Skip to main content Accessibility help

Invariant Jordan curves of Sierpiński carpet rational maps



In this paper, we prove that if $R:\widehat{\mathbb{C}}\rightarrow \widehat{\mathbb{C}}$ is a postcritically finite rational map with Julia set homeomorphic to the Sierpiński carpet, then there is an integer $n_{0}$ , such that, for any $n\geq n_{0}$ , there exists an $R^{n}$ -invariant Jordan curve $\unicode[STIX]{x1D6E4}$ containing the postcritical set of $R$ .



Hide All
[Ber] Bernard, J.. Dynamique des perturbations d’un exemple de Lattès. PhD Thesis, Université de Paris-Sud, Orsay, 1994.
[BLM] Bonk, M., Lyubich, M. and Merenkov, S.. Quasisymmetries of Sierpiński carpet Julia sets. Preprint, 2013.
[BM] Bonk, M. and Meyer, D.. Expanding Thurston maps. Available at
[CFKP] Cannon, J. W., Floyd, W. J., Kenyon, R. and Parry, W. R.. Constructing rational maps from subdivision rules. Conform. Geom. Dyn. 7 (2003), 76102 (electronic).
[CFP] Cannon, J. W., Floyd, W. J. and Parry, W. R.. Constructing subdivision rules from rational maps. Conform. Geom. Dyn. 11 (2007), 128136 (electronic).
[CGNPP] Cordwell, K., Gilbertson, S., Nuechterlein, N., Pilgrim, K. M. and Pinella, S.. On the classification of critically fixed rational maps. Conform. Geom. Dyn. 19 (2015), 5194 (electronic).
[Dav] Daverman, R. J.. Decompositions of Manifolds (Pure and Applied Mathematics, 124) . Academic Press, Orlando, FL, 1986.
[Dev] Devaney, R. L.. Singular perturbations of complex polynomials. Bull. Amer. Math. Soc. (N.S.) 50(3) (2013), 391429.
[DH] Douady, A. and Hubbard, J.. Etude Dynamique des Polynomes Complexes, I, II. Publications Mathématiques d’Orsay, Université de Paris-Sud, Départment de Mathématiques, Orsay, 1984/5.
[Eps] Epstein, A.. Bounded hyperbolic components of quadratic rational maps. Ergod. Th. & Dynam. Sys. 20(3) (2000), 727748.
[HP09] Haïssinsky, P. and Pilgrim, K. M.. Coarse expanding conformal dynamics. Astérisque 325 (2009), viii + 139.
[LMS1] Lodge, R., Mikulich, Y. and Schleicher, D.. Combinatorial properties of Newton maps. Preprint, 2015, arXiv:1510.02761.
[LMS2] Lodge, R., Mikulich, Y. and Schleicher, D.. A classification of postcritically finite Newton maps. Preprint, 2015, arXiv:1510.02771.
[McM1] McMullen, C. T.. The classification of conformal dynamical systems. Current Developments in Mathematics, 1995 (Cambridge, MA). International Press, Cambridge, MA, 1994, pp. 323360.
[McM2] McMullen, C. T.. Automorphisms of rational maps. Holomorphic Functions and Moduli I (Mathematical Sciences Research Institute Publications, 10) . Springer, New York, 1988, pp. 3160.
[Mil1] Milnor, J.. Geometry and dynamics of quadratic rational maps. Experiment. Math. 2(1) (1993), 3783 With an appendix by the author and Tan Lei.
[Mil2] Milnor, J.. Dynamics in One Complex Variable (Introductory Lectures) . Friedr. Vieweg & Sohn, Braunschweig, 1999.
[Ree1] Rees, M.. Views of parameter space: Topographer and Resident. Astérisque 288 (2003), vi+418.
[Ree2] Rees, M.. Persistent Markov partitions for rational maps. Preprint, 2013, arXiv:1306.6166.
[Ros] Rosetti, B.. Sur la détermination des fractions rationnelles postcritiquement finies par des graphes planaires finis. PhD Thesis, Université Paul Sabatier, 2015.
[Why1] Whyburn, G. T.. Topological characterization of the Sierpiński curve. Fund. Math. 45 (1958), 320324.
[Why2] Whyburn, G. T.. Analytic Topology (American Mathematical Society Colloquium Publications, XXVIII) . American Mathematical Society, Providence, RI, 1963.
[Wit] Wittner, B.. On the deformation loci of rational maps of degree two. PhD Thesis, Cornell University, 1988.

Related content

Powered by UNSILO

Invariant Jordan curves of Sierpiński carpet rational maps



Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.