Embedding unicritical connectedness loci

Mukundan, Malavika

doi:10.1090/ecgd/389

Embedding unicritical connectedness loci
HTML articles powered by AMS MathViewer

by Malavika Mukundan;

Conform. Geom. Dyn. 28 (2024), 131-164

DOI: https://doi.org/10.1090/ecgd/389

Published electronically: December 3, 2024

HTML | PDF | Request permission

Abstract:

In this article, for degree $d\geq 1$, we construct an embedding $\Phi _d$ of the connectedness locus $\mathcal {M}_{d+1}$ of the polynomials $z^{d+1}+c$ into the connectedness locus of degree $2d+1$ bicritical odd polynomials.

References

Clara Bodelón, Robert L. Devaney, Michael Hayes, Gareth Roberts, Lisa R. Goldberg, and John H. Hubbard, Dynamical convergence of polynomials to the exponential, J. Differ. Equations Appl. 6 (2000), no. 3, 275–307. MR 1785056, DOI 10.1080/10236190008808229
Bodil Branner and Adrien Douady, Surgery on complex polynomials, Holomorphic dynamics (Mexico, 1986) Lecture Notes in Math., vol. 1345, Springer, Berlin, 1988, pp. 11–72. MR 980952, DOI 10.1007/BFb0081395
Bodil Branner and Núria Fagella, Homeomorphisms between limbs of the Mandelbrot set, J. Geom. Anal. 9 (1999), no. 3, 327–390. MR 1757453, DOI 10.1007/BF02921981
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. MR 945011, DOI 10.1007/BF02392275
A. Douady and J. H. Hubbard, Étude dynamique des polynômes complexes. Partie II, Publications Mathématiques d’Orsay [Mathematical Publications of Orsay], vol. 85, Université de Paris-Sud, Département de Mathématiques, Orsay, 1985 (French). With the collaboration of P. Lavaurs, Tan Lei and P. Sentenac. MR 812271
Adrien Douady, Systèmes dynamiques holomorphes, Bourbaki seminar, Vol. 1982/83, Astérisque, vol. 105, Soc. Math. France, Paris, 1983, pp. 39–63 (French). MR 728980
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
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
Dzmitry Dudko and Mikhail Lyubich, Local connectivity of the Mandelbrot set at some satellite parameters of bounded type, Geom. Funct. Anal. 33 (2023), no. 4, 912–1047. MR 4616693, DOI 10.1007/s00039-023-00637-8
Dzmitry Dudko and Dierk Schleicher, Homeomorphisms between limbs of the Mandelbrot set, Proc. Amer. Math. Soc. 140 (2012), no. 6, 1947–1956. MR 2888182, DOI 10.1090/S0002-9939-2011-11047-5
Dominik Eberlein, Sabyasachi Mukherjee, and Dierk Schleicher, Rational parameter rays of the multibrot sets, Dynamical systems, number theory and applications, World Sci. Publ., Hackensack, NJ, 2016, pp. 49–84. MR 3444240
A. È. Erëmenko and M. Yu. Lyubich, Dynamical properties of some classes of entire functions, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 4, 989–1020 (English, with English and French summaries). MR 1196102, DOI 10.5802/aif.1318
Yan Gao and Giulio Tiozzo, The core entropy for polynomials of higher degree, J. Eur. Math. Soc. (JEMS) 24 (2022), no. 7, 2555–2603. MR 4546475, DOI 10.4171/JEMS/1154
Lisa R. Goldberg and Linda Keen, A finiteness theorem for a dynamical class of entire functions, Ergodic Theory Dynam. Systems 6 (1986), no. 2, 183–192. MR 857196, DOI 10.1017/S0143385700003394
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. MR 1215974
John Hamal Hubbard, Teichmüller theory and applications to geometry, topology, and dynamics. Vol. 2, Matrix Editions, Ithaca, NY, 2016. Surface homeomorphisms and rational functions. MR 3675959
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, The Mandelbrot set is universal, The Mandelbrot set, theme and variations, London Math. Soc. Lecture Note Ser., vol. 274, Cambridge Univ. Press, Cambridge, 2000, pp. 1–17. MR 1765082
John Milnor, Dynamics in one complex variable, 3rd ed., Annals of Mathematics Studies, vol. 160, Princeton University Press, Princeton, NJ, 2006. MR 2193309
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
Rolf Nevanlinna, Analytic functions, Die Grundlehren der mathematischen Wissenschaften, Band 162, Springer-Verlag, New York-Berlin, 1970. Translated from the second German edition by Phillip Emig. MR 279280, DOI 10.1007/978-3-642-85590-0
Ch. Pommerenke, Boundary behaviour of conformal maps, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 299, Springer-Verlag, Berlin, 1992. MR 1217706, DOI 10.1007/978-3-662-02770-7
Johannes Riedl and Dierk Schleicher, On the locus of crossed renormalization, Sūrikaisekikenkyūsho K\B{o}kyūroku 1042 (1998), 11–31. Problems on complex dynamical systems (Japanese) (Kyoto, 1997). MR 1667914
Dierk Schleicher, On fibers and local connectivity of Mandelbrot and Multibrot sets, Fractal geometry and applications: a jubilee of Benoît Mandelbrot. Part 1, Proc. Sympos. Pure Math., vol. 72, Amer. Math. Soc., Providence, RI, 2004, pp. 477–517. MR 2112117, DOI 10.1090/pspum/072.1/2112117