×

zbMATH — the first resource for mathematics

Frontiers in complex dynamics. (English) Zbl 0807.30013
A rational function is said to be hyperbolic if all critical points tend to attracting cycles under iteration. Two equivalent definitions are that the function is expanding on its Julia set or that the postcritical set and the Julia set are disjoint. One of the main conjectures in complex dynamics is that hyperbolic rational functions are dense among all rational functions. This paper describes the background of this conjecture, connects it to a number of equivalent or related conjectures, and surveys recent results in this direction.
Among the topics and results discussed are: the density of structurally stable rational functions (Mañé-Sad-Sullivan), the connection with (the non-existence of) invariant line fields, the special case of quadratic polynomials and the Mandelbrot set, real quadratic polynomials, local connectivity of the Mandelbrot set implies density of hyperbolicity among quadratic polynomials (Doaudy-Hubbard), renormalization, the Mandelbrot set is connected at \(c\) if \(z^ 2+ c\) is not infinitely renormalizable (Yoccoz).
The author also sketches the proof of his result that the Julia set of \(f(z)= z^ 2+ c\) carries no invariant line field if \(f\) is an infinitely renormalizable real quadratic polynomial. Finally, he briefly mentions related recent work of Świątek, Lyubich, and Shishikura.
The paper is well written and highly recommended to anyone interested in this active area of research.

MSC:
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
37F99 Dynamical systems over complex numbers
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Alan F. Beardon, Iteration of rational functions, Graduate Texts in Mathematics, vol. 132, Springer-Verlag, New York, 1991. Complex analytic dynamical systems. · Zbl 0742.30002
[2] Lipman Bers and H. L. Royden, Holomorphic families of injections, Acta Math. 157 (1986), no. 3-4, 259 – 286. · Zbl 0619.30027
[3] Paul Blanchard, Complex analytic dynamics on the Riemann sphere, Bull. Amer. Math. Soc. (N.S.) 11 (1984), no. 1, 85 – 141. · Zbl 0558.58017
[4] Bodil Branner and John H. Hubbard, The iteration of cubic polynomials. I. The global topology of parameter space, Acta Math. 160 (1988), no. 3-4, 143 – 206. · Zbl 0668.30008
[5] Bodil Branner and John H. Hubbard, The iteration of cubic polynomials. II. Patterns and parapatterns, Acta Math. 169 (1992), no. 3-4, 229 – 325. · Zbl 0812.30008
[6] Lennart Carleson and Theodore W. Gamelin, Complex dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. · Zbl 0782.30022
[7] Global analysis, Proceedings of Symposia in Pure Mathematics, Vols. XIV-XVI. Edited by Shiing-shen Chern and Stephen Smale, American Mathematical Society, Providence, R.I., 1970.
[8] Universality in chaos, Adam Hilger, Ltd., Bristol; Heyden & Son, Inc., Philadelphia, PA, 1984. A selection of reprints edited by Predrag Cvitanović.
[9] Adrien Douady, Systèmes dynamiques holomorphes, Bourbaki seminar, Vol. 1982/83, Astérisque, vol. 105, Soc. Math. France, Paris, 1983, pp. 39 – 63 (French). · Zbl 0532.30019
[10] Adrien Douady, Descriptions of compact sets in \?, Topological methods in modern mathematics (Stony Brook, NY, 1991) Publish or Perish, Houston, TX, 1993, pp. 429 – 465. · Zbl 0801.58025
[11] Adrien Douady and John Hamal Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 2, 287 – 343. · Zbl 0587.30028
[12] Peter Doyle and Curt McMullen, Solving the quintic by iteration, Acta Math. 163 (1989), no. 3-4, 151 – 180. · Zbl 0705.65036
[13] 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. · Zbl 0712.58036
[14] P. Fatou, Sur les équations fonctionnelles, Bull. Soc. Math. France 48 (1920), 33 – 94 (French).
[15] J. H. Hubbard, Local connectivity of Julia sets and bifurcation loci: three theorems of J.-C. Yoccoz, Topological methods in modern mathematics (Stony Brook, NY, 1991) Publish or Perish, Houston, TX, 1993, pp. 467 – 511. · Zbl 0797.58049
[16] M. V. Jakobson, Absolutely continuous invariant measures for one-parameter families of one-dimensional maps, Comm. Math. Phys. 81 (1981), no. 1, 39 – 88. · Zbl 0497.58017
[17] S. Lattès, Sur l’iteration des substitutions rationelles et les fonctions de Poincaré, C. R. Acad. Sci. Paris Sér. I Math. 166 (1918), 26-28. · JFM 46.0522.01
[18] M. Lyubich, An analysis of the stability of the dynamics of rational functions, Selecta Math. Sov. 9 (1990), 69-90. · Zbl 0697.30026
[19] -, Geometry of quadratic polynomials: Moduli, rigidity, and local connectivity, Stony Brook IMS Preprint 1993/9.
[20] R. Mañé, A proof of the \( {C^1}\) stability conjecture, Publ. Math. Inst. Hautes Études Sci. 66 (1988), 160-210. · Zbl 0678.58022
[21] R. Mañé, P. Sad, and D. Sullivan, On the dynamics of rational maps, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 2, 193 – 217. · Zbl 0524.58025
[22] Curtis T. McMullen, Complex dynamics and renormalization, Annals of Mathematics Studies, vol. 135, Princeton University Press, Princeton, NJ, 1994. · Zbl 0822.30002
[23] Curtis T. McMullen and Dennis P. Sullivan, Quasiconformal homeomorphisms and dynamics. III. The Teichmüller space of a holomorphic dynamical system, Adv. Math. 135 (1998), no. 2, 351 – 395. · Zbl 0926.30028
[24] John Milnor, Self-similarity and hairiness in the Mandelbrot set, Computers in geometry and topology (Chicago, IL, 1986) Lecture Notes in Pure and Appl. Math., vol. 114, Dekker, New York, 1989, pp. 211 – 257. · Zbl 0676.58036
[25] John Milnor, Dynamics in one complex variable, Friedr. Vieweg & Sohn, Braunschweig, 1999. Introductory lectures. · Zbl 0946.30013
[26] -, Local connectivity of Julia sets: Expository lectures, Stony Brook IMS Preprint 1992/11.
[27] John Milnor, Remarks on iterated cubic maps, Experiment. Math. 1 (1992), no. 1, 5 – 24. · Zbl 0762.58018
[28] John Milnor, Geometry and dynamics of quadratic rational maps, Experiment. Math. 2 (1993), no. 1, 37 – 83. With an appendix by the author and Lei Tan. · Zbl 0922.58062
[29] Mary Rees, Positive measure sets of ergodic rational maps, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 3, 383 – 407. · Zbl 0611.58038
[30] M. Rees, Components of degree two hyperbolic rational maps, Invent. Math. 100 (1990), no. 2, 357 – 382. · Zbl 0712.30022
[31] Mary Rees, A partial description of parameter space of rational maps of degree two. I, Acta Math. 168 (1992), no. 1-2, 11 – 87. · Zbl 0774.58035
[32] Mitsuhiro Shishikura, The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets, Ann. of Math. (2) 147 (1998), no. 2, 225 – 267. · Zbl 0922.58047
[33] S. Smale, Structurally stable systems are not dense, Amer. J. Math. 88 (1966), 491 – 496. · Zbl 0149.20001
[34] Dennis Sullivan, On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 465 – 496.
[35] Dennis Sullivan, Conformal dynamical systems, Geometric dynamics (Rio de Janeiro, 1981) Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 725 – 752.
[36] Dennis Sullivan, Bounds, quadratic differentials, and renormalization conjectures, American Mathematical Society centennial publications, Vol. II (Providence, RI, 1988) Amer. Math. Soc., Providence, RI, 1992, pp. 417 – 466. · Zbl 0936.37016
[37] Dennis P. Sullivan and William P. Thurston, Extending holomorphic motions, Acta Math. 157 (1986), no. 3-4, 243 – 257. · Zbl 0619.30026
[38] G. Światek, Hyperbolicity is dense in the real quadratic family, Stony Brook IMS Preprint 1992/10.
[39] Jean-Christophe Yoccoz, Polynômes quadratiques et attracteur de Hénon, Astérisque 201-203 (1991), Exp. No. 734, 143 – 165 (1992) (French). Séminaire Bourbaki, Vol. 1990/91.
[40] -, Sur la connexité locale des ensembles de Julia et du lieu de connexité des polynômes, in preparation.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.