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Holomorphic dynamics

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References

  1. Beardon A., Iteration of Rational Functions, Springer-Verlag, 1990.

    Google Scholar 

  2. Bielefeld B. (editor), Conformal Dynamics Problem List, SUNY Stony Brook Institute for Mathematical Sciences, preprint #1990/1.

    Google Scholar 

  3. Blanchard P., Complex analytic dynamics on the Riemann sphere, Bull. Amer. Math. Soc. 11 (1984), 85–141.

    Article  MathSciNet  MATH  Google Scholar 

  4. Carleson L., Complex dynamics, UCLA course notes, 1990.

    Google Scholar 

  5. Eremenko, A., Lyubich M., The dynamics of analytic transformations, Leningrad Math J. 1:3 (1990).

    Google Scholar 

  6. Lyubich M., The dynamics of rational transforms: the topological picture, Russian Math. Surveys 41:4 (1986), 43–117.

    Article  MathSciNet  MATH  Google Scholar 

  7. Milnor J., Dynamics in one complex variable: Introductory Lectures, SUNY Stony Brook Institute for Mathematical Sciences, preprint #1990/5.

    Google Scholar 

References

  1. Branner B., Douady A., Surgery on Compex Polynomials, Proc. Symp. of Dynamical Systems Mexico (1986).

    Google Scholar 

  2. Douady A., Hubbard J. H., On the Dynamics of Polynomial Like Mappings, Ann. Sc. E.N.S., 4ème Séries 18 (1966).

    Google Scholar 

  3. Shishikura M., On the Quasiconformal Surgery of Rational Functions, Ann. Sc. E.N.S., 4ème Séries 20 (1987).

    Google Scholar 

  4. Shishikura M., Tan Lei, A Family of Cubic Rational Maps and Matings of Cubic Polynomials, preprint of Max-Plank-Institute, Bonn 50 (1988).

    Google Scholar 

  5. Tan Lei, Accouplements des polynômes quadratiques complexes, CRAS Paris (1986), 635–638.

    MATH  Google Scholar 

  6. Tan Lei, Accouplements des polynômes complexes, Thèse, Orsay (1987).

    Google Scholar 

  7. B. Wittner, On the Bifurcation Loci of Rational Maps of Degree Two, Ph.D thesis, Cornell Univ., Ithaca N.Y. (1986).

    Google Scholar 

References

  1. Bers L., Simultaneous uniformization, Bull. AMS 66 (1960), 94–97.

    Article  MathSciNet  MATH  Google Scholar 

  2. Douady A., Systemes dynamiques holomorphes, Astérisque 105–106 (1983), 39–64.

    MathSciNet  MATH  Google Scholar 

  3. Douady A., Algorithms for computing angles in the Mandelbrot set, Chaotic Dynamics and Fractals (Barnsley M. F., Demko S. G., eds.), Academic Press, 1986, pp. 155–168.

    Google Scholar 

  4. Douady A., Hubbard J., Étude dynamique des polynômes complexes, Pub. Math. d’Orsay, 1984.

    Google Scholar 

  5. Douady A., Hubbard, J., On the dynamics of polynomial-like mappings Ann. Sci. Éc. Norm. Sup. 18 (1985), 287–344.

    MathSciNet  MATH  Google Scholar 

  6. Fathi A., Laudenbach F., Poénaru V., Travaux de Thurston sur les surfaces, volume 66–67, Astérisque, 1979.

    Google Scholar 

  7. Kerckhoff S., Thurston W., Non-continuity of the action of the modular group at Bers’ boundary of Teichmüller space, Invent. math. 100 (1990), 25–48.

    Article  MathSciNet  MATH  Google Scholar 

  8. Lavaurs P., Une description combinatoire de l’involution définie par M sur les rationnels à dénominateur impair, CRAS Paris 303 (1986), 143–146.

    MathSciNet  MATH  Google Scholar 

  9. Mañé R., Sad P., Sullivan D., On the dynamics of rational maps, Ann. Sci. Éc. Norm. Sup. 16 (1983), 193–217.

    MathSciNet  MATH  Google Scholar 

  10. McMullen C., Cusps are dense, Annals of Math. 133 (1991), 217–247.

    Article  MathSciNet  MATH  Google Scholar 

  11. Sullivan D., Quasiconformal homeomorphisms and dynamics III: Topological conjugacy classes of analytic endomorphisms, Preprint.

    Google Scholar 

  12. Thurston W. P., On the combinatorics and dynamics of iterated rational maps Preprint.

    Google Scholar 

References

  1. Douady A., Hubbard J. H., A proof of Thurston’s topological characterization of rational functions, Mittag-Leffler, 1984, preprint.

    Google Scholar 

  2. Shishikura M., On a theorem of M. Rees for the matings of polynomials, IHES, 1990, preprint.

    Google Scholar 

  3. Tan Lei, Accouplements des polynômes complexes, Thèse, Orsay, 1987; Mating of quadratic polynomials (to appear).

    Google Scholar 

References

  1. Douady A., Hubbard J. H., A proof of Thurston’s topological characterization of rational functions, preprint, Mittag-Leffler, 1985.

    Google Scholar 

  2. Rees M., Critically-defined spaces of branched coverings, In preparation.

    Google Scholar 

  3. Sullivan D., Bounds, Quadratic differentials and renormalization conjectures, Preprint, 1990.

    Google Scholar 

  4. Thurston W. P., On the combinatorics of iterated rational maps, Preprint, Princeton University and I.A.S., 1985.

    Google Scholar 

References

  1. Bielefeld B., Conformal dynamics problem list, Stony Brook IMS, Preprint #1990/1.

    Google Scholar 

  2. Cremer H., Zum Zentrumproblem, Math. Ann. 98 (1927), 151–163.

    Article  MathSciNet  MATH  Google Scholar 

  3. Cremer H., Über die Häufigkeit der Nichtzentren, Math. Ann. 115 (1938), 573–580.

    Article  MathSciNet  MATH  Google Scholar 

  4. Douady A., Disques de Siegel et anneaux de Herman, Sém. Bourbaki, 1986–87; vol. 152–153, 1987–88.

    Google Scholar 

  5. Douady A., Hubbard J. H., Systèmes dynamiques holomorphes I,II: itération des polynômes complexes, Math. Orsay, 84.02 and 85.04.

    Google Scholar 

  6. Douady A., Hubbard J. H., On the dynamics of polynomial-like mappings Ann. Sci. Ec. Norm. Sup. 18 (1985), Paris, 287–343.

    MathSciNet  MATH  Google Scholar 

  7. Ghys E., Transformations holomorphes au voisinage d’une courbe de Jordan, CRAS Paris 298 (1984), 385–388.

    MathSciNet  MATH  Google Scholar 

  8. Goldberg L., Milnor J., Fixed point portraits of polynomial maps., Stony Brook IMS, preprint 1990/14.

    Google Scholar 

  9. Herman M., Recent results and some open questions on Siegel’s linearization theorem of germs of complex analytic diffeomorphisms of Cn near a fixed point, Proc 8th Int. Cong. Math. Phys., World Sci., 1986, pp. 138–198.

    Google Scholar 

  10. Hubbard J. H., Puzzles and quadratic tableaux, preprint 1990 (according to Yoccoz).

    Google Scholar 

  11. Lyubich M., An analysis of the stability of the dynamics of rational functions, Funk. Anal. i. Pril., 42 (1984), 72–91 (Russian); Selecta Math. Sovietica 9 (1990), 69–90.

    MathSciNet  MATH  Google Scholar 

  12. Lyubich M., On the Lebesgue measure of the Julia set of a quadratic polynomial, Stony Brook IMS, preprint 1991/10.

    Google Scholar 

  13. Petersen C., On the Poomerenke-Levin-Yoccoz inequality, IHES, 1991, preprint.

    Google Scholar 

  14. Perez-Marco R., Sur la dynamique des germes de difféomorphismes holomorphes de (C,0) et des difféomorphismes analytiques du cercle, Paris-Sud, 1990, Thèse.

    Google Scholar 

  15. Perez-Marco R., Solution complète au Problème de Siegel de linéarisation d’une application holomorphe au voisinage d’un point fixé (d’apres J.-C. Yoccoz), Sém. Bourbaki, Feb. 1992.

    Google Scholar 

  16. Rogers J. T., Singularities in the boundaries of local Siegel disks (to appear).

    Google Scholar 

  17. Shishikura M., The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets, Stony Brook IMS, preprint 1991/7.

    Google Scholar 

  18. Siegel C. L., Iteration of analytic functions, Ann. of Math., 43 (1942), 607–612.

    Article  MathSciNet  MATH  Google Scholar 

  19. Sørensen D. E. K., Local connectivity of quadratic Julia sets, Tech. Univ. Denmark, Lyngby, 1992, preprint.

    Google Scholar 

  20. Sullivan D., Conformal dynamical systems, Geometric Dynamics (Palis, ed.), Lecture Notes Math. 1007, Springer, 1983, pp. 725–752.

    Google Scholar 

  21. Yoccoz J.-C., Linéarisation des germes de difféomorphismes holomorphes de (ℂ,0) CRAS Paris 306 (1988), 55–58.

    MathSciNet  MATH  Google Scholar 

References

  1. Branner B., Hubbard J. H., The iteration of cubic polynomials, Part II: patterns and parapatterns, Acta Math. (to appear).

    Google Scholar 

  2. Blokh A., Lyubich M., The decomposition of one-dimensional dynamical systems into ergodic components, Leningrad Math. J. 1 (1990), 137–155.

    MathSciNet  MATH  Google Scholar 

  3. Blokh A., Lyubich M., Measurable dynamics of S-unimodal maps, Ann. Sci. École Norm. Sup. 24 (1991), no. 4, 545–573.

    MathSciNet  MATH  Google Scholar 

  4. Douady A., Hubbard J. H., Études dynamique des polynômes complexes, I., Math Orsay, 84-02.

    Google Scholar 

  5. Lyubich M., On the typical behavior of trajectories of a rational mapping of the sphere, Soviet Math. Dokl. 27 (1983), 22–25.

    MathSciNet  MATH  Google Scholar 

  6. Lyubich M., On the Lebesgue measure of the Julia set of a quadratic polynomial., 1991/10, Preprint IMS.

    Google Scholar 

  7. Lyubich M., Combinatorics, geometry and attractors of quadratic polynomials, 1992, Preprint.

    Google Scholar 

  8. Lyubich M., Milnor J., The Fibonacci unimodal map, Stony Brook, 1991/15, Preprint IMS.

    Google Scholar 

  9. Mané R., Sad P., Sullivan D., On the dynamics of rational maps, Ann. Sci. École Norm. Sup. 16 (1983), no. 4, 193–217.

    MathSciNet  MATH  Google Scholar 

  10. Rees M., Positive measure sets of ergodic rational maps, Ann. Sci. École Norm. Sup. 19 (1986), no. 4, 383–407.

    MathSciNet  MATH  Google Scholar 

  11. Shishikura M., On the quasiconformal surgery of rational functions, Ann. Sci. École Norm. Sup. 20 (1987), no. 4, 61–77.

    MathSciNet  MATH  Google Scholar 

  12. Sullivan D., The ergodic theory at infinity of a discrete group of hyperbolic isometries, Ann. of Math. Studies 97 (1981), Princeton Univ. Press, 465–497.

    Google Scholar 

References

  1. Aaronson J., Denker M., Urbański M., Ergodic theory for Markov fibred systems and parabolic rational maps, vol. 32, Göttingen, 1990, Preprint.

    Google Scholar 

  2. Bowen R., Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, L. N. Math. 470, Springer-Verlag, Berlin-Heidelberg-New York, 1975.

    MATH  Google Scholar 

  3. Denker M., Urbański M., Ergodic theory of equilibrium states for rational maps, Nonlinearity 4 (1991), 103–134.

    Article  MathSciNet  MATH  Google Scholar 

  4. Eremenko A. E., Levin G. M., On periodic points of polynomials, Ukr. Mat. Journal 41.11 (1989), 1467–1471.

    MathSciNet  MATH  Google Scholar 

  5. Ledrappier, F., Quelques propriétés ergodiques des applications rationelles, Sér. I Math. 299 (1984), C. R. Acad. Sci., Paris, 37–40.

    MATH  Google Scholar 

  6. Przytycki F., On the Perron-Frobenius-Ruelle operator for rational maps on the Riemann sphere and for Hölder continuous functions, Bol. Soc. Bras. Mat. 20.2 (1990), 95–125.

    Article  MathSciNet  MATH  Google Scholar 

  7. Przytycki, F., Urbanski M., Zdunik, A., Harmonic, Gibbs and Hausdorff measures for holomorphic maps, Part 1, Annals of Math. 130 (1989), 1–40; Part 2, Studia Math. 97.3 (1991), 189–225.

    Article  MathSciNet  MATH  Google Scholar 

  8. Przytycki F., Skrzypczak J., Convergence and pre-images of limit points for coding trees for iterations of holomorphic maps, Math. Annalen 290 (1991), 425–440.

    Article  MathSciNet  MATH  Google Scholar 

  9. Rees M., Positive measure sets of ergodic rational maps, Ann. scient. Éc. Norm. Sup. 19 (1986), 383–407.

    MathSciNet  MATH  Google Scholar 

  10. Zdunik A., Parabolic orbifolds and the dimension of the maximal measure for rational maps, Inventiones Math. 99 (1990), 627–649.

    Article  MathSciNet  MATH  Google Scholar 

  11. Zdunik A., Harmonic measure versus Hausdorff measures on repelers for holomorphic maps, Trans. AMS 326.2 (1991), 633–652.

    Article  MathSciNet  MATH  Google Scholar 

References

  1. Aarts J., Oversteegen L., A Characterization of smooth Cantor bouguets, Preprint.

    Google Scholar 

  2. Baker I. N., Wandering domains in the iteration of entire functions, Proc. London. Math. Soc. 49 (1984), 563–576.

    Article  MathSciNet  MATH  Google Scholar 

  3. Baker I. N., Kotus J., Lú Yinian, Iterates of meromorphic functions, I, II, and III, Preprints.

    Google Scholar 

  4. Devaney R. L., Keen L., Dynamics of meromorphic maps: maps with polynomial Schwarzian derivative, Annales Scientifiques de l’Ecole Normale Supérieure. 22 (1989), 55–79.

    MathSciNet  MATH  Google Scholar 

  5. Douady A., Goldberg L., The nonconjugacy of certain exponential functions, Holomorphic functions and moduli I., MSRI Publ., Springer Verlag, 1988, pp. 1–8.

    Google Scholar 

  6. Devaney R. L., Goldberg L., Hubbard J., A dynamical approximation to the exponential map by polynomials, Preprint.

    Google Scholar 

  7. Devaney R. L., Tangerman F., Dynamics of entire functions near the essential singularity, Ergodic Thy. Dynamical Syst 6 (1986), 489–503.

    MathSciNet  MATH  Google Scholar 

  8. Eremenko A., Lyubich M. Yu., Iterates of entire functions, Dokl. Akad. Nauk SSSR 279 (1984), 25–27 (Russian); English translation in Soviet Math. Dokl. 30 (1984), 592–594.

    MathSciNet  MATH  Google Scholar 

  9. Eremenko A. and Lyubich M. Yu., Structural stability in some families of entire functions, Funk. Anal. i Prilozh. 19 (1985), 86–87. (Russian).

    MathSciNet  MATH  Google Scholar 

  10. Goldberg L. R., Keen L., A finiteness theorem for a dynamical class of entire functions, Ergodic Theory and Dynamical Systems 6 (1986), 183–192.

    Article  MathSciNet  MATH  Google Scholar 

  11. Herman M., Exemples de fractions rationelles ayant une orbite dense sur la sphère de Riemann, Bull. Soc. Math. France 112 (1984), 93–142.

    MathSciNet  MATH  Google Scholar 

  12. Haruta M., The dynamics of Newton’s method on the exponential in the complex plane, Dissertation, Boston University, 1992.

    Google Scholar 

  13. von Haesler F., Kriete H., The relaxed Newton’s method for rational functions, Preprint.

    Google Scholar 

  14. LaVaurs P., Le lieu de connexité des polynômes du troisième degré n’est pas localement connexe, Preprint.

    Google Scholar 

  15. Milnor J., Remarks on iterated cubic maps, Preprint.

    Google Scholar 

  16. McMullen C., Area and Hausdorff dimension of Julia sets of entire functions, Trans. A.M.S. 300 (1987), 329–342.

    Article  MathSciNet  MATH  Google Scholar 

  17. Winters R., Dissertation, Boston University, 1990.

    Google Scholar 

References

  1. Baker I. N., Multiply connected domains of normality in iteration theory, Math. Z. 104 (1968), 252–256.

    Article  MathSciNet  Google Scholar 

  2. Baker I. N., Wandering domains in the iteration of entire functions, Proc. London Math. Soc. 49 (1984), 563–576.

    Article  MathSciNet  MATH  Google Scholar 

  3. Carleson L., Complex Dynamics, UCLA Course Notes, 1990, Winter.

    Google Scholar 

  4. Douady A., Hubbard J. H., On the dynamics of polynomial-like mappings, Ann. Sci. ENS 18 (1985), 287–343.

    MathSciNet  MATH  Google Scholar 

  5. Eremenko A., Lyubich M., Examples of entire functions with pathological dynamics, J. London Math. Soc. 36 (1987), 458–468.

    Article  MathSciNet  MATH  Google Scholar 

  6. Eremenko A., Lyubich M., Dynamical properties of some classes of entire functions, Preprint, SUNY Inst Math. Sci., 1990/4.

    Google Scholar 

  7. Fatou P., Sur les équations fonctionnelles, Bull. Soc. Math. France 48 (1920), 33–94, 208–314.

    MathSciNet  Google Scholar 

  8. Goldberg L., Keen, L., A finiteness theorem for a dynamical class of entire functions, Erg. Theory and Dynam. Syst. 6 (1986), 183–192.

    MathSciNet  MATH  Google Scholar 

  9. Herman M., Exemples de fractions rationnelles ayant une orbite dense sur la sphère de Riemann, Bull. Soc. Math. France 112 (1984), 93–142.

    MathSciNet  MATH  Google Scholar 

  10. Levin B. Ja., Distribution of zeros of entire functions, AMS Translations Math. Monographs, 1964, v. 5.

    Google Scholar 

  11. Sullivan D., Quasi conformal homeomorphisms and dynamics I. Solution of Fatou-Julia problem on wandering domains, Annals Math. 122 (1985), 401–418.

    Article  MATH  Google Scholar 

References

  1. Benzinger H., Julia sets and differential equations, Proc. Amer. Math. Soc. (to appear).

    Google Scholar 

  2. Douady A., Hubbard J. H., On the dynamics of polynomial-like mappings, Ann. Sci. Ecolc Norm. Sup., 4e série t.18 (1985), 287–343.

    MathSciNet  MATH  Google Scholar 

  3. Head J., The combinatorics of Newton’s method for cubic polynomials, Thesis, Cornell University, 1989.

    Google Scholar 

  4. Fatou P., Sur les équations fonctionnelles, Bull. Soc. Math. France 47 (1919), 161–271; Bull. Soc. Math. France 48 (1920), 33–94, 208–314.

    MathSciNet  MATH  Google Scholar 

  5. Haessler, F. V., Peitgen H.-O., Newton’s method and complex dynamical systems, Acta Appl. Math. 13 (1988), 3–58.

    Article  MathSciNet  MATH  Google Scholar 

  6. Jongen H. T. H., Jonker P., Twilt F., The continuous desingularized Newton’s method for meromorphic functions, Acta Appl. Math. 13 (1988), 81–121.

    Article  MathSciNet  MATH  Google Scholar 

  7. Saupe D., Discrete versus continuous Newton’s method: a case study, Acta Appl. Math. 13 (1988), 59–80.

    Article  MathSciNet  MATH  Google Scholar 

  8. Schröder E., Über unendlich viele Algorithmen zur Auflösung der Gleichungen, Math. Annalen 2 (1870), 317–365.

    Article  Google Scholar 

  9. Schröder E., Über iterierte Funktionen, Math. Annalen 3 (1871), 296–321.

    Article  Google Scholar 

  10. Shub M., Tischler D., Williams R., The Newtonian graph of a complex polynomial, SIAM J. Math. Anal. 19 (1988), 246–256.

    Article  MathSciNet  MATH  Google Scholar 

  11. Smale S., On the efficiency of algorithms of analysis, Bull. Amer. Math. Soc. 13 (1985), 87–121.

    Article  MathSciNet  MATH  Google Scholar 

  12. Sutherland S., Finding roots of complex polynomials with Newton’s method, Thesis, Boston University, 1989.

    Google Scholar 

  13. Tan Lei, Cubic Newton’s method of Thurston’s type, Ecole Norm. Sup. Lyon, 1991, Preprint.

    Google Scholar 

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Bielefeld, B., Lyubich, M. (1994). Holomorphic dynamics. In: Havin, V.P., Nikolski, N.K. (eds) Linear and Complex Analysis Problem Book 3. Lecture Notes in Mathematics, vol 1574. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0101069

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