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Conformal Geometry and Dynamics

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Iterated function system quasiarcs


Authors: Annina Iseli and Kevin Wildrick
Journal: Conform. Geom. Dyn. 21 (2017), 78-100
MSC (2010): Primary 28A80, 30C65
DOI: https://doi.org/10.1090/ecgd/305
Published electronically: February 3, 2017
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Abstract: We consider a class of iterated function systems (IFSs) of contracting similarities of  $ \mathbb{R}^n$, introduced by Hutchinson, for which the invariant set possesses a natural Hölder continuous parameterization by the unit interval. When such an invariant set is homeomorphic to an interval, we give necessary conditions in terms of the similarities alone for it to possess a quasisymmetric (and as a corollary, bi-Hölder) parameterization. We also give a related necessary condition for the invariant set of such an IFS to be homeomorphic to an interval.


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  • [1] V. V. Aseev, A. V. Tetenov, and A. S. Kravchenko, Self-similar Jordan curves on the plane, Sibirsk. Mat. Zh. 44 (2003), no. 3, 481-492 (Russian, with Russian summary); English transl., Siberian Math. J. 44 (2003), no. 3, 379-386. MR 1984698
  • [2] Kari Astala, Personal communication.
  • [3] -, Self-similar zippers, Holomorphic functions and moduli, Vol. I (Berkeley, CA, 1986), Math. Sci. Res. Inst. Publ., vol. 10, Springer, New York, 1988, pp. 61-73. MR 955808 (89i:30018)
  • [4] Mario Bonk, Quasiconformal geometry of fractals, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 1349-1373. MR 2275649
  • [5] Mario Bonk and Bruce Kleiner, Quasisymmetric parametrizations of two-dimensional metric spheres, Invent. Math. 150 (2002), no. 1, 127-183. MR 1930885
  • [6] Mario Bonk and Daniel Meyer, Expanding thurston maps, (preprint: arXiv:1009.3647).
  • [7] Marc Bourdon and Bruce Kleiner, Combinatorial modulus, the combinatorial Loewner property, and Coxeter groups, Groups Geom. Dyn. 7 (2013), no. 1, 39-107. MR 3019076
  • [8] Matias Carrasco Piaggio, On the conformal gauge of a compact metric space, Ann. Sci. Éc. Norm. Supér. (4) 46 (2013), no. 3, 495-548 (2013) (English, with English and French summaries). MR 3099984
  • [9] Guy David and Stephen Semmes, Fractured fractals and broken dreams, Oxford Lecture Series in Mathematics and its Applications, vol. 7, The Clarendon Press, Oxford University Press, New York, 1997. MR 1616732
  • [10] K. J. Falconer, Dimensions and measures of quasi self-similar sets, Proc. Amer. Math. Soc. 106 (1989), no. 2, 543-554. MR 969315
  • [11] K. J. Falconer and D. T. Marsh, Classification of quasi-circles by Hausdorff dimension, Nonlinearity 2 (1989), no. 3, 489-493. MR 1005062
  • [12] K. J. Falconer and D. T. Marsh, On the Lipschitz equivalence of Cantor sets, Mathematika 39 (1992), no. 2, 223-233. MR 1203278
  • [13] Frederick W. Gehring, Kari Hag, and The ubiquitous quasidisk, Mathematical Surveys and Monographs, vol. 184, American Mathematical Society, Providence, RI, 2012. With contributions by Ole Jacob Broch. MR 2933660
  • [14] Manouchehr Ghamsari and David A. Herron, Higher dimensional Ahlfors regular sets and chordarc curves in $ \mathbf {R}^n$, Rocky Mountain J. Math. 28 (1998), no. 1, 191-222. MR 1639853
  • [15] Masayoshi Hata, On the structure of self-similar sets, Japan J. Appl. Math. 2 (1985), no. 2, 381-414. MR 839336
  • [16] Juha Heinonen, Lectures on analysis on metric spaces, Universitext, Springer-Verlag, New York, 2001. MR 1800917
  • [17] David Herron and Daniel Meyer, Quasicircles and bounded turning circles modulo bi-Lipschitz maps, Rev. Mat. Iberoam. 28 (2012), no. 3, 603-630. MR 2949615
  • [18] John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713-747. MR 625600
  • [19] Marta Llorente and Pertti Mattila, Lipschitz equivalence of subsets of self-conformal sets, Nonlinearity 23 (2010), no. 4, 875-882. MR 2602018
  • [20] John McLaughlin, A note on Hausdorff measures of quasi-self-similar sets, Proc. Amer. Math. Soc. 100 (1987), no. 1, 183-186. MR 883425
  • [21] Daniel Meyer, Snowballs are quasiballs, Trans. Amer. Math. Soc. 362 (2010), no. 3, 1247-1300. MR 2563729
  • [22] Istvan Prause, Holomorphic motions: http://www.math.helsinki.fi/$ {\sim }$prause/qc.html.
  • [23] Hui Rao, Huo-Jun Ruan, and Yang Wang, Lipschitz equivalence of self-similar sets: algebraic and geometric properties, Fractal geometry and dynamical systems in pure and applied mathematics. I. Fractals in pure mathematics, Contemp. Math., vol. 600, Amer. Math. Soc., Providence, RI, 2013, pp. 349-364. MR 3203409
  • [24] Steffen Rohde, Quasicircles modulo bilipschitz maps, Rev. Mat. Iberoamericana 17 (2001), no. 3, 643-659. MR 1900898
  • [25] P. Tukia and J. Väisälä, Quasisymmetric embeddings of metric spaces, Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980), no. 1, 97-114. MR 595180
  • [26] Zhi-Ying Wen and Li-Feng Xi, Relations among Whitney sets, self-similar arcs and quasi-arcs, Israel J. Math. 136 (2003), 251-267. MR 1998112
  • [27] Li-Feng Xi, Lipschitz equivalence of self-conformal sets, J. London Math. Soc. (2) 70 (2004), no. 2, 369-382. MR 2078899

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Additional Information

Annina Iseli
Affiliation: Mathematisches Institut, Universität Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland
Email: annina.iseli@math.unibe.ch

Kevin Wildrick
Affiliation: Department of Mathematical Sciences, Montana State University, P.O. Box 172400 Bozeman, Montana 59717
Email: kevin.wildrick@montana.edu

DOI: https://doi.org/10.1090/ecgd/305
Received by editor(s): November 20, 2015
Received by editor(s) in revised form: November 25, 2016
Published electronically: February 3, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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