# American Institue of Mathematical Sciences

2012, 32(7): 2403-2416. doi: 10.3934/dcds.2012.32.2403

## Examples of coarse expanding conformal maps

 1 Centre de Mathmatiques et Informatique (CMI) et LATP, Aix-Marseille Université, 39, rue F. Joliot Curie 13453 Marseille Cedex 13, France 2 Dept. Mathematics, Indiana University, Bloomington, IN 47405, United States

Received  February 2010 Revised  October 2010 Published  March 2012

In previous work, a class of noninvertible topological dynamical systems $f: X \to X$ was introduced and studied; we called these topologically coarse expanding conformal systems. To such a system is naturally associated a preferred quasisymmetry (indeed, snowflake) class of metrics in which arbitrary iterates distort roundness and ratios of diameters by controlled amounts; we called this metrically coarse expanding conformal. In this note we extend the class of examples to several more familiar settings, give applications of our general methods, and discuss implications for the computation of conformal dimension.
Citation: Peter Haïssinsky, Kevin M. Pilgrim. Examples of coarse expanding conformal maps. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2403-2416. doi: 10.3934/dcds.2012.32.2403
##### References:
 [1] Christoph Bandt, On the Mandelbrot set for pairs of linear maps,, Nonlinearity, 15 (2002), 1127. doi: 10.1088/0951-7715/15/4/309. [2] Christoph Bandt and Karsten Keller, Self-similar sets. II. A simple approach to the topological structure of fractals,, Math. Nachr., 154 (1991), 27. doi: 10.1002/mana.19911540104. [3] Christoph Bandt and Hui Rao, Topology and separation of self-similar fractals in the plane,, Nonlinearity, 20 (2007), 1463. doi: 10.1088/0951-7715/20/6/008. [4] Mladen Bestvina, "Characterizing Universal $k$-Dimensional Menger Compacta,", Memoirs of the American Mathematical Society, 380 (1988). [5] Paul Blanchard, Robert L. Devaney, Daniel M. Look, Pradipta Seal and Yakov Shapiro, Sierpinski-curve Julia sets and singular perturbations of complex polynomials,, Ergodic Theory Dynam. Systems, 25 (2005), 1047. doi: 10.1017/S0143385704000380. [6] Mario Bonk, Quasiconformal geometry of fractals,, In, (2006), 1349. [7] Mario Bonk and Sergiy Merenkov, Quasisymmetric rigidity of Sierpiński carpets,, \arXiv{1102.3224}., (). [8] Matias Carrasco, "Jauge Conforme des Espaces Métriques Compacts,", Ph.D. thesis, (2011). [9] J. W. Cannon, W. J. Floyd and W. R. Parry, Finite subdivision rules,, Conformal Geometry and Dynamics, 5 (2001), 153. doi: 10.1090/S1088-4173-01-00055-8. [10] Adrien Douady and John Hubbard, A Proof of Thurston's topological characterization of rational functions,, Acta. Math., 171 (1993), 263. doi: 10.1007/BF02392534. [11] Allan L. Edmonds, Branched coverings and orbit maps,, Michigan Math. J., 23 (1976), 289. doi: 10.1307/mmj/1029001762. [12] Kemal Eroğlu, Steffen Rohde and Boris Solomyak, Quasisymmetric conjugacy between quadratic dynamics and iterated function systems,, Ergodic Theory Dynam. Systems, 30 (2010), 1665. doi: 10.1017/S0143385709000789. [13] Peter Haïssinsky and Kevin Pilgrim, Thurston obstructions and Ahlfors regular conformal dimension,, Journal de Mathématiques Pures et Appliquées, 90 (2008), 229. doi: 10.1016/j.matpur.2008.04.006. [14] Peter Haïssinsky and Kevin M. Pilgrim, Coarse expanding conformal dynamics,, Astérisque No., 325 (2009). [15] Juha Heinonen, "Lectures on Analysis on Metric Spaces,", Universitext, (2001). doi: 10.1007/978-1-4613-0131-8. [16] Atsushi Kameyama, Julia sets of postcritically finite rational maps and topological self-similar sets,, Nonlinearity, 13 (2000), 165. doi: 10.1088/0951-7715/13/1/308. [17] R. Daniel Mauldin and Stanley C. Williams, Hausdorff dimension in graph directed constructions,, Trans. Amer. Math. Soc., 309 (1988), 811. doi: 10.1090/S0002-9947-1988-0961615-4. [18] Curtis T. McMullen, "Complex Dynamics and Renormalization,", Annals of Mathematics Studies, 135 (1994). [19] Sergiy Merenkov, A Sierpiński carpet with the co-Hopfian property,, Invent. Math., 180 (2010), 361. [20] John Milnor, Geometry and dynamics of quadratic rational maps,, With an appendix by the author and Tan Lei, 2 (1993), 37. [21] Kevin M. Pilgrim, Canonical Thurston obstructions,, Advances in Mathematics, 158 (2001), 154. doi: 10.1006/aima.2000.1971. [22] Christopher W. Stark, Minimal dynamics on Menger manifolds,, Topology Appl., 90 (1998), 21. doi: 10.1016/S0166-8641(97)00185-5. [23] Jeremy T. Tyson and Jang-Mei Wu, Quasiconformal dimensions of self-similar fractals,, Rev. Mat. Iberoam., 22 (2006), 205. doi: 10.4171/RMI/454.

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##### References:
 [1] Christoph Bandt, On the Mandelbrot set for pairs of linear maps,, Nonlinearity, 15 (2002), 1127. doi: 10.1088/0951-7715/15/4/309. [2] Christoph Bandt and Karsten Keller, Self-similar sets. II. A simple approach to the topological structure of fractals,, Math. Nachr., 154 (1991), 27. doi: 10.1002/mana.19911540104. [3] Christoph Bandt and Hui Rao, Topology and separation of self-similar fractals in the plane,, Nonlinearity, 20 (2007), 1463. doi: 10.1088/0951-7715/20/6/008. [4] Mladen Bestvina, "Characterizing Universal $k$-Dimensional Menger Compacta,", Memoirs of the American Mathematical Society, 380 (1988). [5] Paul Blanchard, Robert L. Devaney, Daniel M. Look, Pradipta Seal and Yakov Shapiro, Sierpinski-curve Julia sets and singular perturbations of complex polynomials,, Ergodic Theory Dynam. Systems, 25 (2005), 1047. doi: 10.1017/S0143385704000380. [6] Mario Bonk, Quasiconformal geometry of fractals,, In, (2006), 1349. [7] Mario Bonk and Sergiy Merenkov, Quasisymmetric rigidity of Sierpiński carpets,, \arXiv{1102.3224}., (). [8] Matias Carrasco, "Jauge Conforme des Espaces Métriques Compacts,", Ph.D. thesis, (2011). [9] J. W. Cannon, W. J. Floyd and W. R. Parry, Finite subdivision rules,, Conformal Geometry and Dynamics, 5 (2001), 153. doi: 10.1090/S1088-4173-01-00055-8. [10] Adrien Douady and John Hubbard, A Proof of Thurston's topological characterization of rational functions,, Acta. Math., 171 (1993), 263. doi: 10.1007/BF02392534. [11] Allan L. Edmonds, Branched coverings and orbit maps,, Michigan Math. J., 23 (1976), 289. doi: 10.1307/mmj/1029001762. [12] Kemal Eroğlu, Steffen Rohde and Boris Solomyak, Quasisymmetric conjugacy between quadratic dynamics and iterated function systems,, Ergodic Theory Dynam. Systems, 30 (2010), 1665. doi: 10.1017/S0143385709000789. [13] Peter Haïssinsky and Kevin Pilgrim, Thurston obstructions and Ahlfors regular conformal dimension,, Journal de Mathématiques Pures et Appliquées, 90 (2008), 229. doi: 10.1016/j.matpur.2008.04.006. [14] Peter Haïssinsky and Kevin M. Pilgrim, Coarse expanding conformal dynamics,, Astérisque No., 325 (2009). [15] Juha Heinonen, "Lectures on Analysis on Metric Spaces,", Universitext, (2001). doi: 10.1007/978-1-4613-0131-8. [16] Atsushi Kameyama, Julia sets of postcritically finite rational maps and topological self-similar sets,, Nonlinearity, 13 (2000), 165. doi: 10.1088/0951-7715/13/1/308. [17] R. Daniel Mauldin and Stanley C. Williams, Hausdorff dimension in graph directed constructions,, Trans. Amer. Math. Soc., 309 (1988), 811. doi: 10.1090/S0002-9947-1988-0961615-4. [18] Curtis T. McMullen, "Complex Dynamics and Renormalization,", Annals of Mathematics Studies, 135 (1994). [19] Sergiy Merenkov, A Sierpiński carpet with the co-Hopfian property,, Invent. Math., 180 (2010), 361. [20] John Milnor, Geometry and dynamics of quadratic rational maps,, With an appendix by the author and Tan Lei, 2 (1993), 37. [21] Kevin M. Pilgrim, Canonical Thurston obstructions,, Advances in Mathematics, 158 (2001), 154. doi: 10.1006/aima.2000.1971. [22] Christopher W. Stark, Minimal dynamics on Menger manifolds,, Topology Appl., 90 (1998), 21. doi: 10.1016/S0166-8641(97)00185-5. [23] Jeremy T. Tyson and Jang-Mei Wu, Quasiconformal dimensions of self-similar fractals,, Rev. Mat. Iberoam., 22 (2006), 205. doi: 10.4171/RMI/454.
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