geometric Function Theory, McGraw-Hill, New York, D眉sseldorf, Johannesburg, 1973.
    3. K. Astala, T. Iwaniec and G. J. Martin, Elliptic Partial Differential Equations and Quasi-conformal Mappings in the Plane, Princeton Mathematical Series 48, Princeton University Press, Princeton, NJ, 2009.
    4. I. N. Baker, Repulsive fixpoints of entire functions, Math. Z. 104 (1968), 252鈥?56. CrossRef
    5. K. Bara艅ski, Trees and hairs for some hyperbolic entire maps of finite order, Math. Z. 257 (2007), 33鈥?9. CrossRef
    6. D. Bargmann, Simple proofs of some fundamental properties of the Julia set, Ergodic Theory Dynam. Systems 19 (1999), 553鈥?58. CrossRef
    7. D. Bargmann and W. Bergweiler, Periodic points and normal families, Proc. Amer. Math. Soc. 129 (2001), 2881鈥?888. CrossRef
    8. A. F. Beardon, Iteration of Rational Functions, Graduate Texts in Mathematics 91, Springer-Verlag, New York, 1991. CrossRef
    9. W. Bergweiler, Periodic points of entire functions: proof of a conjecture of Baker, Complex Variables Theory Appl. 17 (1991), 57鈥?2. CrossRef
    10. W. Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc. (N. S.) 29 (1993), 151鈥?88. CrossRef
    11. W. Bergweiler, A new proof of the Ahlfors five islands theorem, J. Analyse Math. 76 (1998), 337鈥?47. CrossRef
    12. W. Bergweiler, The role of the Ahlfors five islands theorem in complex dynamics, Conform. Geom. Dyn. 4 (2000), 22鈥?4. CrossRef
    13. 鈥? Karpi艅ska鈥檚 paradox in dimension three, to appear in / Duke Math. J., arxiv: 0902.2686.
    14. W. Bergweiler and A. Eremenko, Dynamics of a higher dimensional analogue of the trigonometric functions, to appear in / Ann. Acad. Sci. Fenn. Math., arXiv: 1002.4159v1.
    15. W. Bergweiler, A. Fletcher, J. Langley and J. Meyer, The escaping set of a quasiregular mapping, Proc. Amer. Math. Soc. 137 (2009), 641鈥?51. CrossRef
    16. W. Bergweiler, P. J. Rippon and G. M. Stallard, Dynamics of meromorphic functions with direct or logarithmic singularities, Proc. London Math. Soc. 97 (2008), 368鈥?00. CrossRef
    17. F. Berteloot and J. Duval, Une d茅monstration directe de la densit茅 des cycles r茅pulsifs dans l鈥檈nsemble de Julia, in: / Complex Analysis and Geometry (Paris 1997), Progress in Mathematics 188, Birkh盲user, Basel 2000, 221鈥?22.
    18. L. Carleson and T. W. Gamelin, Complex Dynamics, Universitext, Springer-Verlag, New York, 1993. CrossRef
    19. C. T. Chuang, Normal Families of Meromorphic Functions, World Scientific, River Edge, NJ, 1993. CrossRef
    20. H. Cremer, 脺ber die Schr枚dersche Funktionalgleichung und das Schwarzsche Eckenabbil-dungsproblem, Ber. Verh. S盲chs. Akad. Wiss. Leipzig, Math.-Phys. Kl. 84 (1932), 291鈥?24.
    21. R. L. Devaney and M. Krych, iDynamics of exp(z), Ergodic Theory Dynam. Systems 4 (1984), 35鈥?2. CrossRef
    22. R. L. Devaney and F. F. Tangerman, Dynamics of entire functions near the essential singularity, Ergodic Theory Dynam. Systems 6 (1986), 489鈥?03. CrossRef
    23. P. Dom铆nguez, Dynamics of transcendental meromorphic functions, Ann. Acad. Sci. Fenn. Math. 23 (1998), 225鈥?50.
    24. A. Douady and J. H. Hubbard, 脡tude dynamique des polyn么mes complexes I & II, Publ. Math. Orsay 84鈥?2 (1984) & 85鈥?4 (1985).
    25. A. Douady and J. H. Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. 脡cole Norm. Sup. (4) 18 (1985), 287鈥?43.
    26. A. 脠. Erem毛nko, On the iteration of entire functions, in: Dynamical Systems and Ergodic Theory; Banach Center Publications / 23, Polish Scientific Publishers, Warsaw 1989, 339鈥?45.
    27. A. 脠. Erem毛nko and M. Yu. Lyubich, The dynamics of analytic transformations, Leningrad Math. J. 1 (1990), 563鈥?34; translation from / Algebra i Analiz 1 (1989), 1鈥?0.
    28. P. Fatou, Sur les 茅quations fonctionelles, Bull. Soc. Math. France 47 (1919), 161鈥?71; 48 (1920), 33鈥?4, 208鈩?4.
    29. P. Fatou, Sur l鈥檌t茅ration des fonctions transcendantes enti猫res, Acta Math. 47 (1926), 337鈥?60. CrossRef
    30. A. Fletcher and D. A. Nicks, Quasiregular dynamics on the / n-sphere, / Ergodic Theory Dynam. Systems, doi:10.1017/S0143385709001072.
    31. L. Geyer, Quasikonforme Deformation in der Iterationstheorie, Diploma thesis, Technical University Berlin, 1994.
    32. W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964.
    33. W. K. Hayman, The local growth of power series: a survey of the Wiman-Valiron method, Canad. Math. Bull. 17 (1974), 317鈥?58. CrossRef
    34. J. Heinonen and P. Koskela, Weighted Sobolev and Poincar茅 inequalities and quasiregular mappings of polynomial type, Math. Scand. 77 (1995), 251鈥?71.
    35. A. Hinkkanen, Uniformly quasiregular semigroups in two dimensions, Ann. Acad. Sci. Fenn. Math. 21 (1996), 205鈥?22.
    36. A. Hinkkanen and G. J. Martin, Attractors in quasiregular semigroups, in: I. Laine and O. Martio (eds.), XVIth Rolf Nevanlinna Colloquium, Proceedings of the International Colloquium held in Joensuu, August 1鈥?, 1995, Walter de Gruyter & Co., Berlin, 1996, 135鈥?41.
    37. A. Hinkkanen and G. J. Martin, Limit functions for convergence groups and uniformly quasiregular maps, J. London Math. Soc. (2) 73 (2006), 716鈥?26. CrossRef
    38. A. Hinkkanen, G. J. Martin and V. Mayer, Local dynamics of uniformly quasiregular mappings, Math. Scand. 95 (2004), 80鈥?00.
    39. T. Iwaniec and G. J. Martin, Quasiregular semigroups, Ann. Acad. Sci. Fenn. Math. 21 (1996), 241鈥?54.
    40. T. Iwaniec and G. J. Martin, Geometric Function Theory and Non-Linear Analysis, Oxford Mathematical Monographs, Oxford University Press, New York, 2001.
    41. G. Julia, Sur l鈥檌t茅ration des fonctions rationelles, J. Math. Pures Appl. (7) 4 (1918), 47鈥?45.
    42. B. Karpi艅ska, Area and Hausdorff dimension of the set of accessible points of the Julia sets of 位ez and 位sin z, Fund. Math. 159 (1999), 269鈥?87.
    43. B. Karpi艅ska, Hausdorff dimension of the hairs without endpoints for 位expz, C. R. Acad. Sci. Paris S茅r. I Math. 328 (1999), 1039鈥?044. CrossRef
    44. M. Kisaka and M. Shishikura, On multiply connected wandering domains of entire functions, in: P. J. Rippon and G. M. Stallard (eds.), Transcendental Dynamics and Complex Analysis, LMS Lecture Note Series 348, Cambridge University Press, Cambridge, 2008, 217鈥?50. CrossRef
    45. O. Lehto and K. I. Virtanen, Quasiconformal Mappings in the Plane, Die Grundlehren der mathematischen Wissenschaften 126, Springer-Verlag, New York, Heidelberg, 1973. CrossRef
    46. G. J. Martin, Branch sets of uniformly quasiregular maps, Conform. Geom. Dyn. 1 (1997), 24鈥?7. CrossRef
    47. G. J. Martin, Analytic continuation for Beltrami systems, Siegel鈥檚 theorem for UQR maps, and the Hilbert-Smith conjecture, Math. Ann. 324 (2002), 329鈥?40. CrossRef
    48. G. J. Martin and V. Mayer, Rigidity in holomorphic and quasiregular dynamics, Trans. Amer. Math. Soc. 355 (2003), 4349鈥?363. CrossRef
    49. G. J. Martin and K. Peltonen, Sto茂lov factorization for quasiregular maps in all dimensions, Proc. Amer. Math. Soc. 138 (2010), 147鈥?51. CrossRef
    50. O. Martio and U. Srebro, Periodic quasimeromorphic mappings in / Rn, J. Anal. Math. 28 (1975), 20鈥?0. CrossRef
    51. V. Mayer, Uniformly quasiregular mappings of Latt猫s type, Conform. Geom. Dyn. 1 (1997), 104鈥?11. CrossRef
    52. V. Mayer, Quasiregular analogues of critically finite rational functions with parabolic orbifold, J. Anal. Math. 75 (1998), 105鈥?19. CrossRef
    53. V. Mayer, Behavior of quasiregular semigroups near attracting fixed points, Ann. Acad. Sci. Fenn. Math. 25 (2000), 31鈥?9.
    54. C. McMullen, Area and Hausdorff dimension of Julia sets of entire functions, Trans. Amer. Math. Soc. 300 (1987), 329鈥?42. CrossRef
    55. J. Milnor, Dynamics in One Complex Variable, Third edition. Annals of Mathematics Studies 160, Princeton University Press, Princeton, NJ, 2006.
    56. R. Miniowitz, Normal families of quasimeromorphic mappings, Proc. Amer. Math. Soc. 84 (1982), 35鈥?3. CrossRef
    57. P. Montel, Sur les suites infinies des fonctions, Ann. 脡cole Norm. Sup. 24 (1907), 233鈥?34.
    58. S. Morosawa, Y. Nishimura, M. Taniguchi and T. Ueda, Holomorphic Dynamics, Cambridge Studies in Advanced Mathematics 66, Cambridge University Press, Cambridge, 2000.
    59. R. Nevanlinna, Eindeutige analytische Funktionen, Springer-Verlag, Berlin, G枚ttingen, Heidelberg, 1953. CrossRef
    60. K. Peltonen, Examples of uniformly quasiregular maps, Conform. Geom. Dyn. 3 (1999), 158鈥?63. CrossRef
    61. Yu. G. Reshetnyak, Space Mappings with Bounded Distortion, Translations of Mathematical Monographs 73, Amer. Math. Soc., Providence, RI, 1989.
    62. S. Rickman, On the number of omitted values of entire quasiregular mappings, J. Analyse Math. 37 (1980), 100鈥?17. CrossRef
    63. S. Rickman, The analogue of Picard鈥檚 theorem for quasiregular mappings in dimension three, Acta Math. 154 (1985), 195鈥?42. CrossRef
    64. S. Rickman, Quasiregular Mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 26, Springer-Verlag, Berlin, 1993. CrossRef
    65. P. J. Rippon and G. M. Stallard, On questions of Fatou and Eremenko, Proc. Amer. Math. Soc. 133 (2005), 1119鈥?126. CrossRef
    66. G. Rottenfu尾er, J. R眉ckert, L. Rempe and D. Schleicher, Dynamic rays of bounded-type entire functions, to appear in / Ann. of Math., arXiv: 0704.3213v2.
    67. J. L. Schiff, Normal Families, Universitext, Springer-Verlag, New York, 1993. CrossRef
    68. D. Schleicher, The dynamical fine structure of iterated cosine maps and a dimension paradox, Duke Math. J. 136 (2007), 343鈥?56. CrossRef
    69. D. Schleicher and J. Zimmer, Escaping points of exponential maps, J. London Math. Soc. (2) 67 (2003), 380鈥?00. CrossRef
    70. W. Schwick, Repelling periodic points in the Julia set, Bull. London Math. Soc. 29 (1997), 314鈥?16. CrossRef
    71. H. Siebert, Fixpunkte und normale Familien quasiregul盲rer Abbildungen, Dissertation, University of Kiel, 2004; http://e-diss.uni-kiel.de/diss1260.
    72. H. Siebert, Fixed points and normal families of quasiregular mappings, J. Anal. Math. 98 (2006), 145鈥?68. CrossRef
    73. N. Steinmetz, Rational Iteration, de Gruyter Studies in Mathematics 16, Walter de Gruyter & Co., Berlin 1993. CrossRef
    74. D. Sullivan, It茅ration des fonctions analytiques complexes, C. R. Acad. Sci. Paris S茅r. I Math. 294 (1982), 301鈥?03.
    75. D. Sullivan, Conformal dynamical systems, in: Geometric Dynamics (Rio de Janeiro, 1981), Lecture Notes in Math. 1007, Springer, Berlin, 1983, 725鈥?52. CrossRef
    76. D. Sullivan, Quasiconformal homeomorphisms and dynamics I, Solution of the Fatou-Julia problem on wandering domains, Ann. of Math. 122 (1985), 401鈥?18. CrossRef
    77. D. Sun and L. Yang, Value distribution of quasimeromorphic mappings (in Chinese), Sci. China Ser. A 27 (1997), 132鈥?39.
    78. D. Sun and L. Yang, Value distribution of quasiconformal mappings, Complex Variables Theory Appl. 34 (1997), 219鈥?29. CrossRef
    79. D. Sun and L. Yang, Quasirational dynamical systems (in Chinese), Chinese Ann. Math. Ser. A 20 (1999), 673鈥?84.
    80. D. Sun and L. Yang, Quasirational dynamic system, Chinese Science Bull. 45 (2000), 1277鈥?279. CrossRef
    81. D. Sun and L. Yang, Iteration of quasi-rational mapping, Progr. Natur. Sci. (English Ed.) 11 (2001), 16鈥?5.
    82. D. Sun and L. Yang, Normal family of quasimeromorphic mappings, Sci. China Ser. A 46 (2003), 440鈥?49. CrossRef
    83. Z. J. Wu and D. C. Sun, Iteration of quasipolynomial mappings (in Chinese), Acta Math. Sci. Ser. A Chin. Ed. 26 (2006), 493鈥?97.
    84. L. Zalcman, A heuristic principle in complex function theory, Amer. Math. Monthly 82 (1975), 813鈥?17. CrossRef
    85. L. Zalcman, Normal families: new perspectives, Bull. Amer. Math. Soc. (N. S.) 35 (1998), 215鈥?30. CrossRef
    86. V. A. Zorich, A theorem of M. A. Lavrent鈥檈v on quasiconformal space maps, Math. USSR. Sb. 3 (1967), 389鈥?03; translation from / Mat. Sb. (N.S.) 74 (116) (1967), 417鈥?33. CrossRef
  • 作者单位:Walter Bergweiler (1)

    1. Mathematisches Seminar, Christian-Albrechts-Universit盲t zu Kiel, Ludewig-Meyn-Str. 4, D-24098, Kiel, Germany
  • ISSN:2195-3724
文摘
We survey some results on the iteration of quasiregular mappings. In particular we discuss some recent results on the dynamics of quasiregular maps which are not uniformly quasiregular.

© 2004-2018 中国地质图书馆版权所有 京ICP备05064691号 京公网安备11010802017129号

地址:北京市海淀区学院路29号 邮编:100083

电话:办公室:(+86 10)66554848;文献借阅、咨询服务、科技查新:66554700