# American Institute of Mathematical Sciences

October  2011, 29(4): 1405-1417. doi: 10.3934/dcds.2011.29.1405

## Hausdorffization and polynomial twists

 1 Department of Mathematics, Computer Science, and Statistics, University of Illinois at Chicago, Chicago, IL, United States 2 Department of Mathematics, Indiana University, Bloomington, IN, United States

Received  December 2009 Revised  October 2010 Published  December 2010

We study dynamical equivalence relations on the moduli space $\MP_d$ of complex polynomial dynamical systems. Our main result is that the critical-heights quotient $\MP_d \to \cT_d$* of [4] is the Hausdorffization of a relation based on the twisting deformation of the basin of infinity. We also study relations of topological conjugacy and the Branner-Hubbard wringing deformation.
Citation: Laura DeMarco, Kevin Pilgrim. Hausdorffization and polynomial twists. Discrete and Continuous Dynamical Systems, 2011, 29 (4) : 1405-1417. doi: 10.3934/dcds.2011.29.1405
##### References:
 [1] N. Bourbaki, General topology. Chapters 1-4, in "Elements of Mathematics" (Berlin), Springer-Verlag, 1998. (translated from the French, reprint of the 1989 English translation) [2] B. Branner and J. H. Hubbard, The iteration of cubic polynomials. I. The global topology of parameter space, Acta Math., 160 (1988), 143-206. doi: 10.1007/BF02392275. [3] R. J. Daverman, "Decompositions of Manifolds," AMS Chelsea Publishing, Providence, RI, 2007. (reprint of the 1986 original) [4] L. DeMarco and K. Pilgrim, Critical heights on the moduli space of polynomials, Advances in Math., 226 (2011), 350-372. doi: 10.1016/j.aim.2010.06.020. [5] L. DeMarco and K. Pilgrim, Polynomial basins of infinity, preprint, 2009. [6] A. Douady and J. H. Hubbard, "Étude Dynamique des Polynômes Complexes," volume 84 of Publications Mathématiques d'Orsay, Université de Paris-Sud, Département de Mathématiques, Orsay, 1984. [7] Peter Haïssinsky and Tan Lei, Convergence of pinching deformations and matings of geometrically finite polynomials, Fund. Math., 181 (2004), 143-188. doi: 10.4064/fm181-2-4. [8] R. Mañé, P. Sad and D. Sullivan, On the dynamics of rational maps, Ann. Sci. Ec. Norm. Sup., 16 (1983), 193-217. [9] C. T. McMullen and D. P. Sullivan, Quasiconformal homeomorphisms and dynamics. III. The Teichmüller space of a holomorphic dynamical system, Adv. Math., 135 (1998), 351-395. doi: 10.1006/aima.1998.1726.

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##### References:
 [1] N. Bourbaki, General topology. Chapters 1-4, in "Elements of Mathematics" (Berlin), Springer-Verlag, 1998. (translated from the French, reprint of the 1989 English translation) [2] B. Branner and J. H. Hubbard, The iteration of cubic polynomials. I. The global topology of parameter space, Acta Math., 160 (1988), 143-206. doi: 10.1007/BF02392275. [3] R. J. Daverman, "Decompositions of Manifolds," AMS Chelsea Publishing, Providence, RI, 2007. (reprint of the 1986 original) [4] L. DeMarco and K. Pilgrim, Critical heights on the moduli space of polynomials, Advances in Math., 226 (2011), 350-372. doi: 10.1016/j.aim.2010.06.020. [5] L. DeMarco and K. Pilgrim, Polynomial basins of infinity, preprint, 2009. [6] A. Douady and J. H. Hubbard, "Étude Dynamique des Polynômes Complexes," volume 84 of Publications Mathématiques d'Orsay, Université de Paris-Sud, Département de Mathématiques, Orsay, 1984. [7] Peter Haïssinsky and Tan Lei, Convergence of pinching deformations and matings of geometrically finite polynomials, Fund. Math., 181 (2004), 143-188. doi: 10.4064/fm181-2-4. [8] R. Mañé, P. Sad and D. Sullivan, On the dynamics of rational maps, Ann. Sci. Ec. Norm. Sup., 16 (1983), 193-217. [9] C. T. McMullen and D. P. Sullivan, Quasiconformal homeomorphisms and dynamics. III. The Teichmüller space of a holomorphic dynamical system, Adv. Math., 135 (1998), 351-395. doi: 10.1006/aima.1998.1726.
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