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Bulletin of the American Mathematical Society
Bulletin of the American Mathematical Society
ISSN 1088-9485(online) ISSN 0273-0979(print)

 

Complex analytic dynamics on the Riemann sphere


Author: Paul Blanchard
Journal: Bull. Amer. Math. Soc. 11 (1984), 85-141
MSC (1980): Primary 58Fxx; Secondary 30D05
MathSciNet review: 741725
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  • [Al] L. Ahlfors, Complex analysis, McGraw-Hill, 1979. MR 510197
  • [A2] L. Ahlfors, Conformal invariants: Topics in geometric function theory, McGraw-Hill, 1973. MR 357743
  • [A3] L. Ahlfors, Lectures on quasiconformal mappings, Van Nostrand, 1966. MR 200442
  • [Abl] W. Abikoff, The real analytic theory of Teichmüller space, Lecture Notes in Math., vol. 820, Springer-Verlag, 1980. MR 590044
  • [Ab2] W. Abikoff, The uniformization theorem, Amer. Math. Monthly 88 (1981), 574-592. MR 628026
  • [Ar] V. Arnold, Small denominators. I: On the mappings of the circumference onto itself, Amer. Math. Soc. Transl. (2) 46 (1965), 213-284.
  • [B] H. Brolin, Invariant sets under iteration of rational functions, Ark. Mat., 6 (1965), 103-144. MR 194595
  • [Bal] I. Baker, The existence of fixpoints of entire functions, Math. Z. 73 (1960), 280-284. MR 114008
  • [Ba2] I. Baker, Multiply connected domains of normality in iteration theory, Math. Z. 81 (1963), 206-214. MR 153842
  • [Ba3] I. Baker, Fixpoints of polynomials and rational functions, J. London Math. Soc. 39 (1964), 615-622. MR 169989
  • [Ba4] I. Baker, Sets of non-normality in iteration theory, J. London Math. Soc. 40 (1965), 499-502. MR 180668
  • [Ba5] I. Baker, The distribution of fixpoints of entire functions, Proc. London Math. Soc. (3) 16 (1966), 493-506. MR 197725
  • [Ba6] I. Baker, Repulsive fixpoints of entire functions, Math. Z. 104 (1968), 252-256. MR 226009
  • [Ba7] I. Baker, Limit functions and sets of non-normality in iteration theory, Ann. Acad. Sci. Fenn. Ser. A 467 (1970), 1-11. MR 264071
  • [Ba8] I. Baker, Completely invariant domains of entire functions, Mathematical Essays Dedicated to A. J. MacIntyre, Ohio Univ. Press, 1970. MR 271344
  • [Ba9] I. Baker, The domains of normality of an entire function, Ann. Acad. Sci. Fenn. Ser. A 1 (1975), 277-283. MR 402044
  • [Bal0] I. Baker, An entire function which has wandering domains, J. Austral. Math. Soc. (A) 22 (1976), 173-176. MR 419759
  • [Ball] I. Baker, The iteration of polynomials and transcendental entire functions, J. Austral. Math. Soc. (A) 30 (1981), 483-495. MR 621564
  • [BGHl] M. Barnsley, J. Geronimo and A. Harrington, Some treelike Julia sets and Pade approximants (preprint).
  • [BGH2] M. Barnsley, J. Geronimo and A. Harrington, Infinite dimensional Jacobi matrices associated with Julia sets, Proc. Amer. Math. Soc. (to appear). MR 702288
  • [BGH3] M. Barnsley, J. Geronimo and A. Harrington, Geometry, electrostatic measure and orthogonal polynomials on Julia sets for polynomials (preprint). MR 753919
  • [BGH4] M. Barnsley, J. Geronimo and A. Harrington, Orthogonal polynomials associated with invariant measures on Julia sets, Bull. Amer. Math. Soc. (N.S.) 7 (1982), 381-384. MR 663789
  • [BGH5] M. Barnsley, J. Geronimo and A. Harrington, On the invariant sets of a family of quadratic maps, Comm. Math. Phys. (to appear). MR 702565
  • [BGH6] M. Barnsley, J. Geronimo and A. Harrington, Geometrical and electrical properties of some Julia sets (preprint). MR 756242
  • [BH] M. Barnsley and A. Harrington, Moments of balanced measures on Julia sets (preprint). MR 742425
  • [BL] I. Baker and L. Liverpool, The value distribution of entire functions of order at most one, Acta Sci. Math. 41 (1979), 3-14. MR 534493
  • [Bo] Boettcher, Bull. Kasan Math. Soc. 14 (1905), 176.
  • [Bu] R. Burckel, Iterating analytic self-maps of discs, Amer. Math. Monthly 88 (1981), 396-407. MR 622955
  • [C] H. Cremer, Zum Zentrumproblem, Math. Ann. 98 (1928), 151-163. MR 1512397
  • [Ca] C. Camacho, On the local structure of conformal mappings and holomorphic vector fields, Astérisque 59-60 (1978), 83-94. MR 542732
  • [Cd] C. Carathéodory, Theory of functions, Vols. 1, 2, Chelsea, New York, 1960.
  • [CE] P. Collet and J. Eckmann, Iterated maps of the interval as dynamical systems, Birkhäuser, 1980. MR 613981
  • [CH] S. Chow and J. Hale, Methods of bifurcation theory, Springer-Verlag, 1982. MR 660633
  • [CGS] J. Curry, L. Garnett and D. Sullivan, On the iteration of a rational function computer experiments with Newton’s method, Comm. Math. Phys. 91 (1983), 267-277.
  • [Co] H. Cohn, Conformal mapping on Riemann surfaces, McGraw-Hill, 1967. MR 220922
  • [Col] P. Collet, Local C conjugacy on the Julia set for some holomorphic perturbations of z →z2, Preprint No. 18, Inst. Math. and Appl. Minneapolis, Minn., 1983.
  • [D] A. Douady, Systems dynamiques holomorphes, Séminaire Bourbaki, 35e année, 1982/83, No. 599, 1982. MR 728980
  • [De] R. Devaney, The structural instability of Exp(z) (preprint). MR 787910
  • [DH] A. Douady and J. Hubbard, Itération des polynômes quadratiques complexes, R. Acad. Sci. Paris 294 (1982), 123-126. MR 651802
  • [DK] R. Devaney and M. Krych, Dynamics ofexp(z), Ergodic Theory Dynamical Systems (to appear). MR 758892
  • [dL] R. de la Llave, A simple proof of Siegel’s center theorem, Preprint No. 2, Inst. Math. and Appl., Minneapolis, Minn., 1983.
  • [FI] P. Fatou, Sur les equations fonctionnelles, Bull. Soc. Math. France 47 (1919), 161-271. MR 1504787
  • [F2] P. Fatou, Sur les equations fonctionnelles, Bull. Soc. Math. France 48 (1920), 33-94. MR 1504792
  • [F3] P. Fatou, Sur les equations fonctionnelles, Bull. Soc. Math. France 48 (1920), 208-314. MR 1504797
  • [F4] P. Fatou, Sur l’itération des fonctions transcendantes entières, Acta Math. 47 (1926), 337-370.
  • [FK] H. Farkas and I. Kra, Riemann surfaces, Graduate Texts in Math., vol. 71, Springer-Verlag, 1980. MR 583745
  • [G] J. Guckenheimer, Endomorphisms of the Riemann sphere (S. S. Chern and S. Smale, eds.), Proc. Sympos. Pure Math., vol. 14, Amer. Math. Soc., Providence, R. I., 1970. MR 274740
  • [HI] M. Herman, Exemples de fractions rationeles ayant une orbite dense sur la sphere Riemann (preprint).
  • [H2] M. Herman, Lecture notes on an easy proof of Siege’s theorem. See also Appendice VII of [HI].
  • [Hi] E. Hille, Analytic function theory, Vols. 1, 2, Chelsea, New York, 1973.
  • [J] G. Julia, Memoire sur l’iteration des fonctions rationnelles, J. Math. 8 (1918), 47-245. See also Oeuvres de Gaston Julia, Gauthier-Villars, Vol. I, Paris, 121-319.
  • [Jal] M. Jakobson, Structure of polynomial mappings on a singular set, Mat. Sb. 80 (1968), 105-124; English transl. in Math. USSR-Sb. 6 (1968). MR 231984
  • [Ja2] M. Jakobson, On the problem of the classification of polynomial endomorphisms of the plane, Mat. Sb. 80 (1969), 365-387; English transl. in Math. USSR-Sb. 9 (1969). MR 251194
  • [L] S. Lattès, Sur l’itérartion des substitutions rationnelles et les fonctions de Poincaré, R. Acad. Sci. Paris 166 (1918), 26-28.
  • [LV] O. Lehto and K. Virtanen, Quasiconformal mappings in the plane, Springer-Verlag, 1973. MR 344463
  • [Ml] Mandelbrot, The fractal geometry of nature, Freeman, 1982. MR 665254
  • [M2] Mandelbrot, Fractal aspects of the iteration of z →λ z (l-z) and z, Ann. New York Acad. Sci. 357 (1980), 249-259.
  • [MN] N. S. Manton and M. Nauenberg, Universal scaling behavior for iterated maps in the complex plane (preprint). MR 713685
  • [Mi] M. Misiurewicz, On iterates of e, Ergodic Theory Dynamical Systems 1 (1981), 103-106. MR 627790
  • [Mo] P. Montel, Leçons sur les familles normales, Gauthier-Villars, Paris, 1927.
  • [Moe] R. Moeckel, Rotations of the closures of some simply connected domains, Univ. of Minnesota Math. Report 83-121.
  • [MSS] R. Mañé, P. Sad and D. Sullivan, On the dynamics of rational maps (preprint).
  • [Myl] P. Myrberg, Inversion der Iteration fur rationale Funktionen, Ann. Acad. Sci. Fenn. Ser. A 292 (1960). MR 122992
  • [My2] P. Myrberg, Sur l’iteration des polynomes reels quadratiques, J. Math. Pures Appl. (9) 41 (1962), 339-351.
  • [My3] P. Myrberg, Iteration der polynome mit reellen Koeffizienten, Ann. Acad. Sci. Fenn. Ser A 348 (1964). MR 169997
  • [N] M. Narasimhan, Riemann surfaces, Tata Inst. Fund. Res., Math. Pamphlets No. 1, 1963. MR 259108
  • [Ne] M. Newman, Elements of the topology of plane sets, Cambridge Univ. Press, 1939.
  • [Ni] I. Niven, Irrational numbers, Math. Assoc. Amer., Washington, D. C., 1956. MR 80123
  • [R] D. Ruelle, Repellersfor real analytic maps, Ergodic Theory Dynamical Systems 2 (1982), 99-108. MR 684247
  • [Rel] M. Rees, Ergodic rational maps with dense critical point forward orbit, Univ. of Minnesota Math. Report 82-140.
  • [Re2] M. Rees, Positive measure sets of ergodic rational maps, Univ. of Minnesota Math. Report 82-169.
  • [Ri] J. Ritt, On the iteration of rational functions, Trans. Amer. Math. Soc. 23 (1920), 348-356. MR 1501149
  • [Ro] P. Rosenbloom, L’iteration des fonctions entieres, R. Acad. Sci. Paris 9 (1948), 382-383.
  • [Ru] H. Rüssman, Kleine Nenner. II: Bemerkungen zur Newtonschen Method, Nachr. Akad. Wiss. Göttingen Math. Phys. Kl (1972), 1-20. MR 309297
  • [S1] D. Sullivan, Quasiconformal homeomorphisms and dynamics. I (preprint).
  • [S2] D. Sullivan, Quasiconformal homeomorphisms and dynamics. III: Topological conjugacy classes of analytic endomorphisms (preprint).
  • [S3] D. Sullivan, Seminar on conformal and hyperbolic geometry, IHES Seminar notes, March 1982.
  • [S4] D. Sullivan, Conformal dynamical systems (preprint). MR 730296
  • [Si] C. Siegel, Iteration of analytic functions, Ann. of Math. (2) 43 (1942), 607-612. MR 7044
  • [SM] C. Siegel and J. Moser, Lectures on celestial mechanics, Springer-Verlag, 1971. MR 502448
  • [ST] D. Sullivan and W. Thurston, Extending holomorphic motions (preprint). MR 857674
  • [Tl] W. Thurston, Lecture notes from seminar at Princeton University.
  • [T2] W. Thurston, Lecture notes, CBMS Conf. Univ. of Minnesota, Duluth, Minn.
  • [To] H. Töpfer, Über die Iteration der gazen transzendenten Funktionen, insbesondere von sin z undcos z, Math. Ann. 117 (1939), 65-84. MR 1293
  • [W] M. Widom, Renormalization group analysis of quasi-periodicity in analytic maps, Comm. Math. Phys. 92 (1983), 121-136. MR 728450
  • [WBKS] M. Widom, D. Bensimon, L. Kadinoff and S. Shenker, Strange objects in the complex plane (preprint).
  • [Z] E. Zehnder, A simple proof of a generalization of a theorem of L. Siegel, Lecture Notes in Math., vol. 597, Springer-Verlag, 1977, pp. 855-866. MR 461575

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DOI: http://dx.doi.org/10.1090/S0273-0979-1984-15240-6
PII: S 0273-0979(1984)15240-6