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Given any postsingularly finite exponential function $p_\lambda(z) = \lambda \exp(z)$ where $\lambda \in \C^*$, we construct a sequence of postcritically finite unicritical polynomials $p_{d,\lambda_d}(z) = \lambda_d(1+\frac{z}{d})^d$ that converge to $p_\lambda$ locally uniformly in $\C$, with the same postsingular portrait as that of $p_\lambda$.
Misha Gromov, Metric structures for Riemannian and non-Riemannian spaces , english ed., Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2007, Based on the 1981 French original, With appendices by M. Katz, P. Pansu and S. Semmes, Translated from the French by Sean Michael Bates. MR 2307192
1981
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Adrien Douady and John Hamal Hubbard, Itération des polynômes quadratiques complexes , C. R. Acad. Sci. Paris Sér. I Math. 294
1982
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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
1984
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Pierre Lavaurs, Une description combinatoire de l’involution définie par M M sur les rationnels à dénominateur impair , C. R. Acad. Sci. Paris Sér. I Math. 303
1986
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Ben Bielefeld, Yuval Fisher, and John Hubbard, The classification of critically preperiodic polynomials as dynamical systems , J. Amer. Math. Soc. 5
1992
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Adrien Douady and John H. Hubbard, A proof of Thurston’s topological characterization of rational functions , Acta Mathematica 171
1993
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Alfredo Poirier, On post-critically finite polynomials , ProQuest LLC, Ann Arbor, MI, 1993, Thesis (Ph.D.)–State University of New York at Stony Brook. MR 2690269
1993
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John H. Hubbard and Dierk Schleicher, The spider algorithm , Complex dynamical systems (Cincinnati, OH, 1994), Proc. Sympos. Appl. Math., vol. 49, Amer. Math. Soc., Providence, RI, 1994, pp. 155–180. MR 1315537
1994
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Dierk Schleicher, Internal addresses in the Mandelbrot set and irreducibility of polynomials , ProQuest LLC, Ann Arbor, MI, 1994, Thesis (Ph.D.)–Cornell University. MR 2691195
1994
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Masashi Kisaka, Local uniform convergence and convergence of julia sets , Nonlinearity 8
1995
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Kevin M. Pilgrim and Tan Lei, Combining rational maps and controlling obstructions , Ergodic Theory Dynam. Systems 18
1998
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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
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Dierk Schleicher and Johannes Zimmer, Periodic points and dynamic rays of exponential maps , Ann. Acad. Sci. Fenn. Math. 28
2003
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by same author, 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
2004
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Markus Förster and Dierk Schleicher, Parameter rays for the exponential family , 2005
Benson Farb and Dan Margalit, A primer on mapping class groups , Princeton Mathematical Series, vol. 49, Princeton University Press, Princeton, NJ, 2012. MR 2850125
2012
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Helena Mihaljević-Brandt, Dynamical approximation and kernels of non-escaping hyperbolic components , Ergodic Theory Dynam. Systems 32
2012
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Bodil Branner and Núria Fagella, Quasiconformal surgery in holomorphic dynamics , Cambridge Studies in Advanced Mathematics, vol. 141, Cambridge University Press, Cambridge, 2014, With contributions by Xavier Buff, Shaun Bullett, Adam L. Epstein, Peter Haïssinsky, Christian Henriksen, Carsten L. Petersen, Kevin M. Pilgrim, Tan Lei and Michael Yampolsky. MR 3445628
2014
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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
2016
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2005
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Markus Förster, Exponential maps with escaping singular orbits , Ph.D. thesis, 2006
2006
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Alexandra Kaffl, Hubbard trees and kneading sequences for unicritical and cubic polynomials , Ph.D. thesis, 2006
2006
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Markus Förster, Lasse Rempe, and Dierk Schleicher, Classification of escaping exponential maps , Proceedings of the American Mathematical Society 136
2007
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2008
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2009
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Adrien Douady and John Hubbard, Exploring the mandelbrot set. the orsay notes
Cited in the paper.
Bastian Laubner, Dierk Schleicher, and Vlad Vicol, A combinatorial classification of postsingularly finite complex exponential maps
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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
2016
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D. Pfrang, Homotopy hubbard trees for post-singularly finite transcendental entire maps , Ph.D. thesis, Jacobs University, Bremen, 2019
2019
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2020
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David Pfrang, Sören Petrat, and Dierk Schleicher, Dreadlock pairs and dynamic partitions for post-singularly finite entire functions , 2021
2021
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S. Shemyakov, A topological characterization of certain postsingularly finite entire functions: transcendental dynamics and thurston theory , Ph.D. thesis, Université d’Aix-Marseille, 2022
2022
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2023
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