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Quadratic-Like Dynamics of Cubic Polynomials

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Abstract

A small perturbation of a quadratic polynomial f with a non-repelling fixed point gives a polynomial g with an attracting fixed point and a Jordan curve Julia set, on which g acts like angle doubling. However, there are cubic polynomials with a non-repelling fixed point, for which no perturbation results into a polynomial with Jordan curve Julia set. Motivated by the study of the closure of the Cubic Principal Hyperbolic Domain, we describe such polynomials in terms of their quadratic-like restrictions.

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References

  1. Blokh, A., Fokkink, R., Mayer, J., Oversteegen, L., Tymchatyn, E.: Fixed point theorems for plane continua with applications. Memoirs Am. Math. Soc. 224(1053) (2013)

  2. Blokh A., Oversteegen L., Ptacek R., Timorin V.: The main cubioid. Nonlinearity 27, 1879–1897 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Blokh, A., Oversteegen, L., Ptacek, R., Timorin, V.: Complementary components to the cubic principal hyperbolic domain (preprint). arXiv:1411.2535 (2014)

  4. Branner B., Hubbard J.: The iteration of cubic polynomials, Part II: patterns and parapatterns. Acta Math. 169, 229–325 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  5. Buff X., Henriksen C.: Julia sets in parameter spaces. Commun. Math. Phys. 220, 333–375 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Douady, A., Hubbard, J.: Étude dynamique des polynômes complex I & II. Publ. Math. Orsay (1984–85)

  7. Douady A., Hubbard J.: On the dynamics of polynomial-like mappings. Ann. Sci. École Norm. Sup. 18, 287–343 (1985)

    MathSciNet  MATH  Google Scholar 

  8. Hubbard, J.H.: Local connectivity of Julia sets and bifurcation loci: three theorems of Yoccoz. In: Topological Methods in Modern Mathematics, Publish or Perish (1993)

  9. Levin G.: On Pommerenkes inequality for the eigenvalues of fixed points. Colloq. Math. 62, 167–177 (1991)

    MathSciNet  MATH  Google Scholar 

  10. Lyubich M.: Some typical properties of the dynamics of rational mappings. Russian Math. Surv. 38(5), 154–155 (1983)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Mañé R.: Hyperbolicity, sinks and measure in one dimensional dynamics. Commun. Math. Phys. 100, 495–524 (1985)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  12. Mañé R.: On a theorem of Fatou. Bol. Soc. Bras. Mat. 24, 1–11 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  13. Mañé R., Sad, P., Sullivan, D.: On the dynamics of rational maps. Ann. Sci. École Norm. Sup. (4) 16(2), 193–217 (1983)

    MATH  MathSciNet  Google Scholar 

  14. McMullen C.: Complex Dynamics and Renormalization. Princeton University Press, Princeton (1994)

    MATH  Google Scholar 

  15. McMullen, C.: The Mandelbrot set is universal. In: Lei, T. (ed.) The Mandelbrot Set, Theme and Variations. London Mathematical Society Lecture Note Series, vol. 274, pp. 1–18. Cambridge University Press, Cambridge

  16. Petersen C.: On the Pommerenke-Levin-Yoccoz inequality. Ergod. Theor. Dyn. Syst. 13, 785–806 (1993)

    MATH  MathSciNet  Google Scholar 

  17. Petersen, C., Lei, T.: Analytic coordinates recording cubic dynamics. In: Schleicher, D., Peters, A.K.: Complex Dynamics, Families and Friends (2009)

  18. Pilgrim K., Lei T.: Spinning deformations of rational maps. Conform. Geom. Dyn. 8, 52–86 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Pommerenke Ch.: On conformal mapping and iteration of rational functions. Complex Var. Theory Appl. 5, 117–126 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  20. Zakeri S.: Dynamics of cubic Siegel polynomials. Commun. Math. Phys. 206, 185–233 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL, 35294, USA

    Alexander Blokh, Lex Oversteegen & Ross Ptacek

  2. Faculty of Mathematics, National Research University Higher School of Economics, Vavilova St. 7, 112312, Moscow, Russia

    Vladlen Timorin

  3. Independent University of Moscow, Bolshoy Vlasyevskiy Pereulok 11, 119002, Moscow, Russia

    Vladlen Timorin

Authors
  1. Alexander Blokh
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  2. Lex Oversteegen
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  3. Ross Ptacek
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  4. Vladlen Timorin
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Correspondence to Alexander Blokh.

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Communicated by M. Lyubich

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Blokh, A., Oversteegen, L., Ptacek, R. et al. Quadratic-Like Dynamics of Cubic Polynomials. Commun. Math. Phys. 341, 733–749 (2016). https://doi.org/10.1007/s00220-015-2559-6

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  • DOI: https://doi.org/10.1007/s00220-015-2559-6

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