Dynamics of quadratic polynomials, I-II

Acta Mathematica

Volumn 178, Issue 2, 1997, Pages 185-297

  Lyubich, Mikhail ^a  

^a STONY BROOK UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001057166     PISSN: 00015962     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF02392694     Document Type: Article

References (75)

1. ABIKOFF, W., The Real Analytic Theory of Teichmüller Space. Lecture Notes in Math., 820. Springer-Verlag, Berlin-New York, 1980.
2. AHLFORS, L., Lectures on Quasiconformal Mappings. Van Nostrand, Toronto-New York-London, 1966.
3. ATELA, L., Bifurcations of dynamical rays in complex polynomials of degree two. Ergodic Theory Dynamical Systems, 12 (1991), 401-423.
4. BEARDON, A., Iteration of Rational Functions. Graduate Texts in Math., 132. Springer-Verlag. New York, 1991.
5. BELINSKII, P. P., General Properties of Quasiconformal Maps. Nauka, Novosibirsk, 1974 (Russian).
6. BRANNER, B. & HUBBARD, J. H., The iteration of cubic polynomials. Part II: Patterns and parapatterns. Acta Math., 169 (1992), 229-325.
7. BRUIN, H., KELLER, G., NOWICKI, T. & STRIEN, S. VAN, Wild Cantor attractors exist. Ann. of Math., 143 (1996), 97-130.
8. BRANNER, B., The Mandelbrot set, in Chaos and Fractals, pp. 75-106. Proc. Sympos. Appl. Math., 39. Amer. Math. Soc., Providence, RI, 1989.
9. BUFF, X., Points fixes de renormalisation. Thesis, Orsay, 1996.
10. CARLESON, L. & GAMELIN, T. W., Complex Dynamics. Springer-Verlag, New York, 1993.
11. DOUADY, A., Chirurgie sur les applications holomorphes, in Proceedings of the International Congress of Mathematicians (Berkeley, CA, 1986), pp. 724-738. Amer. Math. Soc., Providence, RI, 1987.
12. _ Description of compact sets in C, in Topological Methods in Modern Mathematics, Proceedings of a Symposium in Honor of John Milnor's Sixtieth Birthday (Stony Brook, NY, 1991), pp. 429-465. Publish or Perish, Houston, TX, 1993.
13. DOUADY, A. & HUBBARD, J. H., Étude dynamique des polynômes complexes, I, II. Publications Mathématiques d'Orsay, 84-02 and 85-04.
14. _ On the dynamics of polynomial-like maps. Ann. Sci. École Norm. Sup., 18 (1985), 287-343.
15. _ A proof of Thurston's topological characterization of rational functions. Acta Math., 171 (1993), 263-297.
16. EPSTEIN, H., Fixed points of composition operators. Nonlinearity, 2 (1989), 305-310.
17. GUCKENHEIMER, J. & JOHNSON, S., Distortion of S-unimodal maps. Ann. of Math., 132 (1990), 71-130.
18. GRACZYK, J. & ŚWIA̧TEK, G., Induced expansion for quadratic polynomials. Preprint, IMS at Stony Brook, 1993/8.
19. _ Polynomial-like property for real quadratic polynomials. Preprint, 1995.
20. GOLDBERG, L. & MILNOR, J., Fixed points of polynomial maps. II: Fixed point portraits. Ann. Sci. École Norm. Sup., 26 (1993), 51-98.
21. HU, J. & JIANG, Y., The Julia set of the Feigenbaum quadratic polynomial is locally connected. Preprint, 1993.
22. HERMAN, R. H., PUTNAM, I. F. & SKAU, C. F., Ordered Bratteli diagrams, dimension groups and topological dynamics. Internal. J. Math., 3 (1992), 827-864.
23. HUBBARD, J. H., Local connectivity of Julia sets and bifurcation loci: three theorems of J.-C. Yoccoz, in Topological Methods in Modern Mathematics, Proceedings of a Symposium in Honor of John Milnor's Sixtieth Birthday (Stony Brook, NY, 1991), pp. 467-511 and 375-378 (figures). Publish or Perish, Houston, TX, 1993.
24. JIANG, Y., Infinitely renormalizable quadratic Julia sets. Preprint, 1993.
25. JACOBSON, M. & ŚWIA̧TEK, G., Quasisymmetric conjugacies between unimodal maps. Preprint, IMS at Stony Brook, 1991/16.
26. KAHN, J., Holomorphic removability of Julia sets. Thesis, 1995.
27. KELLER, G. & NOWICKI, T., Fibonacci maps re(aℓ)visited. Ergodic Theory Dynamical Systems, 15 (1995), 99-120.
28. KRUSHKAL, S. L., Quasiconformal Mappings and Riemann Surfaces. John Wiley & Sons, New York-Toronto-London, 1979.
29. LYUBICH, M., The dynamics of rational transforms: the topological picture. Russian Math. Surveys, 41 (1986), 43-117.
30. _ On the Lebesgue measure of the Julia set of a quadratic polynomial. Preprint, IMS at Stony Brook, 1991/10.
31. _ Milnor's attractors, persistent recurrence and renormalization, in Topological Methods in Modern Mathematics, Proceedings of a Symposium in Honor of John Milnor's Sixtieth Birthday (Stony Brook, NY, 1991), pp. 85-92. Publish or Perish, Houston, TX, 1993.
32. _ Combinatorics, geometry and attractors of quasi-quadratic maps. Ann. of Math., 140 (1994), 347-404.
33. _ Teichmüller space of Fibonacci maps. Preprint, IMS at Stony Brook, 1993/12.
34. _ On the borderline of real and complex dynamics, in Proceedings of the International Congress of Mathematicians (Zürich, 1994), pp. 1203-1215. Birkhäuser, Basel, 1995.
35. _ Renormalization ideas in conformal dynamics, in Current Developments in Mathematics, pp. 155-184. International Press, Cambridge, MA, 1995.
36. _ Dynamics of quadratic polynomials. III: Parapuzzle and SBR measures. Preprint, IMS at Stony Brook, 1996/5.
37. _ Feigenbaum-Coullet-Tresser universality and Milnor's hairiness conjecture. Preprint, IHES/M/96/61.
38. _ How big the set of infinitely renormalizable quadratics is? Preprint, 1996.
39. LYUBICH, M. & MILNOR, J., The unimodal Fibonacci map. J. Amer. Math. Soc., 6 (1993), 425-457.
40. LAU, E. & SCHLEICHER, D., Internal addresses in the Mandelbrot set and irreducibility of polynomials. Preprint, IMS at Stony Brook, 1994/19.
41. LEVIN, G. & STRIEN, S. VAN, Local connectivity of Julia sets of real polynomials. Preprint, IMS at Stony Brook, 1995/5.
42. LEHTO, O. & VIRTANEN, K., Quasiconformal Mappings in the Plane. Springer-Verlag, New York-Heidelberg, 1973.
43. LYUBICH, M. & YAMPOLSKY, M., Complex bounds for real maps. Preprint, MSRI, 034-95. To appear in Ann. Inst. Fourier.
44. MANNING, A., Logarithmic capacity and renormalizability for landing on the Mandelbrot set. Preprint, 1995.
45. MARTENS, M., Distortion results and invariant Cantor sets for unimodal maps. Ergodic Theory Dynamical Systems, 14 (1994), 331-349.
46. MCMULLEN, C., Families of rational maps. Ann. of Math., 125 (1987), 467-493.
47. _ Complex Dynamics and Renormalization. Ann. of Math. Stud., 135. Princeton Univ. Press, Princeton, NJ, 1994.
48. _ Renormalization and 3-Manifolds which Fiber over the Circle. Ann. of Math. Stud., 142. Princeton Univ. Press, Princeton, NJ, 1996.
49. _ The classification of conformal dynamical systems, in Current Developments in Mathematics, pp. 323-360. International Press, Cambridge, MA, 1995.
50. MILNOR, J., Dynamics in one complex variable: Introductory lectures. Preprint, IMS at Stony Brook, 1990/5.
51. _ Local connectivity of Julia sets: Expository lectures. Preprint, IMS at Stony Brook, 1992/11.
52. _ Self-similarity and hairiness in the Mandelbrot set, in Computers in Geometry and Topology (Chicago, IL, 1986), pp. 211-257. Lecture Notes in Pure and Appl. Math., 114. Dekker, New York, 1989.
53. _ Periodic orbits, external rays and the Mandelbrot set: An expository account. Preprint, 1995.
54. MARTENS. M. & NOWICKI, N., Invariant measures for Lebesgue typical quadratic maps. Preprint, IMS at Stony Brook, 1996/6.
55. MELO, W. DE & STRIEN, S. VAN. One-Dimensional Dynamics. Springer-Verlag, Berlin, 1993.
56. MILNOR, J. & THURSTON, W., On iterated maps of the interval, in Dynamical Systems, University of Maryland, 1986-87 (J. C. Alexander, ed.), pp. 465-563. Lecture Notes in Math., 1342. Springer-Verlag, Berlin-New York, 1988.
57. PETERSEN, C. L., Local connectivity of some Julia sets containing a circle with an irrational rotation. Acta Math., 177 (1996), 163-224.
58. PRADO, E., Conformal measures in polynomial dynamics. Thesis, Stony Brook, 1995.
59. _ Ergodicity of conformal measures for unimodal polynomials. Preprint, IMS at Stony Brook, 1996/8.
60. PRZYTYCKI, F., Holomorphic Collet-Eckmann maps. Preprint, 1995.
61. REES, M.. A possible approach to a complex renormalization problem, in Linear and Complex Analysis Problem Book 3, Part. II, pp. 437-440. Lecture Notes in Math., 1574. Springer-Verlag, Berlin-New York, 1994.
62. SULLIVAN, D., Quasiconformal homeomorphisms and dynamics, topology and geometry, in Proceedings of the International Congress of Mathematicians (Berkeley, CA, 1986), pp. 1216-1228. Amer. Math. Soc., Providence, RI, 1987.
63. _ Bounds, quadratic differentials, and renormalization conjectures, in American Mathematical Society Centennial Publications, Vol. II (Providence, RI, 1988), pp. 417-466. Amer. Math. Soc., Providence, RI, 1992.
64. SARIO, L. & NAKAI, M., Classification Theory of Riemann Surfaces. Springer-Verlag, New York-Berlin, 1970.
65. SCHLEICHER, D., The structure of the Mandelbrot set. Preprint, 1995.
66. _ Rational external rays of the Mandelbrot set. Preprint, 1996.
67. SHISHIKURA, M., The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets. Preprint, IMS at Stony Brook, 1991/7.
68. _ Topological, geometric and complex analytic properties of Julia sets, in Proceedings of the International Congress of Mathematicians (Zürich, 1994), pp. 886-895. Birkhäuser, Basel, 1995.
69. STRIEN, S. VAN & NOWICKI, T., Polynomial maps with a Julia set of positive measure: Fibonacci maps. Preprint, IMS at Stony Brook, 1994/3.
70. THURSTON, W., The geometry and topology of 3-manifolds. Princeton Univ. Lecture Notes, 1982.
71. VERSHIK, A. M., Uniform algebraic approximation of shift and multiplication operators. Soviet. Math. Dokl., 24 (1981), 97-100.
72. VOLKOVYSKI, L. I., Quasi-conformal mappings. Lvov University, 1954 (Russian).
73. WENSTROM, L., Parameter scaling for the Fibonacci point. Preprint, IMS at Stony Brook, 1996/4.
74. YAMPOLSKY, M., Complex bounds for critical circle maps. Preprint, IMS at Stony Brook, 1995/12.
75. YARRINGTON, B., Local connectivity and Lebesgue measure of polynomial Julia sets. Thesis, Stony Brook, NY, 1995.