# American Institute of Mathematical Sciences

July  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, 2012, 32 (7) : 2403-2416. doi: 10.3934/dcds.2012.32.2403
##### References:
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##### References:
 [1] Christoph Bandt, On the Mandelbrot set for pairs of linear maps, Nonlinearity, 15 (2002), 1127-1147. doi: 10.1088/0951-7715/15/4/309.  Google Scholar [2] Christoph Bandt and Karsten Keller, Self-similar sets. II. A simple approach to the topological structure of fractals, Math. Nachr., 154 (1991), 27-39. doi: 10.1002/mana.19911540104.  Google Scholar [3] Christoph Bandt and Hui Rao, Topology and separation of self-similar fractals in the plane, Nonlinearity, 20 (2007), 1463-1474. doi: 10.1088/0951-7715/20/6/008.  Google Scholar [4] Mladen Bestvina, "Characterizing Universal $k$-Dimensional Menger Compacta," Memoirs of the American Mathematical Society, 380 (1988).  Google Scholar [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), 1047-1055. doi: 10.1017/S0143385704000380.  Google Scholar [6] Mario Bonk, Quasiconformal geometry of fractals, In "International Congress of Mathematicians," Vol. II, Eur. Math. Soc., Zürich, (2006), 1349-1373.  Google Scholar [7] Mario Bonk and Sergiy Merenkov, Quasisymmetric rigidity of Sierpiński carpets,, \arXiv{1102.3224}., ().   Google Scholar [8] Matias Carrasco, "Jauge Conforme des Espaces Métriques Compacts," Ph.D. thesis, Université de Provence, October 25, 2011. Available from: http://tel.archives-ouvertes.fr/docs/00/64/52/84/PDF/TheseCarrasco.pdf. Google Scholar [9] J. W. Cannon, W. J. Floyd and W. R. Parry, Finite subdivision rules, Conformal Geometry and Dynamics, 5 (2001), 153-196 (electronic). doi: 10.1090/S1088-4173-01-00055-8.  Google Scholar [10] Adrien Douady and John Hubbard, A Proof of Thurston's topological characterization of rational functions, Acta. Math., 171 (1993), 263-297. doi: 10.1007/BF02392534.  Google Scholar [11] Allan L. Edmonds, Branched coverings and orbit maps, Michigan Math. J., 23 (1976), 289-301. doi: 10.1307/mmj/1029001762.  Google Scholar [12] Kemal Eroğlu, Steffen Rohde and Boris Solomyak, Quasisymmetric conjugacy between quadratic dynamics and iterated function systems, Ergodic Theory Dynam. Systems, 30 (2010), 1665-1684. doi: 10.1017/S0143385709000789.  Google Scholar [13] Peter Haïssinsky and Kevin Pilgrim, Thurston obstructions and Ahlfors regular conformal dimension, Journal de Mathématiques Pures et Appliquées, 90 (2008), 229-241. doi: 10.1016/j.matpur.2008.04.006.  Google Scholar [14] Peter Haïssinsky and Kevin M. Pilgrim, Coarse expanding conformal dynamics, Astérisque No., 325 (2009), viii+139 pp.  Google Scholar [15] Juha Heinonen, "Lectures on Analysis on Metric Spaces," Universitext, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4613-0131-8.  Google Scholar [16] Atsushi Kameyama, Julia sets of postcritically finite rational maps and topological self-similar sets, Nonlinearity, 13 (2000), 165-188. doi: 10.1088/0951-7715/13/1/308.  Google Scholar [17] R. Daniel Mauldin and Stanley C. Williams, Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc., 309 (1988), 811-829. doi: 10.1090/S0002-9947-1988-0961615-4.  Google Scholar [18] Curtis T. McMullen, "Complex Dynamics and Renormalization," Annals of Mathematics Studies, 135, Princeton University Press, Princeton, NJ, 1994.  Google Scholar [19] Sergiy Merenkov, A Sierpiński carpet with the co-Hopfian property, Invent. Math., 180 (2010), 361-388.  Google Scholar [20] John Milnor, Geometry and dynamics of quadratic rational maps, With an appendix by the author and Tan Lei, Experiment. Math., 2 (1993), 37-83.  Google Scholar [21] Kevin M. Pilgrim, Canonical Thurston obstructions, Advances in Mathematics, 158 (2001), 154-168. doi: 10.1006/aima.2000.1971.  Google Scholar [22] Christopher W. Stark, Minimal dynamics on Menger manifolds, Topology Appl., 90 (1998), 21-30. doi: 10.1016/S0166-8641(97)00185-5.  Google Scholar [23] Jeremy T. Tyson and Jang-Mei Wu, Quasiconformal dimensions of self-similar fractals, Rev. Mat. Iberoam., 22 (2006), 205-258. doi: 10.4171/RMI/454.  Google Scholar
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