American Institute of Mathematical Sciences

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

Hausdorffization and polynomial twists

Laura DeMarco 1, and  Kevin Pilgrim 2,

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.
Keywords: conjugacy classes, Hausdorffization, quasiconformal deformation., equivalence relations, Polynomial dynamics.
Mathematics Subject Classification: Primary: 37F45; Secondary: 54B1.
Citation: Laura DeMarco, Kevin Pilgrim. Hausdorffization and polynomial twists. Discrete & 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)  Google Scholar

[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.  Google Scholar

[3]

R. J. Daverman, "Decompositions of Manifolds," AMS Chelsea Publishing, Providence, RI, 2007. (reprint of the 1986 original)  Google Scholar

[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.  Google Scholar

[5]

L. DeMarco and K. Pilgrim, Polynomial basins of infinity, preprint, 2009. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

R. Mañé, P. Sad and D. Sullivan, On the dynamics of rational maps, Ann. Sci. Ec. Norm. Sup., 16 (1983), 193-217.  Google Scholar

[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.  Google Scholar

show all references

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)  Google Scholar

[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.  Google Scholar

[3]

R. J. Daverman, "Decompositions of Manifolds," AMS Chelsea Publishing, Providence, RI, 2007. (reprint of the 1986 original)  Google Scholar

[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.  Google Scholar

[5]

L. DeMarco and K. Pilgrim, Polynomial basins of infinity, preprint, 2009. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

R. Mañé, P. Sad and D. Sullivan, On the dynamics of rational maps, Ann. Sci. Ec. Norm. Sup., 16 (1983), 193-217.  Google Scholar

[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.  Google Scholar

[1]

Jamshid Moori, Bernardo G. Rodrigues, Amin Saeidi, Seiran Zandi. Designs from maximal subgroups and conjugacy classes of Ree groups. Advances in Mathematics of Communications, 2020, 14 (4) : 603-611. doi: 10.3934/amc.2020033
[2]

Katherine Morrison. An enumeration of the equivalence classes of self-dual matrix codes. Advances in Mathematics of Communications, 2015, 9 (4) : 415-436. doi: 10.3934/amc.2015.9.415
[3]

Kingshook Biswas. Complete conjugacy invariants of nonlinearizable holomorphic dynamics. Discrete & Continuous Dynamical Systems, 2010, 26 (3) : 847-856. doi: 10.3934/dcds.2010.26.847
[4]

Kengo Matsumoto. $ C^* $-algebras associated with asymptotic equivalence relations defined by hyperbolic toral automorphisms. Electronic Research Archive, 2021, 29 (4) : 2645-2656. doi: 10.3934/era.2021006
[5]

Jing Quan, Zhiyou Wu, Guoquan Li. Global optimality conditions for some classes of polynomial integer programming problems. Journal of Industrial & Management Optimization, 2011, 7 (1) : 67-78. doi: 10.3934/jimo.2011.7.67
[6]

Hai Q. Dinh, Hien D. T. Nguyen. On some classes of constacyclic codes over polynomial residue rings. Advances in Mathematics of Communications, 2012, 6 (2) : 175-191. doi: 10.3934/amc.2012.6.175
[7]

Pinhui Ke, Panpan Qiao, Yang Yang. On the equivalence of several classes of quaternary sequences with optimal autocorrelation and length $ 2p$. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020112
[8]

Xu Zhang, Guanrong Chen. Polynomial maps with hidden complex dynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2941-2954. doi: 10.3934/dcdsb.2018293
[9]

Xavier Perrot, Xavier Carton. Point-vortex interaction in an oscillatory deformation field: Hamiltonian dynamics, harmonic resonance and transition to chaos. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 971-995. doi: 10.3934/dcdsb.2009.11.971
[10]

Luciano Pandolfi. Traction, deformation and velocity of deformation in a viscoelastic string. Evolution Equations & Control Theory, 2013, 2 (3) : 471-493. doi: 10.3934/eect.2013.2.471
[11]

Olivier P. Le Maître, Lionel Mathelin, Omar M. Knio, M. Yousuff Hussaini. Asynchronous time integration for polynomial chaos expansion of uncertain periodic dynamics. Discrete & Continuous Dynamical Systems, 2010, 28 (1) : 199-226. doi: 10.3934/dcds.2010.28.199
[12]

Dan Coman. On the dynamics of a class of quadratic polynomial automorphisms of $\mathbb C^3$. Discrete & Continuous Dynamical Systems, 2002, 8 (1) : 55-67. doi: 10.3934/dcds.2002.8.55
[13]

Hiroki Sumi. Dynamics of postcritically bounded polynomial semigroups I: Connected components of the Julia sets. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 1205-1244. doi: 10.3934/dcds.2011.29.1205
[14]

Gaven J. Martin. The Hilbert-Smith conjecture for quasiconformal actions. Electronic Research Announcements, 1999, 5: 66-70.

[15]

Jean-Claude Zambrini. Stochastic deformation of classical mechanics. Conference Publications, 2013, 2013 (special) : 807-813. doi: 10.3934/proc.2013.2013.807
[16]

Rolf Ryham, Chun Liu, Ludmil Zikatanov. Mathematical models for the deformation of electrolyte droplets. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 649-661. doi: 10.3934/dcdsb.2007.8.649
[17]

Eriko Hironaka, Sarah Koch. A disconnected deformation space of rational maps. Journal of Modern Dynamics, 2017, 11: 409-423. doi: 10.3934/jmd.2017016
[18]

Núria Fagella, Christian Henriksen. Deformation of entire functions with Baker domains. Discrete & Continuous Dynamical Systems, 2006, 15 (2) : 379-394. doi: 10.3934/dcds.2006.15.379
[19]

A. Yu. Ol'shanskii and M. V. Sapir. The conjugacy problem for groups, and Higman embeddings. Electronic Research Announcements, 2003, 9: 40-50.

[20]

Kathy Horadam, Russell East. Partitioning CCZ classes into EA classes. Advances in Mathematics of Communications, 2012, 6 (1) : 95-106. doi: 10.3934/amc.2012.6.95

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (48)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences