# American Institute of Mathematical Sciences

July  2012, 6(3): 327-375. doi: 10.3934/jmd.2012.6.327

## The Julia set of a post-critically finite endomorphism of $\mathbb{PC}^2$

 1 Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, United States

Received  April 2011 Published  October 2012

We construct a combinatorial model of the Julia set of the endomorphism $f(z, w)=((1-2z/w)^2, (1-2/w)^2)$ of $\mathbb{PC}^2$.
Citation: Volodymyr Nekrashevych. The Julia set of a post-critically finite endomorphism of $\mathbb{PC}^2$. Journal of Modern Dynamics, 2012, 6 (3) : 327-375. doi: 10.3934/jmd.2012.6.327
 [1] Laurent Bartholdi and Volodymyr V. Nekrashevych, Thurston equivalence of topological polynomials, Acta Math., 197 (2006), 1-51. doi: 10.1007/s11511-006-0007-3. [2] Joan S. Birman, "Braids, Links, and Mapping Class Groups," Annals of Mathematics Studies, 82, Princeton University Press, Princeton, N.J., 1974. [3] Adrien Douady and John H. Hubbard, "Étude Dynamiques des Polynômes Complexes. (Première Partie)," Publications Mathematiques d'Orsay, 2, Université de Paris-Sud, 1984. [4] Adrien Douady and John H. Hubbard, "Étude Dynamiques des Polynômes Complexes. (Deuxième Partie)," Publications Mathematiques d'Orsay, 4, Université de Paris-Sud, 1985. [5] Adrien Douady and John H. Hubbard, A proof of Thurston's topological characterization of rational functions, Acta Math., 171 (1993), 263-297. doi: 10.1007/BF02392534. [6] J. E. Fornæss and N. Sibony, Critically finite rational maps on $\mathbbP^2$, in "The Madison Symposium on Complex Analysis" (Madison, WI, 1991), Contemp. Math., 137, Amer. Math. Soc., Providence, RI, (1992), 245-260. [7] Yutaka Ishii and John Smillie, Homotopy shadowing, Amer. J. Math., 132 (2010), 987-1029. doi: 10.1353/ajm.0.0126. [8] S. Koch, "Teichmüller Theory and Endomorphisms of $\mathbbP^n$," Ph.D. thesis, Université de Provence, 2007. [9] John Milnor, Pasting together Julia sets: A worked out example of mating, Experiment. Math., 13 (2004), 55-92. doi: 10586458.2004.10504523. [10] Volodymyr Nekrashevych, "Self-Similar Groups," Mathematical Surveys and Monographs, 117, Amer. Math. Soc., Providence, RI, 2005. [11] Volodymyr Nekrashevych, A minimal Cantor set in the space of 3-generated groups, Geometriae Dedicata, 124 (2007), 153-190. doi: 10.1007/s10711-006-9118-4. [12] Volodymyr Nekrashevych, Combinatorial models of expanding dynamical systems, preprint, arXiv:0810.4936, 2008. [13] Volodymyr Nekrashevych, Symbolic dynamics and self-similar groups, in "Holomorphic Dynamics and Renormalization" (eds. Mikhail Lyubich and Michael Yampolsky), Fields Institute Communications, 53, Amer. Math. Soc., Providence, RI, (2008), 25-73. [14] Volodymyr Nekrashevych., Combinatorics of polynomial iterations. In D. Schleicher, editor, Complex Dynamics - Families and Friends, pages 169-214. A K Peters, 2009. [15] Volodymyr Nekrashevych, A group of non-uniform exponential growth locally isomorphic to $IMG$$(z^2+i), Transactions of the AMS, 362 (2010), 389-398. [16] Alfredo Poirier, The classification of postcritically finite polynomials II: Hubbard trees, Stony Brook IMS preprint, 1993. [17] John S. Wilson, On exponential growth and uniform exponential growth for groups, Inventiones Mathematicae, 155 (2004), 287-303. doi: 10.1007/s00222-003-0321-8. show all references ##### References:  [1] Laurent Bartholdi and Volodymyr V. Nekrashevych, Thurston equivalence of topological polynomials, Acta Math., 197 (2006), 1-51. doi: 10.1007/s11511-006-0007-3. [2] Joan S. Birman, "Braids, Links, and Mapping Class Groups," Annals of Mathematics Studies, 82, Princeton University Press, Princeton, N.J., 1974. [3] Adrien Douady and John H. Hubbard, "Étude Dynamiques des Polynômes Complexes. (Première Partie)," Publications Mathematiques d'Orsay, 2, Université de Paris-Sud, 1984. [4] Adrien Douady and John H. Hubbard, "Étude Dynamiques des Polynômes Complexes. (Deuxième Partie)," Publications Mathematiques d'Orsay, 4, Université de Paris-Sud, 1985. [5] Adrien Douady and John H. Hubbard, A proof of Thurston's topological characterization of rational functions, Acta Math., 171 (1993), 263-297. doi: 10.1007/BF02392534. [6] J. E. Fornæss and N. Sibony, Critically finite rational maps on \mathbbP^2, in "The Madison Symposium on Complex Analysis" (Madison, WI, 1991), Contemp. Math., 137, Amer. Math. Soc., Providence, RI, (1992), 245-260. [7] Yutaka Ishii and John Smillie, Homotopy shadowing, Amer. J. Math., 132 (2010), 987-1029. doi: 10.1353/ajm.0.0126. [8] S. Koch, "Teichmüller Theory and Endomorphisms of \mathbbP^n," Ph.D. thesis, Université de Provence, 2007. [9] John Milnor, Pasting together Julia sets: A worked out example of mating, Experiment. Math., 13 (2004), 55-92. doi: 10586458.2004.10504523. [10] Volodymyr Nekrashevych, "Self-Similar Groups," Mathematical Surveys and Monographs, 117, Amer. Math. Soc., Providence, RI, 2005. [11] Volodymyr Nekrashevych, A minimal Cantor set in the space of 3-generated groups, Geometriae Dedicata, 124 (2007), 153-190. doi: 10.1007/s10711-006-9118-4. [12] Volodymyr Nekrashevych, Combinatorial models of expanding dynamical systems, preprint, arXiv:0810.4936, 2008. [13] Volodymyr Nekrashevych, Symbolic dynamics and self-similar groups, in "Holomorphic Dynamics and Renormalization" (eds. Mikhail Lyubich and Michael Yampolsky), Fields Institute Communications, 53, Amer. Math. Soc., Providence, RI, (2008), 25-73. [14] Volodymyr Nekrashevych., Combinatorics of polynomial iterations. In D. Schleicher, editor, Complex Dynamics - Families and Friends, pages 169-214. A K Peters, 2009. [15] Volodymyr Nekrashevych, A group of non-uniform exponential growth locally isomorphic to IMG$$(z^2+i)$, Transactions of the AMS, 362 (2010), 389-398. [16] Alfredo Poirier, The classification of postcritically finite polynomials II: Hubbard trees, Stony Brook IMS preprint, 1993. [17] John S. Wilson, On exponential growth and uniform exponential growth for groups, Inventiones Mathematicae, 155 (2004), 287-303. doi: 10.1007/s00222-003-0321-8.
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