Iteration of meromorphic functions

Bergweiler, Walter

doi:10.1090/S0273-0979-1993-00432-4

Iteration of meromorphic functions
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References

Jan M. Aarts and Lex G. Oversteegen, The geometry of Julia sets, Trans. Amer. Math. Soc. 338 (1993), no. 2, 897–918. MR 1182980, DOI 10.1090/S0002-9947-1993-1182980-3
Lars Ahlfors, Zur Theorie der Überlagerungsflächen, Acta Math. 65 (1935), no. 1, 157–194 (German). MR 1555403, DOI 10.1007/BF02420945

Conformal invariants

I. N. Baker, Fixpoints and iterates of entire functions, Math. Z. 71 (1959), 146–153. MR 107015, DOI 10.1007/BF01181396
I. N. Baker, Some entire functions with fixpoints of every order, J. Austral. Math. Soc. 1 (1959/1961), 203–209. MR 0114007, DOI 10.1017/S1446788700025556
I. N. Baker, Some entire functions with fixpoints of every order, J. Austral. Math. Soc. 1 (1959/1961), 203–209. MR 0114007, DOI 10.1017/S1446788700025556
Irvine Noel Baker, Multiply connected domains of normality in iteration theory, Math. Z. 81 (1963), 206–214. MR 153842, DOI 10.1007/BF01111543
I. N. Baker, Fixpoints of polynomials and rational functions, J. London Math. Soc. 39 (1964), 615–622. MR 169989, DOI 10.1112/jlms/s1-39.1.615
I. N. Baker, Sets of non-normality in iteration theory, J. London Math. Soc. 40 (1965), 499–502. MR 180668, DOI 10.1112/jlms/s1-40.1.499
I. N. Baker, Repulsive fixpoints of entire functions, Math. Z. 104 (1968), 252–256. MR 226009, DOI 10.1007/BF01110294
I. N. Baker, Limit functions and sets of non-normality in iteration theory, Ann. Acad. Sci. Fenn. Ser. A I No. 467 (1970), 11. MR 0264071
I. N. Baker, Completely invariant domains of entire functions, Mathematical Essays Dedicated to A. J. Macintyre, Ohio Univ. Press, Athens, Ohio, 1970, pp. 33–35. MR 0271344
I. N. Baker, The domains of normality of an entire function, Ann. Acad. Sci. Fenn. Ser. A I Math. 1 (1975), no. 2, 277–283. MR 0402044, DOI 10.5186/aasfm.1975.0101
I. N. Baker, An entire function which has wandering domains, J. Austral. Math. Soc. Ser. A 22 (1976), no. 2, 173–176. MR 419759, DOI 10.1017/s1446788700015287
I. N. Baker, The iteration of polynomials and transcendental entire functions, J. Austral. Math. Soc. Ser. A 30 (1980/81), no. 4, 483–495. MR 621564, DOI 10.1017/S1446788700017961
I. N. Baker, Wandering domains in the iteration of entire functions, Proc. London Math. Soc. (3) 49 (1984), no. 3, 563–576. MR 759304, DOI 10.1112/plms/s3-49.3.563
I. N. Baker, Some entire functions with multiply-connected wandering domains, Ergodic Theory Dynam. Systems 5 (1985), no. 2, 163–169. MR 796748, DOI 10.1017/S0143385700002832
I. N. Baker, Iteration of entire functions: an introductory survey, Lectures on complex analysis (Xian, 1987) World Sci. Publishing, Singapore, 1988, pp. 1–17. MR 996465
I. N. Baker, Wandering domains for maps of the punctured plane, Ann. Acad. Sci. Fenn. Ser. A I Math. 12 (1987), no. 2, 191–198. MR 951969, DOI 10.5186/aasfm.1987.1204
I. N. Baker, Infinite limits in the iteration of entire functions, Ergodic Theory Dynam. Systems 8 (1988), no. 4, 503–507. MR 980793, DOI 10.1017/S014338570000465X
I. N. Baker, J. Kotus, and Lü Yinian, Iterates of meromorphic functions. I, Ergodic Theory Dynam. Systems 11 (1991), no. 2, 241–248. MR 1116639, DOI 10.1017/S014338570000612X
I. N. Baker, J. Kotus, and Yi Nian Lü, Iterates of meromorphic functions. II. Examples of wandering domains, J. London Math. Soc. (2) 42 (1990), no. 2, 267–278. MR 1083445, DOI 10.1112/jlms/s2-42.2.267
I. N. Baker, J. Kotus, and Yi Nian Lü, Iterates of meromorphic functions. III. Preperiodic domains, Ergodic Theory Dynam. Systems 11 (1991), no. 4, 603–618. MR 1145612, DOI 10.1017/S0143385700006386
I. N. Baker, J. Kotus, and Lü Yinian, Iterates of meromorphic functions. IV. Critically finite functions, Results Math. 22 (1992), no. 3-4, 651–656. MR 1189754, DOI 10.1007/BF03323112
I. N. Baker and P. J. Rippon, Iteration of exponential functions, Ann. Acad. Sci. Fenn. Ser. A I Math. 9 (1984), 49–77. MR 752391, DOI 10.5186/aasfm.1984.0903
Béla Barna, Über die Divergenzpunkte des Newtonschen Verfahrens zur Bestimmung von Wurzeln algebraischen Gleichungen. II, Publ. Math. Debrecen 4 (1956), 384–397 (German). MR 78956
Alan F. Beardon, Iteration of rational functions, Graduate Texts in Mathematics, vol. 132, Springer-Verlag, New York, 1991. Complex analytic dynamical systems. MR 1128089, DOI 10.1007/978-1-4612-4422-6
Walter Bergweiler, On the number of fix-points of iterated entire functions, Arch. Math. (Basel) 55 (1990), no. 6, 558–563. MR 1078277, DOI 10.1007/BF01191691
Walter Bergweiler, Periodic points of entire functions: proof of a conjecture of Baker, Complex Variables Theory Appl. 17 (1991), no. 1-2, 57–72. MR 1123803, DOI 10.1080/17476939108814495
Walter Bergweiler, On the existence of fixpoints of composite meromorphic functions, Proc. Amer. Math. Soc. 114 (1992), no. 3, 879–880. MR 1091176, DOI 10.1090/S0002-9939-1992-1091176-X

Newton’s method and a class of meromorphic functions without wandering domains

Newton’s method for meromorphic functions

Walter Bergweiler, Mako Haruta, Hartje Kriete, Hans-Günter Meier, and Norbert Terglane, On the limit functions of iterates in wandering domains, Ann. Acad. Sci. Fenn. Ser. A I Math. 18 (1993), no. 2, 369–375. MR 1234740
Walter Bergweiler and Steffen Rohde, Omitted values in domains of normality, Proc. Amer. Math. Soc. 123 (1995), no. 6, 1857–1858. MR 1249869, DOI 10.1090/S0002-9939-1995-1249869-X
Walter Bergweiler and Norbert Terglane, Weakly repelling fixpoints and the connectivity of wandering domains, Trans. Amer. Math. Soc. 348 (1996), no. 1, 1–12. MR 1327252, DOI 10.1090/S0002-9947-96-01511-5

Iteration of analytic functions

Paul Blanchard, Complex analytic dynamics on the Riemann sphere, Bull. Amer. Math. Soc. (N.S.) 11 (1984), no. 1, 85–141. MR 741725, DOI 10.1090/S0273-0979-1984-15240-6

Beiträge zu der Theorie der Iterationsrechnung

D. A. Brannan and W. K. Hayman, Research problems in complex analysis, Bull. London Math. Soc. 21 (1989), no. 1, 1–35. MR 967787, DOI 10.1112/blms/21.1.1
Hans Brolin, Invariant sets under iteration of rational functions, Ark. Mat. 6 (1965), 103–144 (1965). MR 194595, DOI 10.1007/BF02591353
Witold D. Bula and Lex G. Oversteegen, A characterization of smooth Cantor bouquets, Proc. Amer. Math. Soc. 108 (1990), no. 2, 529–534. MR 991691, DOI 10.1090/S0002-9939-1990-0991691-9
Lennart Carleson and Theodore W. Gamelin, Complex dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR 1230383, DOI 10.1007/978-1-4612-4364-9
Nikolaj Tschebotareff, Über die Realität von Nullstellen ganzer transzendenter Funktionen, Math. Ann. 99 (1928), no. 1, 660–686 (German). MR 1512472, DOI 10.1007/BF01459119
Chi Tai Chuang, A simple proof of a theorem of Fatou on the iteration and fix-points of transcendental entire functions, Analytic functions of one complex variable, Contemp. Math., vol. 48, Amer. Math. Soc., Providence, RI, 1985, pp. 65–69. MR 838099, DOI 10.1090/conm/048/838099
J. Clunie, The composition of entire and meromorphic functions, Mathematical Essays Dedicated to A. J. Macintyre, Ohio Univ. Press, Athens, Ohio, 1970, pp. 75–92. MR 0271352

Über die Iteration rationaler Funktionen

33

Zum Zentrumsproblem

98

Über die Schrödersche Funktionalgleichung und das Schwarzsche Eckenabbildungsproblem

84

Hubert Cremer, Über die Häufigkeit der Nichtzentren, Math. Ann. 115 (1938), no. 1, 573–580 (German). MR 1513203, DOI 10.1007/BF01448957
Robert L. Devaney, Julia sets and bifurcation diagrams for exponential maps, Bull. Amer. Math. Soc. (N.S.) 11 (1984), no. 1, 167–171. MR 741732, DOI 10.1090/S0273-0979-1984-15253-4
Robert L. Devaney, Structural instability of $\textrm {exp}(z)$, Proc. Amer. Math. Soc. 94 (1985), no. 3, 545–548. MR 787910, DOI 10.1090/S0002-9939-1985-0787910-2
Robert L. Devaney, Dynamics of entire maps, Dynamical systems and ergodic theory (Warsaw, 1986) Banach Center Publ., vol. 23, PWN, Warsaw, 1989, pp. 221–228. MR 1102716

An introduction to chaotic dynamical systems

Robert L. Devaney, Film and video as a tool in mathematical research, Math. Intelligencer 11 (1989), no. 2, 33–38. MR 994962, DOI 10.1007/BF03023821
Robert L. Devaney, Dynamics of simple maps, Chaos and fractals (Providence, RI, 1988) Proc. Sympos. Appl. Math., vol. 39, Amer. Math. Soc., Providence, RI, 1989, pp. 1–24. MR 1010233, DOI 10.1090/psapm/039/1010233
Robert L. Devaney, $e^z$: dynamics and bifurcations, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 1 (1991), no. 2, 287–308. MR 1120198, DOI 10.1142/S0218127491000221
Robert L. Devaney and Marilyn B. Durkin, The exploding exponential and other chaotic bursts in complex dynamics, Amer. Math. Monthly 98 (1991), no. 3, 217–233. MR 1093952, DOI 10.2307/2325024
Robert L. Devaney (ed.), Complex dynamical systems, Proceedings of Symposia in Applied Mathematics, vol. 49, American Mathematical Society, Providence, RI, 1994. The mathematics behind the Mandelbrot and Julia sets; Lecture notes prepared for the short course held at the Annual Meeting of the American Mathematical Society, Cincinnati, Ohio, January 10–11, 1994; AMS Short Course Lecture Notes. MR 1315531, DOI 10.1090/psapm/049
Robert L. Devaney and Linda Keen, Dynamics of meromorphic maps: maps with polynomial Schwarzian derivative, Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 1, 55–79. MR 985854, DOI 10.24033/asens.1575
Robert L. Devaney and MichałKrych, Dynamics of $\textrm {exp}(z)$, Ergodic Theory Dynam. Systems 4 (1984), no. 1, 35–52. MR 758892, DOI 10.1017/S014338570000225X
Robert L. Devaney and Folkert Tangerman, Dynamics of entire functions near the essential singularity, Ergodic Theory Dynam. Systems 6 (1986), no. 4, 489–503. MR 873428, DOI 10.1017/S0143385700003655
Adrien Douady and John Hamal Hubbard, Itération des polynômes quadratiques complexes, C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), no. 3, 123–126 (French, with English summary). MR 651802

Étude dynamique des polynômes complexes

Nasser Doual, James L. Howland, and Rémi Vaillancourt, Selective solutions to transcendental equations, Comput. Math. Appl. 22 (1991), no. 9, 61–76. MR 1136170, DOI 10.1016/0898-1221(91)90207-K
Albert Edrei, Meromorphic functions with three radially distributed values, Trans. Amer. Math. Soc. 78 (1955), 276–293. MR 67982, DOI 10.1090/S0002-9947-1955-0067982-9
A. È. Erëmenko, On the iteration of entire functions, Dynamical systems and ergodic theory (Warsaw, 1986) Banach Center Publ., vol. 23, PWN, Warsaw, 1989, pp. 339–345. MR 1102727
A. È. Erëmenko and M. Yu. Lyubich, Iterations of entire functions, Dokl. Akad. Nauk SSSR 279 (1984), no. 1, 25–27 (Russian). MR 769199
A. È. Erëmenko and M. Ju. Ljubich, Examples of entire functions with pathological dynamics, J. London Math. Soc. (2) 36 (1987), no. 3, 458–468. MR 918638, DOI 10.1112/jlms/s2-36.3.458
A. È. Erëmenko and M. Yu. Lyubich, The dynamics of analytic transformations, Algebra i Analiz 1 (1989), no. 3, 1–70 (Russian); English transl., Leningrad Math. J. 1 (1990), no. 3, 563–634. MR 1015124
A. È. Erëmenko and M. Yu. Lyubich, Dynamical properties of some classes of entire functions, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 4, 989–1020 (English, with English and French summaries). MR 1196102, DOI 10.5802/aif.1318

Sur les équations fonctionelles

47

48

Sur l’itération des fonctions transcendantes entières

47

Marguerite Flexor and Pierrette Sentenac, Algorithmes de Newton généralisés, C. R. Acad. Sci. Paris Sér. I Math. 308 (1989), no. 14, 445–448 (French, with English summary). MR 991879
Étienne Ghys, Lisa R. Goldberg, and Dennis P. Sullivan, On the measurable dynamics of $z\mapsto e^z$, Ergodic Theory Dynam. Systems 5 (1985), no. 3, 329–335. MR 805833, DOI 10.1017/S0143385700002984
Lisa R. Goldberg and Linda Keen, A finiteness theorem for a dynamical class of entire functions, Ergodic Theory Dynam. Systems 6 (1986), no. 2, 183–192. MR 857196, DOI 10.1017/S0143385700003394

The relaxed Newton’s method for polynomials

F. von Haeseler and H.-O. Peitgen, Newton’s method and complex dynamical systems, Acta Appl. Math. 13 (1988), no. 1-2, 3–58. MR 979816, DOI 10.1007/BF00047501

The dynamics of Newton’s method on the exponential function in the complex domain

W. K. Hayman, Meromorphic functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964. MR 0164038
W. K. Hayman, The local growth of power series: a survey of the Wiman-Valiron method, Canad. Math. Bull. 17 (1974), no. 3, 317–358. MR 385095, DOI 10.4153/CMB-1974-064-0
Michael-R. Herman, Exemples de fractions rationnelles ayant une orbite dense sur la sphère de Riemann, Bull. Soc. Math. France 112 (1984), no. 1, 93–142 (French, with English summary). MR 771920, DOI 10.24033/bsmf.2002
Michael-R. Herman, Are there critical points on the boundaries of singular domains?, Comm. Math. Phys. 99 (1985), no. 4, 593–612. MR 796014, DOI 10.1007/BF01215911
A. Hinkkanen, Iteration and the zeros of the second derivative of a meromorphic function, Proc. London Math. Soc. (3) 65 (1992), no. 3, 629–650. MR 1182104, DOI 10.1112/plms/s3-65.3.629

On the size of Julia sets

James Lucien Howland and Rémi Vaillancourt, Attractive cycles in the iteration of meromorphic functions, Numer. Math. 46 (1985), no. 3, 323–337. MR 791694, DOI 10.1007/BF01389489
James L. Howland, Alan Thompson, and Rémi Vaillancourt, On the dynamics of a meromorphic function, Appl. Math. Notes 15 (1990), 7–37. MR 1102745
Gerhard Jank and Lutz Volkmann, Einführung in die Theorie der ganzen und meromorphen Funktionen mit Anwendungen auf Differentialgleichungen, UTB für Wissenschaft: Grosse Reihe. [UTB for Science: Large Series], Birkhäuser Verlag, Basel, 1985 (German). MR 820202, DOI 10.1007/978-3-0348-6621-7
H. Th. Jongen, P. Jonker, and F. Twilt, The continuous, desingularized Newton method for meromorphic functions, Acta Appl. Math. 13 (1988), no. 1-2, 81–121. MR 979818, DOI 10.1007/BF00047503

Sur l’itération des fonctions rationelles

4

Sur des problèmes concernant l’itération des fonctions rationelles

166

Linda Keen, Dynamics of holomorphic self-maps of $\textbf {C}^*$, Holomorphic functions and moduli, Vol. I (Berkeley, CA, 1986) Math. Sci. Res. Inst. Publ., vol. 10, Springer, New York, 1988, pp. 9–30. MR 955806, DOI 10.1007/978-1-4613-9602-4_{2}
Linda Keen, Topology and growth of a special class of holomorphic self-maps of $\textbf {C}^*$, Ergodic Theory Dynam. Systems 9 (1989), no. 2, 321–328. MR 1007413, DOI 10.1017/S0143385700004995
Linda Keen, Julia sets, Chaos and fractals (Providence, RI, 1988) Proc. Sympos. Appl. Math., vol. 39, Amer. Math. Soc., Providence, RI, 1989, pp. 57–74. MR 1010236, DOI 10.1090/psapm/039/1010236
J. Kotus, Iterated holomorphic maps on the punctured plane, Dynamical systems (Sopron, 1985) Lecture Notes in Econom. and Math. Systems, vol. 287, Springer, Berlin, 1987, pp. 10–28. MR 1120038, DOI 10.1007/978-3-662-00748-8_{2}
Janina Kotus, The domains of normality of holomorphic self-maps of $\textbf {C}^*$, Ann. Acad. Sci. Fenn. Ser. A I Math. 15 (1990), no. 2, 329–340. MR 1087340, DOI 10.5186/aasfm.1990.1519

Convergence of Julia sets in the approximation of

On the efficiency of relaxed Newton’s method

Sur l’itération des substitutions rationelles et les fonctions de Poincaré

166

O. Lehto and K. I. Virtanen, Quasikonforme Abbildungen, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band 126, Springer-Verlag, Berlin-New York, 1965 (German). MR 0188434, DOI 10.1007/978-3-662-42594-7
M. Yu. Lyubich, Dynamics of rational transformations: topological picture, Uspekhi Mat. Nauk 41 (1986), no. 4(250), 35–95, 239 (Russian). MR 863874
M. Yu. Lyubich, Measurable dynamics of an exponential, Dokl. Akad. Nauk SSSR 292 (1987), no. 6, 1301–1304 (Russian). MR 880612
M. Yu. Lyubich, The measurable dynamics of the exponential, Sibirsk. Mat. Zh. 28 (1987), no. 5, 111–127 (Russian). MR 924986
Curt McMullen, Area and Hausdorff dimension of Julia sets of entire functions, Trans. Amer. Math. Soc. 300 (1987), no. 1, 329–342. MR 871679, DOI 10.1090/S0002-9947-1987-0871679-3
P. M. Makienko, Iterations of analytic functions in $\textbf {C}^*$, Dokl. Akad. Nauk SSSR 297 (1987), no. 1, 35–37 (Russian); English transl., Soviet Math. Dokl. 36 (1988), no. 3, 418–420. MR 916928

Simple mathematical models with very complicated dynamics

261

John C. Mayer, An explosion point for the set of endpoints of the Julia set of $\lambda \exp (z)$, Ergodic Theory Dynam. Systems 10 (1990), no. 1, 177–183. MR 1053806, DOI 10.1017/S0143385700005460
Hans-Günter Meier, The relaxed Newton-iteration for rational functions: the limiting case, Complex Variables Theory Appl. 16 (1991), no. 4, 239–260. MR 1105199, DOI 10.1080/17476939108814482
John Milnor, Dynamics in one complex variable, Friedr. Vieweg & Sohn, Braunschweig, 1999. Introductory lectures. MR 1721240
MichałMisiurewicz, On iterates of $e^{z}$, Ergodic Theory Dynam. Systems 1 (1981), no. 1, 103–106. MR 627790, DOI 10.1017/s014338570000119x

Leçons sur les familles normales de fonctions analytiques et leurs applications

Rolf Nevanlinna, Analytic functions, Die Grundlehren der mathematischen Wissenschaften, Band 162, Springer-Verlag, New York-Berlin, 1970. Translated from the second German edition by Phillip Emig. MR 0279280, DOI 10.1007/978-3-642-85590-0
H.-O. Peitgen and P. H. Richter, The beauty of fractals, Springer-Verlag, Berlin, 1986. Images of complex dynamical systems. MR 852695, DOI 10.1007/978-3-642-61717-1

Solution complète au problème de Siegel de linéarisation d’une application holomorphe au voisinage d’un point fixe (d’après J.-C. Yoccoz)

753

Hans Rådström, On the iteration of analytic functions, Math. Scand. 1 (1953), 85–92. MR 56702, DOI 10.7146/math.scand.a-10367
Mary Rees, The exponential map is not recurrent, Math. Z. 191 (1986), no. 4, 593–598. MR 832817, DOI 10.1007/BF01162349
J. F. Ritt, On the iteration of rational functions, Trans. Amer. Math. Soc. 21 (1920), no. 3, 348–356. MR 1501149, DOI 10.1090/S0002-9947-1920-1501149-6
Paul Charles Rosenbloom, L’itération des fonctions entières, C. R. Acad. Sci. Paris 227 (1948), 382–383 (French). MR 26691
David Ruelle, Elements of differentiable dynamics and bifurcation theory, Academic Press, Inc., Boston, MA, 1989. MR 982930
Carl Ludwig Siegel, Iteration of analytic functions, Ann. of Math. (2) 43 (1942), 607–612. MR 7044, DOI 10.2307/1968952
M. Viana da Silva, The differentiability of the hairs of $\textrm {exp}(Z)$, Proc. Amer. Math. Soc. 103 (1988), no. 4, 1179–1184. MR 955004, DOI 10.1090/S0002-9939-1988-0955004-1
Mitsuhiro Shishikura, On the quasiconformal surgery of rational functions, Ann. Sci. École Norm. Sup. (4) 20 (1987), no. 1, 1–29. MR 892140, DOI 10.24033/asens.1522

The connectivity of the Julia set and fixed point

Steve Smale, On the efficiency of algorithms of analysis, Bull. Amer. Math. Soc. (N.S.) 13 (1985), no. 2, 87–121. MR 799791, DOI 10.1090/S0273-0979-1985-15391-1
Gwyneth M. Stallard, Entire functions with Julia sets of zero measure, Math. Proc. Cambridge Philos. Soc. 108 (1990), no. 3, 551–557. MR 1068456, DOI 10.1017/S0305004100069437
Gwyneth M. Stallard, A class of meromorphic functions with no wandering domains, Ann. Acad. Sci. Fenn. Ser. A I Math. 16 (1991), no. 2, 211–226. MR 1139794, DOI 10.5186/aasfm.1991.1626
Gwyneth M. Stallard, The Hausdorff dimension of Julia sets of entire functions, Ergodic Theory Dynam. Systems 11 (1991), no. 4, 769–777. MR 1145621, DOI 10.1017/S0143385700006477
Norbert Steinmetz, On Sullivan’s classification of periodic stable domains, Complex Variables Theory Appl. 14 (1990), no. 1-4, 211–214. MR 1048721, DOI 10.1080/17476939008814420
Norbert Steinmetz, Rational iteration, De Gruyter Studies in Mathematics, vol. 16, Walter de Gruyter & Co., Berlin, 1993. Complex analytic dynamical systems. MR 1224235, DOI 10.1515/9783110889314
Dennis Sullivan, Itération des fonctions analytiques complexes, C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), no. 9, 301–303 (French, with English summary). MR 658395
Dennis Sullivan, Quasiconformal homeomorphisms and dynamics. I. Solution of the Fatou-Julia problem on wandering domains, Ann. of Math. (2) 122 (1985), no. 3, 401–418. MR 819553, DOI 10.2307/1971308

Quasiconformal homeomorphisms and dynamics

Topological conjugacy classes of analytic endomorphisms

Finding roots of complex polynomials with Newton’s method

Hans Töpfer, Über die Iteration der ganzen transzendenten Funktionen, insbesondere von $\sin z$ und $\cos z$, Math. Ann. 117 (1939), 65–84 (German). MR 1293, DOI 10.1007/BF01450008
Hans Töpfer, Komplexe Iterationsindizes ganzer und rationaler Funktionen, Math. Ann. 121 (1949), 191–222 (German). MR 31549, DOI 10.1007/BF01329624

Lectures on the general theory of integral functions

Jian Hua Zheng and Chung-Chun Yang, Further results on fixpoints and zeros of entire functions, Trans. Amer. Math. Soc. 347 (1995), no. 1, 37–50. MR 1179403, DOI 10.1090/S0002-9947-1995-1179403-9
Zhuan Ye, Structural instability of exponential functions, Trans. Amer. Math. Soc. 344 (1994), no. 1, 379–389. MR 1242788, DOI 10.1090/S0002-9947-1994-1242788-8
Jean-Christophe Yoccoz, Théorème de Siegel, nombres de Bruno et polynômes quadratiques, Astérisque 231 (1995), 3–88 (French). Petits diviseurs en dimension $1$. MR 1367353
Jean-Christophe Yoccoz, Linéarisation des germes de difféomorphismes holomorphes de $(\textbf {C}, 0)$, C. R. Acad. Sci. Paris Sér. I Math. 306 (1988), no. 1, 55–58 (French, with English summary). MR 929279
Jian Ying Zhou and Zhong Li, Structural instability of the mapping $z\to \lambda \exp (z)\ (\lambda >e^{-1})$, Sci. China Ser. A 32 (1989), no. 10, 1153–1163. MR 1057994

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