Remote Access 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
Full-text PDF

References | Similar Articles | Additional Information

References [Enhancements On Off] (What's this?)

  • Lars V. Ahlfors, Complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1978. An introduction to the theory of analytic functions of one complex variable; International Series in Pure and Applied Mathematics. MR 510197
  • Lars V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. McGraw-Hill Series in Higher Mathematics. MR 0357743
  • Lars V. Ahlfors, Lectures on quasiconformal mappings, Manuscript prepared with the assistance of Clifford J. Earle, Jr. Van Nostrand Mathematical Studies, No. 10, D. Van Nostrand Co., Inc., Toronto, Ont.-New York-London, 1966. MR 0200442
  • William Abikoff, The real analytic theory of Teichmüller space, Lecture Notes in Mathematics, vol. 820, Springer, Berlin, 1980. MR 590044
  • William Abikoff, The uniformization theorem, Amer. Math. Monthly 88 (1981), no. 8, 574–592. MR 628026, 10.2307/2320507
  • [Ar] V. Arnold, Small denominators. I: On the mappings of the circumference onto itself, Amer. Math. Soc. Transl. (2) 46 (1965), 213-284.
  • Hans Brolin, Invariant sets under iteration of rational functions, Ark. Mat. 6 (1965), 103–144 (1965). MR 0194595
  • Irvine Noel Baker, The existence of fixpoints of entire functions, Math. Z. 73 (1960), 280–284. MR 0114008
  • Irvine Noel Baker, Multiply connected domains of normality in iteration theory, Math. Z. 81 (1963), 206–214. MR 0153842
  • I. N. Baker, Fixpoints of polynomials and rational functions, J. London Math. Soc. 39 (1964), 615–622. MR 0169989
  • I. N. Baker, Sets of non-normality in iteration theory, J. London Math. Soc. 40 (1965), 499–502. MR 0180668
  • I. N. Baker, The distribution of fixpoints of entire functions, Proc. London Math. Soc. (3) 16 (1966), 493–506. MR 0197725
  • I. N. Baker, Repulsive fixpoints of entire functions, Math. Z. 104 (1968), 252–256. MR 0226009
  • 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
  • I. N. Baker, An entire function which has wandering domains, J. Austral. Math. Soc. Ser. A 22 (1976), no. 2, 173–176. MR 0419759
  • 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
  • [BGHl] M. Barnsley, J. Geronimo and A. Harrington, Some treelike Julia sets and Pade approximants (preprint).
  • M. F. Barnsley, J. S. Geronimo, and A. N. Harrington, Infinite-dimensional Jacobi matrices associated with Julia sets, Proc. Amer. Math. Soc. 88 (1983), no. 4, 625–630. MR 702288, 10.1090/S0002-9939-1983-0702288-6
  • M. F. Barnsley, J. S. Geronimo, and A. N. Harrington, Geometry, electrostatic measure and orthogonal polynomials on Julia sets for polynomials, Ergodic Theory Dynam. Systems 3 (1983), no. 4, 509–520. MR 753919, 10.1017/S0143385700002108
  • M. F. Barnsley, J. S. Geronimo, and A. N. Harrington, Orthogonal polynomials associated with invariant measures on Julia sets, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 2, 381–384. MR 663789, 10.1090/S0273-0979-1982-15043-1
  • M. F. Barnsley, J. S. Geronimo, and A. N. Harrington, On the invariant sets of a family of quadratic maps, Comm. Math. Phys. 88 (1983), no. 4, 479–501. MR 702565
  • M. F. Barnsley, J. S. Geronimo, and A. N. Harrington, Geometrical and electrical properties of some Julia sets, Classical quantum models and arithmetic problems, Lecture Notes in Pure and Appl. Math., vol. 92, Dekker, New York, 1984, pp. 1–68. MR 756242
  • M. F. Barnsley and A. N. Harrington, Moments of balanced measures on Julia sets, Trans. Amer. Math. Soc. 284 (1984), no. 1, 271–280. MR 742425, 10.1090/S0002-9947-1984-0742425-6
  • I. N. Baker and L. S. O. Liverpool, The value distribution of entire functions of order at most one, Acta Sci. Math. (Szeged) 41 (1979), no. 1-2, 3–14. MR 534493
  • [Bo] Boettcher, Bull. Kasan Math. Soc. 14 (1905), 176.
  • R. B. Burckel, Iterating analytic self-maps of discs, Amer. Math. Monthly 88 (1981), no. 6, 396–407. MR 622955, 10.2307/2321822
  • Hubert Cremer, Zum Zentrumproblem, Math. Ann. 98 (1928), no. 1, 151–163 (German). MR 1512397, 10.1007/BF01451586
  • César Camacho, On the local structure of conformal mappings and holomorphic vector fields in 𝐶², Journées Singulières de Dijon (Univ. Dijon, Dijon, 1978) Astérisque, vol. 59, Soc. Math. France, Paris, 1978, pp. 3, 83–94. MR 542732
  • [Cd] C. Carathéodory, Theory of functions, Vols. 1, 2, Chelsea, New York, 1960.
  • Pierre Collet and Jean-Pierre Eckmann, Iterated maps on the interval as dynamical systems, Progress in Physics, vol. 1, Birkhäuser, Boston, Mass., 1980. MR 613981
  • Shui Nee Chow and Jack K. Hale, Methods of bifurcation theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 251, Springer-Verlag, New York-Berlin, 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.
  • Harvey Cohn, Conformal mapping on Riemann surfaces, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1967. MR 0220922
  • [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.
  • Adrien Douady, Systèmes dynamiques holomorphes, Bourbaki seminar, Vol. 1982/83, Astérisque, vol. 105, Soc. Math. France, Paris, 1983, pp. 39–63 (French). MR 728980
  • Robert L. Devaney, Structural instability of 𝑒𝑥𝑝(𝑧), Proc. Amer. Math. Soc. 94 (1985), no. 3, 545–548. MR 787910, 10.1090/S0002-9939-1985-0787910-2
  • 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
  • Robert L. Devaney and Michał Krych, Dynamics of 𝑒𝑥𝑝(𝑧), Ergodic Theory Dynam. Systems 4 (1984), no. 1, 35–52. MR 758892, 10.1017/S014338570000225X
  • [dL] R. de la Llave, A simple proof of Siegel’s center theorem, Preprint No. 2, Inst. Math. and Appl., Minneapolis, Minn., 1983.
  • P. Fatou, Sur les équations fonctionnelles, Bull. Soc. Math. France 47 (1919), 161–271 (French). MR 1504787
  • P. Fatou, Sur les équations fonctionnelles, Bull. Soc. Math. France 48 (1920), 33–94 (French). MR 1504792
  • P. Fatou, Sur les équations fonctionnelles, Bull. Soc. Math. France 48 (1920), 208–314 (French). MR 1504797
  • [F4] P. Fatou, Sur l’itération des fonctions transcendantes entières, Acta Math. 47 (1926), 337-370.
  • Hershel M. Farkas and Irwin Kra, Riemann surfaces, Graduate Texts in Mathematics, vol. 71, Springer-Verlag, New York-Berlin, 1980. MR 583745
  • John Guckenheimer, Endomorphisms of the Riemann sphere, Global Analysis (Proc. Sympos. Pure Math. Vol, XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 95–123. MR 0274740
  • [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.
  • M. V. Jakobson, The structure of polynomial mappings on a special set, Mat. Sb. (N.S.) 77 (119) (1968), 105–124 (Russian). MR 0231984
  • M. V. Jakobson, On the question of classification of polynomial endomorphisms of the plane, Mat. Sb. (N.S.) 80 (122) (1969), 365–387 (Russian). MR 0251194
  • [L] S. Lattès, Sur l’itérartion des substitutions rationnelles et les fonctions de Poincaré, R. Acad. Sci. Paris 166 (1918), 26-28.
  • O. Lehto and K. I. Virtanen, Quasiconformal mappings in the plane, 2nd ed., Springer-Verlag, New York-Heidelberg, 1973. Translated from the German by K. W. Lucas; Die Grundlehren der mathematischen Wissenschaften, Band 126. MR 0344463
  • Benoit B. Mandelbrot, The fractal geometry of nature, W. H. Freeman and Co., San Francisco, Calif., 1982. Schriftenreihe für den Referenten. [Series for the Referee]. 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.
  • N. S. Manton and M. Nauenberg, Universal scaling behaviour for iterated maps in the complex plane, Comm. Math. Phys. 89 (1983), no. 4, 555–570. MR 713685
  • Michał Misiurewicz, On iterates of 𝑒^{𝑧}, Ergodic Theory Dynamical Systems 1 (1981), no. 1, 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).
  • P. J. Myrberg, Inversion der Iteration für rationale Funktionen, Ann. Acad. Sci. Fenn. Ser. A I No. 292 (1960), 14 (German). MR 0122992
  • [My2] P. Myrberg, Sur l’iteration des polynomes reels quadratiques, J. Math. Pures Appl. (9) 41 (1962), 339-351.
  • P. J. Myrberg, Iteration der Binome beliebigen Grades, Ann. Acad. Sci. Fenn. Ser. A I No. 348 (1964), 14. MR 0169997
  • M. S. Narasimhan, R. R. Simha, Raghavan Narasimhan, and C. S. Seshadri, Riemann surfaces, Mathematical Pamphlets, vol. 1, Tata Institute of Fundamental Research, Bombay, 1963. MR 0259108
  • [Ne] M. Newman, Elements of the topology of plane sets, Cambridge Univ. Press, 1939.
  • Ivan Niven, Irrational numbers, The Carus Mathematical Monographs, No. 11, The Mathematical Association of America. Distributed by John Wiley and Sons, Inc., New York, N.Y., 1956. MR 0080123
  • David Ruelle, Repellers for real analytic maps, Ergodic Theory Dynamical Systems 2 (1982), no. 1, 99–107. 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.
  • J. F. Ritt, On the iteration of rational functions, Trans. Amer. Math. Soc. 21 (1920), no. 3, 348–356. MR 1501149, 10.1090/S0002-9947-1920-1501149-6
  • [Ro] P. Rosenbloom, L’iteration des fonctions entieres, R. Acad. Sci. Paris 9 (1948), 382-383.
  • Helmut Rüssmann, Kleine Nenner. II. Bemerkungen zur Newtonschen Methode, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II (1972), 1–10 (German). MR 0309297
  • [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.
  • Dennis Sullivan, Conformal dynamical systems, Geometric dynamics (Rio de Janeiro, 1981) Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 725–752. MR 730296, 10.1007/BFb0061443
  • Carl Ludwig Siegel, Iteration of analytic functions, Ann. of Math. (2) 43 (1942), 607–612. MR 0007044
  • Carl Ludwig Siegel and Jürgen K. Moser, Lectures on celestial mechanics, Springer-Verlag, New York-Heidelberg, 1971. Translation by Charles I. Kalme; Die Grundlehren der mathematischen Wissenschaften, Band 187. MR 0502448
  • Dennis P. Sullivan and William P. Thurston, Extending holomorphic motions, Acta Math. 157 (1986), no. 3-4, 243–257. MR 857674, 10.1007/BF02392594
  • [Tl] W. Thurston, Lecture notes from seminar at Princeton University.
  • [T2] W. Thurston, Lecture notes, CBMS Conf. Univ. of Minnesota, Duluth, Minn.
  • Hans Töpfer, Über die Iteration der ganzen transzendenten Funktionen, insbesondere von sin𝑧 und cos𝑧, Math. Ann. 117 (1939), 65–84 (German). MR 0001293
  • Michael Widom, Renormalization group analysis of quasiperiodicity in analytic maps, Comm. Math. Phys. 92 (1983), no. 1, 121–136. MR 728450
  • [WBKS] M. Widom, D. Bensimon, L. Kadinoff and S. Shenker, Strange objects in the complex plane (preprint).
  • Eduard Zehnder, A simple proof of a generalization of a theorem by C. L. Siegel, Geometry and topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976) Springer, Berlin, 1977, pp. 855–866. Lecture Notes in Math., Vol. 597. MR 0461575

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (1980): 58Fxx, 30D05

Retrieve articles in all journals with MSC (1980): 58Fxx, 30D05

Additional Information