From Cantor to semi-hyperbolic parameters along external rays

Chen, Yi-Chiuan; Kawahira, Tomoki

doi:10.1090/tran/7839

From Cantor to semi-hyperbolic parameters along external rays
HTML articles powered by AMS MathViewer

by Yi-Chiuan Chen and Tomoki Kawahira PDF

Trans. Amer. Math. Soc. 372 (2019), 7959-7992 Request permission

Abstract:

For the quadratic family $f_{c}(z) = z^2+c$ with $c$ in the exterior of the Mandelbrot set, it is known that every point in the Julia set moves holomorphically. Let $\hat {c}$ be a semi-hyperbolic parameter in the boundary of the Mandelbrot set. In this paper we prove that for each $z = z(c)$ in the Julia set, the derivative $dz(c)/dc$ is uniformly $O(1/\sqrt {|c-\hat {c}|})$ when $c$ belongs to a parameter ray that lands on $\hat {c}$. We also characterize the degeneration of the dynamics along the parameter ray.

References

Lars V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0357743
P. Atela and H. Kriete, Cantor goes Julia, Mathematica Gottingensis, no. 3 (1998), 1–15.
Christoph Bandt and Karsten Keller, Symbolic dynamics for angle-doubling on the circle. I. The topology of locally connected Julia sets, Ergodic theory and related topics, III (Güstrow, 1990) Lecture Notes in Math., vol. 1514, Springer, Berlin, 1992, pp. 1–23. MR 1179168, DOI 10.1007/BFb0097524
Lipman Bers and H. L. Royden, Holomorphic families of injections, Acta Math. 157 (1986), no. 3-4, 259–286. MR 857675, DOI 10.1007/BF02392595
Lennart Carleson and Theodore W. Gamelin, Complex dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR 1230383, DOI 10.1007/978-1-4612-4364-9
Lennart Carleson, Peter W. Jones, and Jean-Christophe Yoccoz, Julia and John, Bol. Soc. Brasil. Mat. (N.S.) 25 (1994), no. 1, 1–30. MR 1274760, DOI 10.1007/BF01232933
Yi-Chiuan Chen and Tomoki Kawahira, Simple proofs for the derivative estimates of the holomorphic motion near two boundary points of the Mandelbrot set, J. Math. Anal. Appl. 473 (2019), no. 1, 345–356. MR 3912825, DOI 10.1016/j.jmaa.2018.12.052
Yi-Chiuan Chen, Tomoki Kawahira, Hua-Lun Li, and Juan-Ming Yuan, Family of invariant Cantor sets as orbits of differential equations. II. Julia sets, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 21 (2011), no. 1, 77–99. MR 2786813, DOI 10.1142/S0218127411028295
Adrien Douady, Does a Julia set depend continuously on the polynomial?, Complex dynamical systems (Cincinnati, OH, 1994) Proc. Sympos. Appl. Math., vol. 49, Amer. Math. Soc., Providence, RI, 1994, pp. 91–138. MR 1315535, DOI 10.1090/psapm/049/1315535
Adrien Douady, Conjectures about the Branner-Hubbard motion of Cantor sets in C, Dynamics on the Riemann sphere, Eur. Math. Soc., Zürich, 2006, pp. 209–222. MR 2348963, DOI 10.4171/011-1/11
A. Douady and J. H. Hubbard, Étude dynamique des polynômes complexes. Partie I, Publications Mathématiques d’Orsay [Mathematical Publications of Orsay], vol. 84, Université de Paris-Sud, Département de Mathématiques, Orsay, 1984 (French). MR 762431
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. MR 816367, DOI 10.24033/asens.1491
Peter L. Duren, Univalent functions, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 259, Springer-Verlag, New York, 1983. MR 708494
M. Yu. Lyubich, Some typical properties of the dynamics of rational mappings, Uspekhi Mat. Nauk 38 (1983), no. 5(233), 197–198 (Russian). MR 718838
Jan Kiwi, Wandering orbit portraits, Trans. Amer. Math. Soc. 354 (2002), no. 4, 1473–1485. MR 1873015, DOI 10.1090/S0002-9947-01-02896-3
Jan Kiwi, $\Bbb R$eal laminations and the topological dynamics of complex polynomials, Adv. Math. 184 (2004), no. 2, 207–267. MR 2054016, DOI 10.1016/S0001-8708(03)00144-0
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. MR 732343, DOI 10.24033/asens.1446
Curtis T. McMullen, Complex dynamics and renormalization, Annals of Mathematics Studies, vol. 135, Princeton University Press, Princeton, NJ, 1994. MR 1312365
John Milnor, Periodic orbits, externals rays and the Mandelbrot set: an expository account, Astérisque 261 (2000), xiii, 277–333 (English, with English and French summaries). Géométrie complexe et systèmes dynamiques (Orsay, 1995). MR 1755445
Feliks Przytycki, Juan Rivera-Letelier, and Stanislav Smirnov, Equivalence and topological invariance of conditions for non-uniform hyperbolicity in the iteration of rational maps, Invent. Math. 151 (2003), no. 1, 29–63. MR 1943741, DOI 10.1007/s00222-002-0243-x
Juan Rivera-Letelier, On the continuity of Hausdorff dimension of Julia sets and similarity between the Mandelbrot set and Julia sets, Fund. Math. 170 (2001), no. 3, 287–317. MR 1880905, DOI 10.4064/fm170-3-6
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. MR 1626737, DOI 10.2307/121009
Sebastian van Strien, Misiurewicz maps unfold generically (even if they are critically non-finite), Fund. Math. 163 (2000), no. 1, 39–54. MR 1750334, DOI 10.4064/fm-163-1-39-54
William P. Thurston, On the geometry and dynamics of iterated rational maps, Complex dynamics, A K Peters, Wellesley, MA, 2009, pp. 3–137. Edited by Dierk Schleicher and Nikita Selinger and with an appendix by Schleicher. MR 2508255, DOI 10.1201/b10617-3