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.

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)

[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.

[1]

Ho Law, Gary P. T. Choi, Ka Chun Lam, Lok Ming Lui. Quasiconformal model with CNN features for large deformation image registration. Inverse Problems and Imaging, 2022, 16 (4) : 1019-1046. doi: 10.3934/ipi.2022010

[2]

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

[3]

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

[4]

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

[5]

Dirk Frettlöh, Alexey Garber, Lorenzo Sadun. Number of bounded distance equivalence classes in hulls of repetitive Delone sets. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1403-1414. doi: 10.3934/dcds.2021157

[6]

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

[7]

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

[8]

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

[9]

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, 2022, 16 (2) : 285-302. doi: 10.3934/amc.2020112

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

Stephen Doty and Anthony Giaquinto. Generators and relations for Schur algebras. Electronic Research Announcements, 2001, 7: 54-62.

[20]

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

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (67)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]